Dependencies

\(A_0(n,d,w)\) is the maximum size of an \((n,d,w)\)-admissible code.

For a finite field \(\mathbb {F}_q\) and \(n \in \mathbb {N}\), the symplectic space is the vector space \(V = \mathbb {F}_q^n \times \mathbb {F}_q^n\), equipped with the symplectic form

A subspace \(S \le V\) is isotropic if \(\omega (u,v)=0\) for all \(u,v\in S\).

Classical codes \(C_Z, C_X \le \mathbb {F}_q^n\) satisfy the CSS condition if

Given \(C_Z \perp C_X\), the CSS stabilizer is the subspace of \(V\) generated by the \(X\)-type vectors \((x,0)\) for \(x \in C_Z\) and \(Z\)-type vectors \((0,z)\) for \(z \in C_X\).

The CSS stabilizer is isotropic if and only if the CSS condition holds.

If \(C_Z \perp C_X\), then \(\dim (\mathrm{cssStabilizer}) = \dim (C_Z) + \dim (C_X)\).

The CSS code \(Q(C_Z, C_X)\) encodes \(k = n - \dim (C_Z) - \dim (C_X)\) logical qudits, and has distance \(d = \min (d_Z, d_X)\) where \(d_Z\) (resp. \(d_X\)) is the minimum weight of a vector in \(C_Z^\perp \setminus C_X\) (resp. \(C_X^\perp \setminus C_Z\)).

For a real function \(f\) on the uniform Boolean cube \(\{ 0,1\} ^n\), write \(D_i f(x)=(f(x^{i\leftarrow 0})-f(x^{i\leftarrow 1}))/2\) and \(B(f)=\mathbb E\sqrt{\overline I(f)^2+\sum _i(D_i f)^2}\), where \(\overline I\) is the Gaussian profile extended by zero outside \((0,1)\). The Bobkov functional assigns the value \(B(f)\) to \(f\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. For real \(a,b\) with \(a+b,a-b\in [0,1]\), \(\overline I(a)\le \bigl(\sqrt{\overline I(a+b)^2+b^2}+\sqrt{\overline I(a-b)^2+b^2}\bigr)/2\).

If \(f:\{ 0,1\} ^n\to [0,1]\), then its uniform mean satisfies \(\mathbb E f\in [0,1]\).

For a real function \(f\) on the uniform Boolean cube \(\{ 0,1\} ^n\), write \(D_i f(x)=(f(x^{i\leftarrow 0})-f(x^{i\leftarrow 1}))/2\) and \(B(f)=\mathbb E\sqrt{\overline I(f)^2+\sum _i(D_i f)^2}\), where \(\overline I\) is the Gaussian profile extended by zero outside \((0,1)\). In dimension \(n=0\), \(B(f)=\overline I(\mathbb E f)\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. Let \(f:\{ 0,1\} ^{n+1}\to [0,1]\), put \(f_b(x)=f(x,b)\) for \(b\in \{ 0,1\} \), and set \(d=(f_0-f_1)/2\). Then \(\overline I(\mathbb E f)\le \bigl(\sqrt{\overline I(\mathbb E f_0)^2+(\mathbb E d)^2}+\sqrt{\overline I(\mathbb E f_1)^2+(\mathbb E d)^2}\bigr)/2\).

For a real function \(f\) on the uniform Boolean cube \(\{ 0,1\} ^n\), write \(D_i f(x)=(f(x^{i\leftarrow 0})-f(x^{i\leftarrow 1}))/2\) and \(B(f)=\mathbb E\sqrt{\overline I(f)^2+\sum _i(D_i f)^2}\), where \(\overline I\) is the Gaussian profile extended by zero outside \((0,1)\). If \(d\in \mathbb R\) and \(\overline I(\mathbb E f)\le B(f)\), then \(\sqrt{\overline I(\mathbb E f)^2+d^2}\le \sqrt{B(f)^2+d^2}\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. Let \(f:\{ 0,1\} ^{n+1}\to \mathbb R\), \(f_b(x)=f(x,b)\), and \(d=(f_0-f_1)/2\). For every \(b\in \{ 0,1\} \) and \(x\in \{ 0,1\} ^n\), \(\sqrt{(\sqrt{\overline I(f_b(x))^2+\sum _{i\lt n}(D_i f_b(x))^2})^2+d(x)^2}=\sqrt{\overline I(f(x,b))^2+\sum _{i\lt n+1}(D_i f(x,b))^2}\), where \(D_i\) denotes the coordinate half-difference.

For a real function \(h\) on a uniform Boolean cube, put \(B(h)=\mathbb E\sqrt{\overline I(h)^2+\sum _i(D_i h)^2}\), where \(D_i\) is the coordinate half-difference and \(\overline I\) is the Gaussian profile extended by zero outside \((0,1)\). For \(f:\{ 0,1\} ^{n+1}\to \mathbb R\), put \(f_b(x)=f(x,b)\) and \(d=(f_0-f_1)/2\). If \(\overline I(\mathbb E f_b)\le B(f_b)\), then \(\sqrt{\overline I(\mathbb E f_b)^2+(\mathbb E d)^2}\le \mathbb E_x\sqrt{\overline I(f(x,b))^2+\sum _{i\lt n+1}(D_i f(x,b))^2}\).

For a real function \(f\) on the uniform Boolean cube \(\{ 0,1\} ^n\), write \(D_i f(x)=(f(x^{i\leftarrow 0})-f(x^{i\leftarrow 1}))/2\) and \(B(f)=\mathbb E\sqrt{\overline I(f)^2+\sum _i(D_i f)^2}\), where \(\overline I\) is the Gaussian profile extended by zero outside \((0,1)\). For every \(n\ge 0\) and \(f:\{ 0,1\} ^n\to [0,1]\), \(\overline I(\mathbb E f)\le B(f)\).

For a real function \(f\) on the uniform Boolean cube \(\{ 0,1\} ^n\), write \(D_i f(x)=(f(x^{i\leftarrow 0})-f(x^{i\leftarrow 1}))/2\) and \(B(f)=\mathbb E\sqrt{\overline I(f)^2+\sum _i(D_i f)^2}\), where \(\overline I\) is the Gaussian profile extended by zero outside \((0,1)\). For \(f:\{ 0,1\} ^n\to [0,1]\) and \(0\le \rho \le 1\), \(B(T_\rho f)\le B(f)\), where \(T_\rho \) is the Boolean noise operator with correlation \(\rho \).

For a real function \(f\) on the uniform Boolean cube \(\{ 0,1\} ^n\), write \(D_i f(x)=(f(x^{i\leftarrow 0})-f(x^{i\leftarrow 1}))/2\) and \(B(f)=\mathbb E\sqrt{\overline I(f)^2+\sum _i(D_i f)^2}\), where \(\overline I\) is the Gaussian profile extended by zero outside \((0,1)\). For \(f:\{ 0,1\} ^n\to [0,1]\), the map \(\rho \mapsto B(T_\rho f)\) is increasing on \([0,1]\), where \(T_\rho \) is Boolean noise with correlation \(\rho \).

Let \(f:\{ 0,1\} ^{n+1}\to \mathbb R\) and \(f_b(x)=f(x,b)\). For \(i\lt n\), \(b\in \{ 0,1\} \), and \(x\in \{ 0,1\} ^n\), \(D_i f_b(x)=D_i f(x,b)\), where \(D_i\) is the coordinate half-difference.

Let \(f:\{ 0,1\} ^{n+1}\to \mathbb R\), \(f_b(x)=f(x,b)\), and \(d=(f_0-f_1)/2\). Then \(D_n f(x,b)=d(x)\) for all \(x\) and \(b\), where \(D_n\) is the last-coordinate half-difference.

Let \(f:\{ 0,1\} ^{n+1}\to \mathbb R\), \(f_b(x)=f(x,b)\), and \(d=(f_0-f_1)/2\). Then \(\sum _{i\lt n+1}(D_i f(x,b))^2=\sum _{i\lt n}(D_i f_b(x))^2+d(x)^2\) for every \(x,b\), with \(D_i\) the coordinate half-difference.

Let \(f:\{ 0,1\} ^{n+1}\to \mathbb R\), \(f_b(x)=f(x,b)\), and \(d=(f_0-f_1)/2\). Uniform expectation satisfies \(\mathbb E d=(\mathbb E f_0-\mathbb E f_1)/2\).

For any real functions \(u,v\) on a uniform Boolean cube, \(\sqrt{(\mathbb E u)^2+(\mathbb E v)^2}\le \mathbb E\sqrt{u^2+v^2}\).

Let \(m\ge 0\), \(u,v\in \mathbb R^m\), and \(A,B,C\in \mathbb R\). If \(0\le A\le (B+C)/2\), then \(\sqrt{A^2+\sum _j u_j^2}\le \bigl(\sqrt{B^2+\sum _j(u_j+v_j)^2}+\sqrt{C^2+\sum _j(u_j-v_j)^2}\bigr)/2\).

For any real functions \(f,g\) on \(\{ 0,1\} ^n\), real scalars \(c,d\), and a coordinate \(i\lt n\), \(D_i(cf+dg)=cD_i f+dD_i g\), where \(D_i f(x)=(f(x^{i\leftarrow 0})-f(x^{i\leftarrow 1}))/2\).

Let \(E\) be a real normed vector space and \(f:E\to \mathbb R\) be differentiable everywhere with derivative \(D(x):E\to \mathbb R\). Suppose \(x\mapsto D(x)\) is \(C\)-Lipschitz in operator norm, where \(C\ge 0\). For all \(x,y\in E\), \(|f(y)-f(x)-D(x)(y-x)|\le C\| y-x\| ^2\).

Let \(E\) be a real normed vector space and \(f:E\to \mathbb R\) be differentiable everywhere with derivative \(D(x):E\to \mathbb R\). Suppose \(x\mapsto D(x)\) is \(C\)-Lipschitz in operator norm, where \(C\ge 0\). For \(x,v\in E\) and \(h\in \mathbb R\), \(|(f(x+hv)-f(x))/2-(h/2)D(x)v|\le (C/2)h^2\| v\| ^2\).

For \(n,m\ge 0\) and \(x\in \{ 0,1\} ^{n\times m}\), define \(S_m(x)_k=m^{-1/2}\sum _{l=1}^m(-1)^{x_{kl}}\), interpreting division by zero as zero. If \(x^{ij\leftarrow b}\) replaces bit \((i,j)\) by \(b\), then \(S_m(x^{ij\leftarrow b})=S_m(x)+\bigl((-1)^b-(-1)^{x_{ij}}\bigr)m^{-1/2}e_i\), where \(e_i\) is the \(i\)th coordinate vector.

For \(n,m\ge 0\) and \(x\in \{ 0,1\} ^{n\times m}\), define \(S_m(x)_k=m^{-1/2}\sum _{l=1}^m(-1)^{x_{kl}}\), interpreting division by zero as zero. Let \(f:\mathbb R^n\to \mathbb R\) have derivative family \(D(z)\) that is globally \(C\)-Lipschitz in operator norm for the maximum norm on \(\mathbb R^n\), with \(C\ge 0\). For any bit \((i,j)\), \(\left|\frac{f(S_m(x^{ij\leftarrow 0}))-f(S_m(x^{ij\leftarrow 1}))}{2}-\frac{D(S_m(x))e_i}{\sqrt m}\right|\le 2C/m\).

For a profile \(P:\mathbb R\to \mathbb R\), \(a,b,t,c\in \mathbb R\), and \(u,v\in \mathbb R^m\), define \(L(t,c)=\frac12\bigl(\sqrt{P(a+tb)^2+c+\| u+tv\| _2^2}+\sqrt{P(a-tb)^2+c+\| u-tv\| _2^2}\bigr)\). The symmetric average length is \(L(t,c)\).

For a profile \(P:\mathbb R\to \mathbb R\), \(a,b,t,c\in \mathbb R\), and \(u,v\in \mathbb R^m\), define \(L(t,c)=\frac12\bigl(\sqrt{P(a+tb)^2+c+\| u+tv\| _2^2}+\sqrt{P(a-tb)^2+c+\| u-tv\| _2^2}\bigr)\). The interpolation deficit is \(\Delta (t,c)=L(t,c)-L(1,c+b^2(1-t^2))\).

Let \(s\) be a subset of a set \(X\) and \(f:X\to \mathbb R\). If \(f(s)\) is bounded above and every \(x\in s\) admits \(y\in s\) with \(f(x)\le f(y)/2\), then \(f(x)\le 0\) for all \(x\in s\).

For any real \(x,y\ge 0\), \(\sqrt{x+y}\le \sqrt x+\sqrt y\).

For a profile \(P:\mathbb R\to \mathbb R\), \(a,b,t,c\in \mathbb R\), and \(u,v\in \mathbb R^m\), define \(L(t,c)=\frac12\bigl(\sqrt{P(a+tb)^2+c+\| u+tv\| _2^2}+\sqrt{P(a-tb)^2+c+\| u-tv\| _2^2}\bigr)\). For all parameters, \(L(-t,c)=L(t,c)\).

For a profile \(P:\mathbb R\to \mathbb R\), \(a,b,t,c\in \mathbb R\), and \(u,v\in \mathbb R^m\), define \(L(t,c)=\frac12\bigl(\sqrt{P(a+tb)^2+c+\| u+tv\| _2^2}+\sqrt{P(a-tb)^2+c+\| u-tv\| _2^2}\bigr)\). For all parameters, \(L(|t|,c)=L(t,c)\).

If \(a+b,a-b\in [0,1]\) and \(-1\le t\le 1\), then \(a+tb\in [0,1]\).

For a profile \(P:\mathbb R\to \mathbb R\), \(a,b,t,c\in \mathbb R\), and \(u,v\in \mathbb R^m\), define \(L(t,c)=\frac12\bigl(\sqrt{P(a+tb)^2+c+\| u+tv\| _2^2}+\sqrt{P(a-tb)^2+c+\| u-tv\| _2^2}\bigr)\). For every \(P,a,b,u,v\), the function \(c\mapsto L(1,c)\) is concave on \([0,\infty )\).

For a profile \(P:\mathbb R\to \mathbb R\), \(a,b,t,c\in \mathbb R\), and \(u,v\in \mathbb R^m\), define \(L(t,c)=\frac12\bigl(\sqrt{P(a+tb)^2+c+\| u+tv\| _2^2}+\sqrt{P(a-tb)^2+c+\| u-tv\| _2^2}\bigr)\). If \(P\) is continuous on \([0,1]\) and \(a+b,a-b\in [0,1]\), then \(t\mapsto L(t,0)\) is continuous on \([0,1]\).

For a profile \(P:\mathbb R\to \mathbb R\), \(a,b,t,c\in \mathbb R\), and \(u,v\in \mathbb R^m\), define \(L(t,c)=\frac12\bigl(\sqrt{P(a+tb)^2+c+\| u+tv\| _2^2}+\sqrt{P(a-tb)^2+c+\| u-tv\| _2^2}\bigr)\). For any real \(t\) and \(c\ge 0\), \(L(t,c)\le \sqrt c+L(t,0)\).

For a profile \(P:\mathbb R\to \mathbb R\), \(a,b,t,c\in \mathbb R\), and \(u,v\in \mathbb R^m\), define \(L(t,c)=\frac12\bigl(\sqrt{P(a+tb)^2+c+\| u+tv\| _2^2}+\sqrt{P(a-tb)^2+c+\| u-tv\| _2^2}\bigr)\). Assume \(P\ge 0\) on \([0,1]\) and, whenever \(z\pm d\in [0,1]\), \(P(z)\le \bigl(\sqrt{P(z+d)^2+d^2}+\sqrt{P(z-d)^2+d^2}\bigr)/2\). If \(a\pm b\in [0,1]\), \(0\le t\le 1\), and \(c\ge 0\), then \(L(t,c)\le \bigl(L(1,c+((1-t)b)^2)+L(2t-1,c+((1-t)b)^2)\bigr)/2\).

For a profile \(P:\mathbb R\to \mathbb R\), \(a,b,t,c\in \mathbb R\), and \(u,v\in \mathbb R^m\), define \(L(t,c)=\frac12\bigl(\sqrt{P(a+tb)^2+c+\| u+tv\| _2^2}+\sqrt{P(a-tb)^2+c+\| u-tv\| _2^2}\bigr)\). For every real \(t,c\), \(\sqrt c\le L(t,c)\), with the real square root taken to be zero on negative arguments.

For a profile \(P:\mathbb R\to \mathbb R\), \(a,b,t,c\in \mathbb R\), and \(u,v\in \mathbb R^m\), define \(L(t,c)=\frac12\bigl(\sqrt{P(a+tb)^2+c+\| u+tv\| _2^2}+\sqrt{P(a-tb)^2+c+\| u-tv\| _2^2}\bigr)\). If \(P\) is continuous on \([0,1]\) and \(a\pm b\in [0,1]\), the set \(\{ L(t,c)-L(1,c+b^2(1-t^2)):0\le t\le 1,\ c\ge 0\} \) is bounded above.

For a profile \(P:\mathbb R\to \mathbb R\), \(a,b,t,c\in \mathbb R\), and \(u,v\in \mathbb R^m\), define \(L(t,c)=\frac12\bigl(\sqrt{P(a+tb)^2+c+\| u+tv\| _2^2}+\sqrt{P(a-tb)^2+c+\| u-tv\| _2^2}\bigr)\). Assume \(P\ge 0\) on \([0,1]\) and, whenever \(z\pm d\in [0,1]\), \(P(z)\le \bigl(\sqrt{P(z+d)^2+d^2}+\sqrt{P(z-d)^2+d^2}\bigr)/2\). Put \(\Delta (t,c)=L(t,c)-L(1,c+b^2(1-t^2))\). If \(a\pm b\in [0,1]\), \(0\le t\le 1\), and \(c\ge 0\), then \(\Delta (t,c)\le \Delta (|2t-1|,c+((1-t)b)^2)/2\).

For a profile \(P:\mathbb R\to \mathbb R\), \(a,b,t,c\in \mathbb R\), and \(u,v\in \mathbb R^m\), define \(L(t,c)=\frac12\bigl(\sqrt{P(a+tb)^2+c+\| u+tv\| _2^2}+\sqrt{P(a-tb)^2+c+\| u-tv\| _2^2}\bigr)\). Assume \(P\ge 0\) on \([0,1]\) and, whenever \(z\pm d\in [0,1]\), \(P(z)\le \bigl(\sqrt{P(z+d)^2+d^2}+\sqrt{P(z-d)^2+d^2}\bigr)/2\). If \(P\) is also continuous on \([0,1]\), \(a\pm b\in [0,1]\), \(0\le t\le 1\), and \(c\ge 0\), then \(L(t,c)\le L(1,c+b^2(1-t^2))\).

For a profile \(P:\mathbb R\to \mathbb R\), \(a,b,t,c\in \mathbb R\), and \(u,v\in \mathbb R^m\), define \(L(t,c)=\frac12\bigl(\sqrt{P(a+tb)^2+c+\| u+tv\| _2^2}+\sqrt{P(a-tb)^2+c+\| u-tv\| _2^2}\bigr)\). Assume \(P\ge 0\) on \([0,1]\) and, whenever \(z\pm d\in [0,1]\), \(P(z)\le \bigl(\sqrt{P(z+d)^2+d^2}+\sqrt{P(z-d)^2+d^2}\bigr)/2\). If \(P\) is also continuous on \([0,1]\), \(a\pm b\in [0,1]\), and \(0\le t\le 1\), then \(L(t,(tb)^2)\le L(1,b^2)\).

Let \(T_\rho \) denote Boolean noise with real correlation \(\rho \), whose Fourier multiplier on a set of size \(k\) is \(\rho ^k\). If \(f:\{ 0,1\} ^n\to [0,1]\) and \(0\le \rho \le 1\), then \((T_\rho f)(x)\in [0,1]\) for every \(x\).

Let \(T_\rho \) denote Boolean noise with real correlation \(\rho \), whose Fourier multiplier on a set of size \(k\) is \(\rho ^k\). For real \(c,d,\rho \) and real functions \(f,g\) on \(\{ 0,1\} ^n\), \(T_\rho (cf+dg)=cT_\rho f+dT_\rho g\).

Let \(T_\rho \) denote Boolean noise with real correlation \(\rho \), whose Fourier multiplier on a set of size \(k\) is \(\rho ^k\). For \(f:\{ 0,1\} ^{n+1}\to \mathbb R\), put \(f_b(x)=f(x,b)\), \(a_f=(f_0+f_1)/2\), and \(d_f=(f_0-f_1)/2\), and let \(s_0=1\), \(s_1=-1\). For every \(x,b\) and real \(\rho \), \((T_\rho f_b)(x)=(T_\rho a_f)(x)+s_b(T_\rho d_f)(x)\), where noise on the right acts on the first \(n\) coordinates.

Let \(T_\rho \) denote Boolean noise with real correlation \(\rho \), whose Fourier multiplier on a set of size \(k\) is \(\rho ^k\). For \(f:\{ 0,1\} ^{n+1}\to \mathbb R\), put \(f_b(x)=f(x,b)\), \(a_f=(f_0+f_1)/2\), and \(d_f=(f_0-f_1)/2\), and let \(s_0=1\), \(s_1=-1\). For every \(x,b\), \(D_n(T_\rho f)(x,b)=\rho (T_\rho d_f)(x)\), where \(D_n\) is the last-coordinate half-difference.

Let \(T_\rho \) denote Boolean noise with real correlation \(\rho \), whose Fourier multiplier on a set of size \(k\) is \(\rho ^k\). For \(f:\{ 0,1\} ^{n+1}\to \mathbb R\), put \(f_b(x)=f(x,b)\), \(a_f=(f_0+f_1)/2\), and \(d_f=(f_0-f_1)/2\), and let \(s_0=1\), \(s_1=-1\). For either bit \(b\), \((T_\rho f)_b=T_\rho a_f+s_b\rho T_\rho d_f\) as functions on \(\{ 0,1\} ^n\).

Let \(T_\rho \) denote Boolean noise with real correlation \(\rho \), whose Fourier multiplier on a set of size \(k\) is \(\rho ^k\). For \(f:\{ 0,1\} ^{n+1}\to \mathbb R\), put \(f_b(x)=f(x,b)\), \(a_f=(f_0+f_1)/2\), and \(d_f=(f_0-f_1)/2\), and let \(s_0=1\), \(s_1=-1\). For either bit \(b\), \(T_\rho f_b=T_\rho a_f+s_bT_\rho d_f\) as functions on \(\{ 0,1\} ^n\).

Let \(T_\rho \) denote Boolean noise with real correlation \(\rho \), whose Fourier multiplier on a set of size \(k\) is \(\rho ^k\). For \(f:\{ 0,1\} ^{n+1}\to \mathbb R\), put \(f_b(x)=f(x,b)\), \(a_f=(f_0+f_1)/2\), and \(d_f=(f_0-f_1)/2\), and let \(s_0=1\), \(s_1=-1\). If \(f\) takes values in \([0,1]\) and \(0\le \rho \le 1\), then \((T_\rho a_f)(x)\pm (T_\rho d_f)(x)\in [0,1]\) for every \(x\).

For \(P:\mathbb R\to \mathbb R\), \(f:\{ 0,1\} ^n\to \mathbb R\), and \(g=(g_1,\ldots ,g_m)\) of real functions on the cube, write \(V(f,g)(x)=(D_1f(x),\ldots ,D_nf(x),g_1(x),\ldots ,g_m(x))\) and \(F_P(f,g)=\mathbb E\sqrt{P(f)^2+\| V(f,g)\| _2^2}\), with \(D_i\) the coordinate half-difference. The extended functional is \(F_P(f,g)\) for arbitrary \(n,m\ge 0\).

For \(f:\{ 0,1\} ^n\to \mathbb R\), \(m\ge 0\), and real cube functions \(g_1,\ldots ,g_m\), define \(V(f,g)(x)=(D_1f(x),\ldots ,D_nf(x),g_1(x),\ldots ,g_m(x))\), where \(D_i\) is the coordinate half-difference.

For \(P:\mathbb R\to \mathbb R\), \(f:\{ 0,1\} ^n\to \mathbb R\), and \(g=(g_1,\ldots ,g_m)\) of real functions on the cube, write \(V(f,g)(x)=(D_1f(x),\ldots ,D_nf(x),g_1(x),\ldots ,g_m(x))\) and \(F_P(f,g)=\mathbb E\sqrt{P(f)^2+\| V(f,g)\| _2^2}\), with \(D_i\) the coordinate half-difference. For every \(x\), \(\| V(f,g)(x)\| _2^2=\sum _i(D_i f(x))^2+\sum _jg_j(x)^2\).

For a profile \(P:\mathbb R\to \mathbb R\), a real function \(h\) on a uniform Boolean cube, and a vector \(k\) of real functions on that cube, put \(V(h,k)(x)=((D_i h(x))_i,(k_j(x))_j)\) and \(F_P(h,k)=\mathbb E\sqrt{P(h)^2+\| V(h,k)\| _2^2}\), where \(D_i\) is the coordinate half-difference. Let \(f\) be a real function and \(g\) a vector of \(m\) real functions on \(\{ 0,1\} ^{n+1}\). Put \(f_b(x)=f(x,b)\), \(g_b(x)=g(x,b)\), \(a_f=(f_0+f_1)/2\), \(d_f=(f_0-f_1)/2\), and define \(a_g,d_g\) componentwise. For every \(x,b\), \(\sqrt{P(f(x,b))^2+\| V(f,g)(x,b)\| _2^2}=\sqrt{P(f_b(x))^2+d_f(x)^2+\| V(f_b,g_b)(x)\| _2^2}\).

For a profile \(P:\mathbb R\to \mathbb R\), a real function \(h\) on a uniform Boolean cube, and a vector \(k\) of real functions on that cube, put \(V(h,k)(x)=((D_i h(x))_i,(k_j(x))_j)\) and \(F_P(h,k)=\mathbb E\sqrt{P(h)^2+\| V(h,k)\| _2^2}\), where \(D_i\) is the coordinate half-difference. For real \(c,d\), functions \(f_1,f_2\) on the cube, and vectors \(g_1,g_2\) of \(m\) cube functions, \(V(cf_1+df_2,cg_1+dg_2)(x)=cV(f_1,g_1)(x)+dV(f_2,g_2)(x)\) for every \(x\).

For a profile \(P:\mathbb R\to \mathbb R\), a real function \(h\) on a uniform Boolean cube, and a vector \(k\) of real functions on that cube, put \(V(h,k)(x)=((D_i h(x))_i,(k_j(x))_j)\) and \(F_P(h,k)=\mathbb E\sqrt{P(h)^2+\| V(h,k)\| _2^2}\), where \(D_i\) is the coordinate half-difference. Let \(f\) be a real function and \(g\) a vector of \(m\) real functions on \(\{ 0,1\} ^{n+1}\). Put \(f_b(x)=f(x,b)\), \(g_b(x)=g(x,b)\), \(a_f=(f_0+f_1)/2\), \(d_f=(f_0-f_1)/2\), and define \(a_g,d_g\) componentwise. Let \(T_\rho \) denote Boolean noise with real correlation \(\rho \), whose Fourier multiplier on a set of size \(k\) is \(\rho ^k\). For \(s_0=1\), \(s_1=-1\), \(V(T_\rho f_b,T_\rho g_b)(x)=V(T_\rho a_f,T_\rho a_g)(x)+s_bV(T_\rho d_f,T_\rho d_g)(x)\) for every \(x,b\).

For a profile \(P:\mathbb R\to \mathbb R\), a real function \(h\) on a uniform Boolean cube, and a vector \(k\) of real functions on that cube, put \(V(h,k)(x)=((D_i h(x))_i,(k_j(x))_j)\) and \(F_P(h,k)=\mathbb E\sqrt{P(h)^2+\| V(h,k)\| _2^2}\), where \(D_i\) is the coordinate half-difference. Let \(f\) be a real function and \(g\) a vector of \(m\) real functions on \(\{ 0,1\} ^{n+1}\). Put \(f_b(x)=f(x,b)\), \(g_b(x)=g(x,b)\), \(a_f=(f_0+f_1)/2\), \(d_f=(f_0-f_1)/2\), and define \(a_g,d_g\) componentwise. Let \(T_\rho \) denote Boolean noise with real correlation \(\rho \), whose Fourier multiplier on a set of size \(k\) is \(\rho ^k\). For \(s_0=1\), \(s_1=-1\), \(V((T_\rho f)_b,(T_\rho g)_b)(x)=V(T_\rho a_f,T_\rho a_g)(x)+s_b\rho V(T_\rho d_f,T_\rho d_g)(x)\) for every \(x,b\).

For a profile \(P:\mathbb R\to \mathbb R\), a real function \(h\) on a uniform Boolean cube, and a vector \(k\) of real functions on that cube, put \(V(h,k)(x)=((D_i h(x))_i,(k_j(x))_j)\) and \(F_P(h,k)=\mathbb E\sqrt{P(h)^2+\| V(h,k)\| _2^2}\), where \(D_i\) is the coordinate half-difference. Let \(f\) be a real function and \(g\) a vector of \(m\) real functions on \(\{ 0,1\} ^{n+1}\). Put \(f_b(x)=f(x,b)\), \(g_b(x)=g(x,b)\), \(a_f=(f_0+f_1)/2\), \(d_f=(f_0-f_1)/2\), and define \(a_g,d_g\) componentwise. Then \(F_P(f,g)=\bigl(F_P(f_0,(d_f,g_0))+F_P(f_1,(d_f,g_1))\bigr)/2\), where \((d_f,g_b)\) adjoins \(d_f\) to the additional coordinates.

For a profile \(P:\mathbb R\to \mathbb R\), a real function \(h\) on a uniform Boolean cube, and a vector \(k\) of real functions on that cube, put \(V(h,k)(x)=((D_i h(x))_i,(k_j(x))_j)\) and \(F_P(h,k)=\mathbb E\sqrt{P(h)^2+\| V(h,k)\| _2^2}\), where \(D_i\) is the coordinate half-difference. Let \(f\) be a real function and \(g\) a vector of \(m\) real functions on \(\{ 0,1\} ^{n+1}\). Put \(f_b(x)=f(x,b)\), \(g_b(x)=g(x,b)\), \(a_f=(f_0+f_1)/2\), \(d_f=(f_0-f_1)/2\), and define \(a_g,d_g\) componentwise. Let \(T_\rho \) denote Boolean noise with real correlation \(\rho \), whose Fourier multiplier on a set of size \(k\) is \(\rho ^k\). For each \(x\), put \(a=T_\rho a_f(x)\), \(b=T_\rho d_f(x)\), \(u=V(T_\rho a_f,T_\rho a_g)(x)\), and \(v=V(T_\rho d_f,T_\rho d_g)(x)\). Then \(F_P(T_\rho f,T_\rho g)=\mathbb E_x\frac12\sum _{\epsilon \in \{ -1,1\} }\sqrt{P(a+\epsilon \rho b)^2+(\rho b)^2+\| u+\epsilon \rho v\| _2^2}\).

For a profile \(P:\mathbb R\to \mathbb R\), a real function \(h\) on a uniform Boolean cube, and a vector \(k\) of real functions on that cube, put \(V(h,k)(x)=((D_i h(x))_i,(k_j(x))_j)\) and \(F_P(h,k)=\mathbb E\sqrt{P(h)^2+\| V(h,k)\| _2^2}\), where \(D_i\) is the coordinate half-difference. Let \(f\) be a real function and \(g\) a vector of \(m\) real functions on \(\{ 0,1\} ^{n+1}\). Put \(f_b(x)=f(x,b)\), \(g_b(x)=g(x,b)\), \(a_f=(f_0+f_1)/2\), \(d_f=(f_0-f_1)/2\), and define \(a_g,d_g\) componentwise. Let \(T_\rho \) denote Boolean noise with real correlation \(\rho \), whose Fourier multiplier on a set of size \(k\) is \(\rho ^k\). For each \(x\), put \(a=T_\rho a_f(x)\), \(b=T_\rho d_f(x)\), \(u=V(T_\rho a_f,T_\rho a_g)(x)\), and \(v=V(T_\rho d_f,T_\rho d_g)(x)\). Then \(\mathbb E_x\frac12\sum _{\epsilon \in \{ -1,1\} }\sqrt{P(a+\epsilon b)^2+b^2+\| u+\epsilon v\| _2^2}=\bigl(F_P(T_\rho f_0,(T_\rho d_f,T_\rho g_0))+F_P(T_\rho f_1,(T_\rho d_f,T_\rho g_1))\bigr)/2\).

For \(P:\mathbb R\to \mathbb R\), \(f:\{ 0,1\} ^n\to \mathbb R\), and \(g=(g_1,\ldots ,g_m)\) of real functions on the cube, write \(V(f,g)(x)=(D_1f(x),\ldots ,D_nf(x),g_1(x),\ldots ,g_m(x))\) and \(F_P(f,g)=\mathbb E\sqrt{P(f)^2+\| V(f,g)\| _2^2}\), with \(D_i\) the coordinate half-difference. Assume \(P\ge 0\) on \([0,1]\) and, whenever \(z\pm d\in [0,1]\), \(P(z)\le \bigl(\sqrt{P(z+d)^2+d^2}+\sqrt{P(z-d)^2+d^2}\bigr)/2\). If \(P\) is continuous on \([0,1]\), \(f\) takes values in \([0,1]\), and \(0\le \rho \le 1\), then \(F_P(T_\rho f,T_\rho g)\le F_P(f,g)\), where \(T_\rho \) is Boolean noise applied to each component.

Let \(c,A,B\in \mathbb R\) with \(A,B\ge 0\). If \((A-B)^2\le 8c^2(A+B-2c^2)\), then \(c\le (\sqrt A+\sqrt B)/2\).

Let \(a,b,t\in \mathbb R\) with \(b\ge 0\), \(a-b\gt 0\), \(a+b\lt 1\), and \(0\le t\le b\). Then \(a+t,a-t\in (0,1)\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). If \(a,t,k\in \mathbb R\) and \(a\pm t\in (0,1)\), then the derivative of \(x\mapsto I(a+x)^2+kI(a-x)^2\) at \(t\) is \(-2I(a+t)Q(a+t)+2kI(a-t)Q(a-t)\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). If \(a,t,k\in \mathbb R\) and \(a\pm t\in (0,1)\), then the derivative at \(t\) of \(x\mapsto -2I(a+x)Q(a+x)+2kI(a-x)Q(a-x)\) is \(2(Q(a+t)^2+kQ(a-t)^2-1-k)\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). If \(b\ge 0\), \(a-b\gt 0\), \(a+b\lt 1\), and \(a\le 1/2\), then \(x\mapsto I(a+x)^2-I(a-x)^2\) is concave on \([0,b]\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). If \(b\ge 0\), \(a-b\gt 0\), \(a+b\lt 1\), and \(a\le 1/2\), then \((I(a+b)^2-I(a-b)^2)^2\le 16I(a)^2Q(a)^2b^2\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). If \(a,b\in \mathbb R\) satisfy \(a-b\gt 0\) and \(a+b\lt 1\), then \(x\mapsto I(a+x)^2+I(a-x)^2+2(1-Q(a)^2)x^2\) is convex on \([-b,b]\). Here \(Q(p)=0\) and \(I(p)=\varphi (0)\) outside \((0,1)\), and \([-b,b]\) may be empty.

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). If \(b\ge 0\), \(a-b\gt 0\), and \(a+b\lt 1\), then \(2Q(a)^2b^2\le I(a+b)^2+I(a-b)^2+2b^2-2I(a)^2\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). If \(b\ge 0\), \(a-b\gt 0\), \(a+b\lt 1\), and \(a\le 1/2\), then \(I(a)\le \bigl(\sqrt{I(a+b)^2+b^2}+\sqrt{I(a-b)^2+b^2}\bigr)/2\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). If \(b\ge 0\) and \(a-b,a+b\in (0,1)\), then \(I(a)\le \bigl(\sqrt{I(a+b)^2+b^2}+\sqrt{I(a-b)^2+b^2}\bigr)/2\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). If \(a,b\in \mathbb R\) and \(a-b,a+b\in (0,1)\), then \(I(a)\le \bigl(\sqrt{I(a+b)^2+b^2}+\sqrt{I(a-b)^2+b^2}\bigr)/2\).

If \(a\in (0,1)\), \(a+b\in [0,1]\), and \(0\le \rho \lt 1\), then \(a+\rho b\in (0,1)\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. If \(a,a+b\in [0,1]\), then \(\lim _{\rho \to 1^-}\overline I(a+\rho b)=\overline I(a+b)\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. If \(a\in (0,1)\) and \(a+b,a-b\in [0,1]\), then \(\overline I(a)\le \bigl(\sqrt{\overline I(a+b)^2+b^2}+\sqrt{\overline I(a-b)^2+b^2}\bigr)/2\).

Let \(I,J,B\) be finite sets, let \(B\) carry a measurable structure with measurable singletons, and let \(\mu _{ij}\) be sigma-finite measures on \(B\). The bijection \(B^{I\times J}\to (B^J)^I\) sending \(x\) to the array \(i\mapsto (j\mapsto x(i,j))\) pushes \(\bigotimes _{(i,j)}\mu _{ij}\) to \(\bigotimes _i\bigotimes _j\mu _{ij}\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. Let \(I\) be finite and \(f:\{ 0,1\} ^I\to [0,1]\). Under the independent uniform bit law, \(\overline I(\mathbb E f)\le \mathbb E\sqrt{\overline I(f)^2+\sum _{i\in I}(D_i f)^2}\), where \(D_i f(x)=(f(x^{i\leftarrow 0})-f(x^{i\leftarrow 1}))/2\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. Let \(I,J\) be finite and \(f:\{ 0,1\} ^{I\times J}\to [0,1]\). Under independent uniform bits, \(\overline I(\mathbb E f)\le \mathbb E\sqrt{\overline I(f)^2+\sum _{i\in I}\sum _{j\in J}(D_{ij}f)^2}\), where \(D_{ij}f(x)=(f(x^{ij\leftarrow 0})-f(x^{ij\leftarrow 1}))/2\).

Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). For any set \(A\subseteq \mathbb R^n\) and real \(t\), its rotation sensitivity is the real value of the product-measure mass of the event \(\mathbf1_A(G)\ne \mathbf1_A(\cos t\, G+\sin t\, G\prime )\); for measurable \(A\), this is the disagreement probability.

Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). For any \(A\subseteq \mathbb R^n\), its Gaussian surface area is the extended nonnegative quantity \(P_\gamma (A)=\liminf _{t\to 0^+}[\sqrt{\pi /2}\, s_A(t)/t]_+\), where \([u]_+=\max (u,0)\).

For a real number \(\alpha \), the closed Gaussian profile is \(U(\alpha )=\varphi (\Phi ^{-1}(\alpha ))\) if \(0\lt \alpha \lt 1\), and \(U(\alpha )=0\) otherwise, where \(\varphi \) and \(\Phi \) are the standard Gaussian density and distribution function.

For a finite set \(I\), a real scalar \(P\), and vectors \(d,g\in \mathbb R^I\), \(\sqrt{P^2+\sum _i d_i^2}\le \sqrt{P^2+\sum _i g_i^2}+\sqrt{\sum _i(d_i-g_i)^2}\).

Let \(n\ge 0\), \(m\gt 0\), \(P\in \mathbb R\), \(d\in \mathbb R^{n\times m}\), \(g\in \mathbb R^n\), and \(e\ge 0\). If \(|d_{ij}-g_i/\sqrt m|\le e\) for every \(i,j\), then \(\sqrt{P^2+\sum _{i,j}d_{ij}^2}\le \sqrt{P^2+\sum _i g_i^2}+\sqrt{nm}\, e\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. Let \(f:\mathbb R^n\to [0,1]\) be continuous, and let \(D(z)\) be a continuous family of real linear functionals on \(\mathbb R^n\) with \(\| D(z)\| \le M\) for all \(z\), for some real \(M\). Then \(z\mapsto \sqrt{\overline I(f(z))^2+\sum _i(D(z)e_i)^2}\) is a bounded continuous function.

For every \(n\ge 0\) and \(C\in \mathbb R\), \(\sqrt{n(m+1)}\, 2C/(m+1)\to 0\) as \(m\to \infty \).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. Let \(f:\mathbb R^n\to [0,1]\) be differentiable with derivative family \(D(z)\), globally \(C\)-Lipschitz in operator norm for the maximum norm, where \(C\ge 0\). Write \(|D(z)|_2=\sqrt{\sum _{i=1}^n(D(z)e_i)^2}\). For independent uniform bits \(X_{ij}\), put \(S_r(X)_i=r^{-1/2}\sum _{j=1}^r(-1)^{X_{ij}}\). For every \(m\gt 0\), \(\overline I(\mathbb E f(S_m(X)))\le \mathbb E\sqrt{\overline I(f(S_m(X)))^2+|D(S_m(X))|_2^2}+\sqrt{nm}\, 2C/m\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). Let \(f:\mathbb R^n\to [0,1]\) be differentiable with derivative family \(D(z)\), globally \(C\)-Lipschitz in operator norm for the maximum norm, where \(C\ge 0\). Write \(|D(z)|_2=\sqrt{\sum _{i=1}^n(D(z)e_i)^2}\). If also \(\| D(z)\| \le M\) for every \(z\), for some \(M\in \mathbb R\), then \(\overline I(\int f\, d\gamma _n)\le \int \sqrt{\overline I(f(z))^2+|D(z)|_2^2}\, d\gamma _n(z)\).

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). Let \(D(z)\) be a continuous family of real linear functionals on \(\mathbb R^n\) and let \(M\in \mathbb R\) satisfy \(\| D(z)\| \le M\) for all \(z\). Then \(z\mapsto \sqrt{\sum _i(D(z)e_i)^2}\) is \(\gamma _n\)-integrable.

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). Let \(f:\mathbb R^n\to [0,1]\) be differentiable with derivative family \(D(z)\), globally \(C\)-Lipschitz in operator norm for the maximum norm, where \(C\ge 0\). Write \(|D(z)|_2=\sqrt{\sum _{i=1}^n(D(z)e_i)^2}\). If also \(\| D(z)\| \le M\) for every \(z\), for some \(M\in \mathbb R\), then \(\overline I(\int f\, d\gamma _n)\le \int \overline I(f)\, d\gamma _n+\int |D(z)|_2\, d\gamma _n(z)\).

Let \(A\subseteq \mathbb R\) be measurable, \(c,s\in \mathbb R\), and \(s\ne 0\). For \(Z\sim N(0,1)\) and the Gaussian density \(p_{s^2}(c,z)\) with mean \(c\) and variance \(s^2\), \(\mathbb E1_A(c+sZ)=\int _{\mathbb R}p_{s^2}(c,z)1_A(z)\, dz\).

For \(v\ge 0\), put \(p_v(c,z)=(2\pi v)^{-1/2}\exp (-(z-c)^2/(2v))\) when \(v\gt 0\), and \(p_0(c,z)=0\); take division by zero to be zero. For all real \(c,z\), the derivative at \(c\) of \(m\mapsto p_v(m,z)\) is \(((z-c)/v)p_v(c,z)\).

Let \(v\gt 0\) and \(p_v(c,z)=(2\pi v)^{-1/2}\exp (-(z-c)^2/(2v))\). For every \(M\ge 0\), there exists a Lebesgue-integrable \(b:\mathbb R\to \mathbb R\) such that \(|((z-m)/v)p_v(m,z)|\le b(z)\) for all \(z\in \mathbb R\) and all \(m\) with \(|m|\le M\).

Let \(v\gt 0\) and \(p_v(c,z)=(2\pi v)^{-1/2}\exp (-(z-c)^2/(2v))\). Let \(f:\mathbb R\to \mathbb R\) be measurable with \(|f(z)|\le 1\) for all \(z\). For every \(c\in \mathbb R\), the derivative at \(c\) of \(m\mapsto \int p_v(m,z)f(z)\, dz\) is \(\int ((z-c)/v)p_v(c,z)f(z)\, dz\).

For \(v\ge 0\), put \(p_v(c,z)=(2\pi v)^{-1/2}\exp (-(z-c)^2/(2v))\) when \(v\gt 0\), and \(p_0(c,z)=0\); take division by zero to be zero. For real \(c,z\), the derivative at \(c\) of \(m\mapsto ((z-m)/v)p_v(m,z)\) is \(((z-c)^2/v^2-1/v)p_v(c,z)\).

Let \(v\gt 0\) and \(p_v(c,z)=(2\pi v)^{-1/2}\exp (-(z-c)^2/(2v))\). For every \(M\ge 0\), there is a Lebesgue-integrable \(b:\mathbb R\to \mathbb R\) such that \(|((z-m)^2/v^2-1/v)p_v(m,z)|\le b(z)\) for all real \(z\) and all \(m\) with \(|m|\le M\).

Let \(v\gt 0\) and \(p_v(c,z)=(2\pi v)^{-1/2}\exp (-(z-c)^2/(2v))\). If \(f:\mathbb R\to \mathbb R\) is measurable and \(|f|\le 1\), then for each real \(c\) the derivative at \(c\) of \(m\mapsto \int ((z-m)/v)p_v(m,z)f(z)\, dz\) is \(\int ((z-c)^2/v^2-1/v)p_v(c,z)f(z)\, dz\).

For every \(c\in \mathbb R\) and \(v\ge 0\), a random variable \(Z\sim N(c,v)\) satisfies \(\mathbb E(Z-c)^2=v\), including the point-mass law when \(v=0\).

Let \(v\gt 0\) and \(p_v(c,z)=(2\pi v)^{-1/2}\exp (-(z-c)^2/(2v))\). Let \(c\in \mathbb R\) and \(g,b:\mathbb R\to \mathbb R\). If \(b\) is integrable for \(N(c,v)\) and \(|g(z)|\le b(z)\) for every \(z\), then \(|\int p_v(c,z)g(z)\, dz|\le \int b(z)\, dN(c,v)(z)\), with a nonintegrable integral on the left assigned value zero.

Let \(v\gt 0\) and \(p_v(c,z)=(2\pi v)^{-1/2}\exp (-(z-c)^2/(2v))\). For any function \(f:\mathbb R\to \mathbb R\) with \(|f|\le 1\) and any real \(c\), \(|\int ((z-c)^2/v^2-1/v)p_v(c,z)f(z)\, dz|\le 2/v\), where nonintegrable integrals are assigned zero.

Let \(v\gt 0\) and \(p_v(c,z)=(2\pi v)^{-1/2}\exp (-(z-c)^2/(2v))\). For any function \(f:\mathbb R\to \mathbb R\) with \(|f|\le 1\) and any real \(c\), \(|\int ((z-c)/v)p_v(c,z)f(z)\, dz|\le 1+1/v\), where nonintegrable integrals are assigned zero.

Let \(v\gt 0\) and \(p_v(c,z)=(2\pi v)^{-1/2}\exp (-(z-c)^2/(2v))\). For measurable \(f:\mathbb R\to \mathbb R\) with \(|f|\le 1\), the function \(c\mapsto \int ((z-c)/v)p_v(c,z)f(z)\, dz\) is globally \(2/v\)-Lipschitz.

Let \(v\gt 0\) and \(p_v(c,z)=(2\pi v)^{-1/2}\exp (-(z-c)^2/(2v))\). For measurable \(f:\mathbb R\to \mathbb R\) with \(|f|\le 1\), the function \(c\mapsto \int p_v(c,z)f(z)\, dz\) is globally \((1+1/v)\)-Lipschitz.

Let \(v\gt 0\) and \(p_v(c,z)=(2\pi v)^{-1/2}\exp (-(z-c)^2/(2v))\). For measurable \(f:\mathbb R\to \mathbb R\) with \(|f|\le 1\), the function \(c\mapsto \int p_v(c,z)f(z)\, dz\) is continuously differentiable on \(\mathbb R\).

For every \(a\in \mathbb R^n\) with \(\sum _i a_i^2=1\), the map \(z\mapsto \sum _i a_i z_i\) pushes standard Gaussian measure on \(\mathbb R^n\) to \(N(0,1)\).

Let \(\Phi ,\varphi \) denote the standard Gaussian distribution function and density. For \(t,y\in \mathbb R\), define \(q_{t,y}(\theta )=\Phi ((t-\sin \theta \, y)/\cos \theta )-\Phi (t)\). For every \(t,y\), \(q_{t,y}'(0)=-\varphi (t)y\).

Let \(\Phi ,\varphi \) denote the standard Gaussian distribution function and density. For \(t,y\in \mathbb R\), define \(q_{t,y}(\theta )=\Phi ((t-\sin \theta \, y)/\cos \theta )-\Phi (t)\). For every \(t,y\), \(|q_{t,y}(\theta )|/\theta \to \varphi (t)|y|\) as \(\theta \to 0^+\).

The standard Gaussian density satisfies \(\varphi (x)\le \varphi (0)\) for every \(x\in \mathbb R\).

The standard Gaussian distribution function and density satisfy \(|\Phi (x)-\Phi (y)|\le \varphi (0)|x-y|\) for all \(x,y\in \mathbb R\).

Let \(\Phi ,\varphi \) denote the standard Gaussian distribution function and density. For \(t,y\in \mathbb R\), define \(q_{t,y}(\theta )=\Phi ((t-\sin \theta \, y)/\cos \theta )-\Phi (t)\). For all \(t,y\in \mathbb R\) and \(0\lt \theta \lt 1\), \(|q_{t,y}(\theta )|/\theta \le 2\varphi (0)(|t|+|y|)\).

Let \(\Phi ,\varphi \) denote the standard Gaussian distribution function and density. For \(t,y\in \mathbb R\), define \(q_{t,y}(\theta )=\Phi ((t-\sin \theta \, y)/\cos \theta )-\Phi (t)\). For every real \(t\), if \(Y\sim N(0,1)\), then \(\mathbb E_Y(|q_{t,Y}(\theta )|/\theta )\to \varphi (t)\mathbb E|Y|\) as \(\theta \to 0^+\).

Let \(\gamma \) be standard Gaussian measure on \(\mathbb R\) and \(\Phi \) its distribution function. For real \(a,b\), \(\gamma (({-\infty },a]\mathbin {\triangle }({-\infty },b])=|\Phi (b)-\Phi (a)|\).

For a random variable \(Y\sim N(0,1)\) and the standard Gaussian density \(\varphi \), \(\mathbb E|Y|=2\varphi (0)\).

For every real \(t,\theta \), the set \(\{ (x,y)\in \mathbb R^2:\neg (x\le t\ \Longleftrightarrow \ \cos \theta \, x+\sin \theta \, y\le t)\} \) is Borel measurable.

Let \(\Phi ,\varphi \) denote the standard Gaussian distribution function and density. For \(t,y\in \mathbb R\), define \(q_{t,y}(\theta )=\Phi ((t-\sin \theta \, y)/\cos \theta )-\Phi (t)\). If \(\cos \theta \gt 0\) and \(X,Y\) are independent standard Gaussian variables, then \(\Pr (1_{(-\infty ,t]}(X)\ne 1_{(-\infty ,t]}(\cos \theta \, X+\sin \theta \, Y))=\mathbb E_Y|q_{t,Y}(\theta )|\).

The standard Gaussian density satisfies \(\sqrt{\pi /2}\, 2\varphi (0)=1\).

Let \(X,Y\) be independent standard Gaussian variables and \(\varphi \) their density. For every real \(t\), \(\sqrt{\pi /2}\, \Pr (1_{(-\infty ,t]}(X)\ne 1_{(-\infty ,t]}(\cos \theta \, X+\sin \theta \, Y))/\theta \to \varphi (t)\) as \(\theta \to 0^+\).

For a measurable \(A\subseteq \mathbb R^n\) and independent standard Gaussian vectors \(X,Y\), put \(R_A(\theta )=\Pr (1_A(X)\ne 1_A(\cos \theta \, X+\sin \theta \, Y))\). If \(a\in \mathbb R^n\) satisfies \(\sum _i a_i^2=1\) and \(H=\{ z:\sum _i a_i z_i\le t\} \), then \(R_H(\theta )=\Pr (1_{(-\infty ,t]}(U)\ne 1_{(-\infty ,t]}(\cos \theta \, U+\sin \theta \, V))\) for every real \(t,\theta \), where \(U,V\) are independent standard Gaussian scalars.

For a measurable \(A\subseteq \mathbb R^n\) and independent standard Gaussian vectors \(X,Y\), put \(R_A(\theta )=\Pr (1_A(X)\ne 1_A(\cos \theta \, X+\sin \theta \, Y))\). If \(a\in \mathbb R^n\) satisfies \(\sum _i a_i^2=1\) and \(H=\{ z:\sum _i a_i z_i\le t\} \) for \(t\in \mathbb R\), then \(\sqrt{\pi /2}\, R_H(\theta )/\theta \to \varphi (t)\) as \(\theta \to 0^+\), where \(\varphi \) is the standard Gaussian density.

For every \(n\ge 0\) and \(a\in \mathbb R^n\), if \(Z\) is a standard Gaussian vector and \(\varphi \) is the one-dimensional standard Gaussian density, then \(\mathbb E|\sum _i a_i Z_i|=\sqrt{\sum _i a_i^2}\, 2\varphi (0)\).

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For \(A\subseteq \mathbb R^n\) and \(\theta \in \mathbb R\), put \(u_\theta (x)=\int 1_A(\cos \theta \, x+\sin \theta \, y)\, d\gamma _n(y)\). If \(A\) is measurable and \(\sin u\ne 0\), then for all \(x\in \mathbb R^n\), \(u_u(x)=\int _{\mathbb R^n}\prod _i p_{\sin ^2u}(\cos u\, x_i,z_i)1_A(z)\, dz\), where \(p_v(c,z)\) is the \(N(c,v)\) density.

For every \(n\ge 0\) and \(u\in \mathbb R\), the linear map \(x\mapsto \cos u\, x\) on \(\mathbb R^n\) has operator norm at most one for the maximum norm.

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For \(A\subseteq \mathbb R^n\) and \(\theta \in \mathbb R\), put \(u_\theta (x)=\int 1_A(\cos \theta \, x+\sin \theta \, y)\, d\gamma _n(y)\). If \(A\) is measurable and \(\sin u\ne 0\), \(u_u\) is differentiable at every \(x\in \mathbb R^n\), with \(D u_u(x)[h]=\cos u\int 1_A(z)\, \partial _c[\prod _i p_{\sin ^2u}(c_i,z_i)]_{c=\cos u\, x}[h]\, dz\), where \(p_v\) is the Gaussian density and \(\partial _c\) is differentiation in the mean vector.

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For \(A\subseteq \mathbb R^n\) and \(\theta \in \mathbb R\), put \(u_\theta (x)=\int 1_A(\cos \theta \, x+\sin \theta \, y)\, d\gamma _n(y)\). If \(A\) is measurable and \(\sin u\ne 0\), then \(\| D u_u(x)\| \le n(1+1/\sin ^2u)\) for every \(x\in \mathbb R^n\), with operator norm for the maximum norm.

Let \(E\) be a real normed vector space, \(D:E\to \mathcal L(E,\mathbb R)\) be \(C\)-Lipschitz in operator norm with \(C\ge 0\), and \(L:E\to E\) be continuous linear with \(\| L\| \le 1\). Then \(x\mapsto D(Lx)\circ L\) is \(C\)-Lipschitz.

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For \(A\subseteq \mathbb R^n\) and \(\theta \in \mathbb R\), put \(u_\theta (x)=\int 1_A(\cos \theta \, x+\sin \theta \, y)\, d\gamma _n(y)\). If \(A\) is measurable and \(\sin u\ne 0\), then \(x\mapsto D u_u(x)\) is globally \(2n^2/\sin ^2u\)-Lipschitz in operator norm for the maximum norm on \(\mathbb R^n\).

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For \(A\subseteq \mathbb R^n\) and \(\theta \in \mathbb R\), put \(u_\theta (x)=\int 1_A(\cos \theta \, x+\sin \theta \, y)\, d\gamma _n(y)\). The averaged indicator is the function \(u_\theta :\mathbb R^n\to \mathbb R\); for a nonintegrable integrand the integral is assigned zero.

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For \(A\subseteq \mathbb R^n\) and \(\theta \in \mathbb R\), put \(u_\theta (x)=\int 1_A(\cos \theta \, x+\sin \theta \, y)\, d\gamma _n(y)\). If \(A\) is measurable, then \(u_\theta (x)=\gamma _n\{ y:\cos \theta \, x+\sin \theta \, y\in A\} \) for every real \(\theta \) and every \(x\).

Let \(\Omega ,\Xi \) be measurable spaces, \(A\subseteq \Xi \) a measurable set, and \(f,g:\Omega \to \Xi \) measurable maps. Then \(\{ z:\neg (f(z)\in A\ \Longleftrightarrow \ g(z)\in A)\} \) is measurable.

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For \(A\subseteq \mathbb R^n\) and \(\theta \in \mathbb R\), put \(u_\theta (x)=\int 1_A(\cos \theta \, x+\sin \theta \, y)\, d\gamma _n(y)\). For independent \(X,Y\sim \gamma _n\), put \(R_A(\theta )=\Pr (1_A(X)\ne 1_A(\cos \theta \, X+\sin \theta \, Y))\). If \(A\) is measurable, then \(\int |u_\theta (x)-1_A(x)|\, d\gamma _n(x)=R_A(\theta )\) for every real \(\theta \).

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For \(A\subseteq \mathbb R^n\) and \(\theta \in \mathbb R\), put \(u_\theta (x)=\int 1_A(\cos \theta \, x+\sin \theta \, y)\, d\gamma _n(y)\). If \(A\) is measurable, then \(0\le u_\theta (x)\le 1\) for every real \(\theta \) and every \(x\).

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For \(A\subseteq \mathbb R^n\) and \(\theta \in \mathbb R\), put \(u_\theta (x)=\int 1_A(\cos \theta \, x+\sin \theta \, y)\, d\gamma _n(y)\). If \(A\) is measurable, then \(\int |u_\theta -1_A|\, d\gamma _n\to 0\) as \(\theta \to 0\) through real values.

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For \(A\subseteq \mathbb R^n\) and \(\theta \in \mathbb R\), put \(u_\theta (x)=\int 1_A(\cos \theta \, x+\sin \theta \, y)\, d\gamma _n(y)\). For measurable \(A\) and every real \(\theta \), the function \(u_\theta \) is measurable.

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For \(A\subseteq \mathbb R^n\) and \(\theta \in \mathbb R\), put \(u_\theta (x)=\int 1_A(\cos \theta \, x+\sin \theta \, y)\, d\gamma _n(y)\). For measurable \(A\) and every real \(\theta \), the function \(u_\theta \) is \(\gamma _n\)-integrable.

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For \(A\subseteq \mathbb R^n\) and \(\theta \in \mathbb R\), put \(u_\theta (x)=\int 1_A(\cos \theta \, x+\sin \theta \, y)\, d\gamma _n(y)\). For measurable \(A\) and every real \(\theta \), \(\int u_\theta \, d\gamma _n=\gamma _n(A)\).

Let \(E\) be a measurable space and \(\mu \) an s-finite measure on \(E\). The map \(((a,b),(c,d))\mapsto ((a,c),(b,d))\) preserves the measure \((\mu \otimes \mu )\otimes (\mu \otimes \mu )\).

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For any real \(u\), if \(X_1,X_2,Y_1,Y_2\) are independent with law \(\gamma _n\), then \((\cos u\, X_1+\sin u\, Y_1,\cos u\, X_2+\sin u\, Y_2)\) has law \(\gamma _n\otimes \gamma _n\).

Let \(\mu \) be a finite measure on a measurable space \(E\), and let \(C,D\subseteq E\) be arbitrary sets. Then \(|\mu (C)-\mu (D)|\le \mu (C\mathbin {\triangle } D)\), where all values are viewed as finite real numbers.

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For \(A\subseteq \mathbb R^n\) and \(\theta \in \mathbb R\), put \(u_\theta (x)=\int 1_A(\cos \theta \, x+\sin \theta \, y)\, d\gamma _n(y)\). Let \(A\) be measurable, \(u,v\in \mathbb R\), and \(x,x\prime \in \mathbb R^n\). For independent \(Y,Y\prime \sim \gamma _n\), \(|u_u(x)-u_u(\cos v\, x+\sin v\, x\prime )|\le \Pr (1_A(\cos u\, x+\sin u\, Y)\ne 1_A(\cos u(\cos v\, x+\sin v\, x\prime )+\sin u(\cos v\, Y+\sin v\, Y\prime )))\).

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For \(A\subseteq \mathbb R^n\) and \(\theta \in \mathbb R\), put \(u_\theta (x)=\int 1_A(\cos \theta \, x+\sin \theta \, y)\, d\gamma _n(y)\). For independent \(X,Y\sim \gamma _n\), put \(R_A(\theta )=\Pr (1_A(X)\ne 1_A(\cos \theta \, X+\sin \theta \, Y))\). If \(A\) is measurable, then for every \(u,v\in \mathbb R\), \(\mathbb E_{X,Y}|u_u(X)-u_u(\cos v\, X+\sin v\, Y)|\le R_A(v)\).

If \(Z\) is a standard real Gaussian random variable, then \(\mathbb E[Z^4]=3\).

If \(Z\) is a standard real Gaussian random variable, then \(\mathbb E[Z^3]=0\).

If \(Z\) is a standard real Gaussian random variable, then \(\mathbb E[Z^2]=1\).

A standard real Gaussian random variable satisfies the invariance moment conditions: it has finite fourth absolute moment, its first three moments are \(0,1,0\), and its fourth moment is at most \(9\).

Let \(X\) be a real random variable on a probability space, measurable up to a null set, with \(\mathbb EX=0\) and \(X(\omega )^2=1\) for every \(\omega \). Then \(X\) has finite fourth absolute moment and satisfies \(\mathbb EX^2=1\), \(\mathbb EX^3=0\), and \(\mathbb EX^4\le 9\), so it satisfies the invariance moment conditions.

Let \(B\) be a uniform bit and put \(X=1\) when \(B\) is false and \(X=-1\) when \(B\) is true. Then \(X\) satisfies the invariance moment conditions, including finite fourth absolute moment, first three moments \(0,1,0\), and fourth moment at most \(9\).

Let \(X\) be a real function on a measured space \((\Omega ,\mu )\). It satisfies the invariance moment conditions when \(\mu \) is a probability measure, \(X\) is measurable up to a null set with finite fourth absolute moment, \(\mathbb E X=\mathbb E X^3=0\), \(\mathbb E X^2=1\), and \(\mathbb E X^4\le 9\).

Let \(X\) and \(Y\) be real random variables on measured spaces \((\Omega ,\mu )\) and \((\Xi ,\nu )\), where \(\nu \) is a probability measure. If \(X\) satisfies the invariance moment conditions and \(X\) and \(Y\) have the same distribution, then \(Y\) is measurable up to a null set with finite fourth absolute moment and moments \(\mathbb EY=\mathbb EY^3=0\), \(\mathbb EY^2=1\), and \(\mathbb EY^4\le 9\); thus \(Y\) satisfies the same moment conditions.

Let \(F(x)=\sum _{S\subseteq \{ 1,\ldots ,n\} }a_S\prod _{i\in S}x_i\) be a real multilinear polynomial of degree at most \(k\); let \(I_i(F)=\sum _{S\ni i}a_S^2\). Let \(X=(X_i)\) and \(Y=(Y_i)\) be families of independent real inputs on probability spaces \((\Omega ,\mu )\) and \((\Xi ,\nu )\), respectively. Each input is measurable up to a null set, has finite fourth absolute moment, and has first, second, and third moments \(0,1,0\) and fourth moment at most \(9\). If \(\psi \colon \mathbb R\to \mathbb R\) is \(C^4\) and \(C\ge 0\) satisfies \(|\psi ^{(4)}(t)|\le C\) for every \(t\), then \(|\mathbb E_\mu \psi (F(X))-\mathbb E_\nu \psi (F(Y))|\le (C/12)9^k\sum _i I_i(F)^2\).

Let \(F(x)=\sum _{S\subseteq \{ 1,\ldots ,n\} }a_S\prod _{i\in S}x_i\) be a real multilinear polynomial of degree at most \(k\); let \(I_i(F)=\sum _{S\ni i}a_S^2\). Let \(X=(X_i)\) and \(Y=(Y_i)\) be families of independent real inputs on probability spaces \((\Omega ,\mu )\) and \((\Xi ,\nu )\), respectively. Each input is measurable up to a null set, has finite fourth absolute moment, and has first, second, and third moments \(0,1,0\) and fourth moment at most \(9\). Suppose \(\sum _{S\ne \varnothing }a_S^2\le 1\), \(\varepsilon \ge 0\), and \(I_i(F)\le \varepsilon \) for every \(i\). If \(\psi \in C^4(\mathbb R)\) and \(C\ge 0\) bounds \(|\psi ^{(4)}|\) everywhere, then \(|\mathbb E_\mu \psi (F(X))-\mathbb E_\nu \psi (F(Y))|\le (C/12)k9^k\varepsilon \).

For every integer \(k\ge 0\), the real-number inequality \(k9^k\le 16^k\) holds.

For every integer \(k\ge 0\) and real \(\varepsilon \ge 0\), the inequality \((k9^k\varepsilon )^{1/4}\le 2^k\varepsilon ^{1/4}\) holds.

There exists a universal real constant \(K\gt 0\) with the following property. Let \(F(x)=\sum _{S\subseteq \{ 1,\ldots ,n\} }a_S\prod _{i\in S}x_i\) be a real multilinear polynomial of degree at most \(k\); let \(I_i(F)=\sum _{S\ni i}a_S^2\). Let \(X=(X_i)\) and \(Y=(Y_i)\) be families of independent real inputs on probability spaces \((\Omega ,\mu )\) and \((\Xi ,\nu )\), respectively. Each input is measurable up to a null set, has finite fourth absolute moment, and has first, second, and third moments \(0,1,0\) and fourth moment at most \(9\). If \(\sum _{S\ne \varnothing }a_S^2\le 1\), \(\varepsilon \ge 0\), \(I_i(F)\le \varepsilon \) for every \(i\), and \(\psi \colon \mathbb R\to \mathbb R\) is \(c\)-Lipschitz with \(c\ge 0\), then \(|\mathbb E_\mu \psi (F(X))-\mathbb E_\nu \psi (F(Y))|\le Kc2^k\varepsilon ^{1/4}\).

Let \(F(x)=\sum _{S\subseteq \{ 1,\ldots ,n\} }a_S\prod _{j\in S}x_j\) be a real multilinear polynomial and fix \(i\in \{ 1,\ldots ,n\} \). The coordinate-free part at \(i\) is the multilinear polynomial with coefficient \(0\) on sets containing \(i\) and coefficient \(a_S\) on every set \(S\) not containing \(i\).

Let \(F(x)=\sum _{S\subseteq \{ 1,\ldots ,n\} }a_S\prod _{j\in S}x_j\) be a real multilinear polynomial and fix \(i\in \{ 1,\ldots ,n\} \). The coordinate derivative polynomial has, at each set \(S\), the Fourier coefficient indexed by \(S\) of the Boolean-cube derivative of \(F\) restricted to sign inputs.

Let \(F(x)=\sum _{S\subseteq \{ 1,\ldots ,n\} }a_S\prod _{j\in S}x_j\) be a real multilinear polynomial and fix \(i\in \{ 1,\ldots ,n\} \). The coefficient indexed by \(S\) in its coordinate derivative is \(0\) if \(i\in S\), and \(a_{S\cup \{ i\} }\) if \(i\notin S\).

Let \(F(x)=\sum _{S\subseteq \{ 1,\ldots ,n\} }a_S\prod _{j\in S}x_j\) be a real multilinear polynomial and fix \(i\in \{ 1,\ldots ,n\} \). Define \(A_i(x)=\sum _{S:i\notin S}a_S\prod _{j\in S}x_j\) and \(D_i(x)=\sum _{S:i\notin S}a_{S\cup \{ i\} }\prod _{j\in S}x_j\). If \(F\) has degree at most \(k+1\), where \(k\ge 0\) is an integer, then \(D_i\) has degree at most \(k\).

Let \(F(x)=\sum _{S\subseteq \{ 1,\ldots ,n\} }a_S\prod _{j\in S}x_j\) be a real multilinear polynomial and fix \(i\in \{ 1,\ldots ,n\} \). On \(\{ -1,1\} ^n\), evaluation of its coordinate derivative polynomial equals the discrete derivative of the cube function \(F\), where the discrete derivative at \(i\) is \(\tfrac 12(F|_{x_i=1}-F|_{x_i=-1})\).

Let \(F(x)=\sum _{S\subseteq \{ 1,\ldots ,n\} }a_S\prod _{j\in S}x_j\) be a real multilinear polynomial and fix \(i\in \{ 1,\ldots ,n\} \). Define \(A_i(x)=\sum _{S:i\notin S}a_S\prod _{j\in S}x_j\) and \(D_i(x)=\sum _{S:i\notin S}a_{S\cup \{ i\} }\prod _{j\in S}x_j\). The sum of the squared coefficients of \(D_i\) is the influence of coordinate \(i\) on the uniform sign-cube function \(F\): \(\sum _{S:i\notin S}a_{S\cup \{ i\} }^2=I_i(F)=\sum _{S\ni i}a_S^2\).

Let \(F(x)=\sum _{S\subseteq \{ 1,\ldots ,n\} }a_S\prod _{j\in S}x_j\) be a real multilinear polynomial and fix \(i\in \{ 1,\ldots ,n\} \). Let \(X_j\colon \Omega \to \mathbb R\) be arbitrary functions. For each \(\omega \in \Omega \), evaluation of the coordinate-free part of \(F\) on \(X(\omega )\) equals \(\sum _{S\subseteq \{ 1,\ldots ,n\} \setminus \{ i\} }a_S\prod _{j\in S}X_j(\omega )\).

Let \(F(x)=\sum _{S\subseteq \{ 1,\ldots ,n\} }a_S\prod _{j\in S}x_j\) be a real multilinear polynomial and fix \(i\in \{ 1,\ldots ,n\} \). Let \(X_j\colon \Omega \to \mathbb R\) be arbitrary functions. For each \(\omega \in \Omega \), evaluation of the coordinate derivative on \(X(\omega )\) equals \(\sum _{S\subseteq \{ 1,\ldots ,n\} \setminus \{ i\} }a_{S\cup \{ i\} }\prod _{j\in S}X_j(\omega )\).

Let \(F(x)=\sum _{S\subseteq \{ 1,\ldots ,n\} }a_S\prod _{j\in S}x_j\) be a real multilinear polynomial and fix \(i\in \{ 1,\ldots ,n\} \). Define \(A_i(x)=\sum _{S:i\notin S}a_S\prod _{j\in S}x_j\) and \(D_i(x)=\sum _{S:i\notin S}a_{S\cup \{ i\} }\prod _{j\in S}x_j\). For any functions \(X_j\colon \Omega \to \mathbb R\) and \(\omega \in \Omega \), \(F(X(\omega ))=A_i(X(\omega ))+X_i(\omega )D_i(X(\omega ))\).

Let \(F(x)=\sum _{S\subseteq \{ 1,\ldots ,n\} }a_S\prod _{j\in S}x_j\) be a real multilinear polynomial and fix \(i\in \{ 1,\ldots ,n\} \). Define \(A_i(x)=\sum _{S:i\notin S}a_S\prod _{j\in S}x_j\) and \(D_i(x)=\sum _{S:i\notin S}a_{S\cup \{ i\} }\prod _{j\in S}x_j\). If \((X_j)\) is an independent family of real functions on a measured space \((\Omega ,\mu )\) and each \(X_j\) is measurable up to a null set, then \(X_i\) is independent of the joint pair \((A_i(X),D_i(X))\).

Let \(F(x)=\sum _{S\subseteq \{ 1,\ldots ,n\} }a_S\prod _{i\in S}x_i\) be a real multilinear polynomial of degree at most \(k\); let \(I_i(F)=\sum _{S\ni i}a_S^2\). Let \((X_i)\) be independent real inputs on a probability space, each measurable up to a null set with finite fourth absolute moment, first three moments \(0,1,0\), and fourth moment at most \(9\). Then \(\mathbb E[F(X)^4]\le 9^k(\sum _S a_S^2)^2\).

Let \(F(x)=\sum _{S\subseteq \{ 1,\ldots ,n\} }a_S\prod _{j\in S}x_j\) be a real multilinear polynomial and fix \(i\in \{ 1,\ldots ,n\} \). Define \(A_i(x)=\sum _{S:i\notin S}a_S\prod _{j\in S}x_j\) and \(D_i(x)=\sum _{S:i\notin S}a_{S\cup \{ i\} }\prod _{j\in S}x_j\). Suppose \(F\) has degree at most \(k+1\) and \((X_j)\) are independent real inputs on a probability space, each measurable up to a null set with finite fourth absolute moment, first three moments \(0,1,0\), and fourth moment at most \(9\). Then \(\mathbb E[D_i(X)^4]\le 9^k I_i(F)^2\), where \(I_i(F)=\sum _{S\ni i}a_S^2\).

Let \(A,D,W\) be real functions on a probability space \((\Omega ,\mu )\) such that \((A,D)\) and \(W\) are measurable up to null sets and \(W\) is independent of \((A,D)\). If \(\psi \colon \mathbb R\to \mathbb R\) is continuous and \(\psi (A+DW)\) is integrable, then \(q(a,d)=\mathbb E_\mu [\psi (a+dW)]\) is integrable under the law \(\lambda \) of \((A,D)\) and \(\mathbb E_\mu [\psi (A+DW)]=\int q(a,d)\, d\lambda (a,d)\).

Let \(A,D,U,V\) be real random variables on a probability space, with \(A,D\in L^4\). Suppose each of \(U,V\) is independent of the pair \((A,D)\) and is measurable up to a null set with finite fourth absolute moment, first three moments \(0,1,0\), and fourth moment at most \(9\). If \(\psi \in C^4(\mathbb R)\), \(C\ge 0\) bounds \(|\psi ^{(4)}|\) everywhere, and both \(\psi (A+DU)\) and \(\psi (A+DV)\) are integrable, then \(|\mathbb E\psi (A+DU)-\mathbb E\psi (A+DV)|\le (3C/4)\mathbb E[D^4]\).

Let \(F(x)=\sum _{S\subseteq \{ 1,\ldots ,n\} }a_S\prod _{j\in S}x_j\) be a real multilinear polynomial and fix \(i\in \{ 1,\ldots ,n\} \). Suppose \(F\) has degree at most \(k+1\), and \(X,Y\) are real input families on one probability space, each independent within itself, agreeing as functions at every coordinate other than \(i\). Every input is measurable up to a null set with finite fourth absolute moment, first three moments \(0,1,0\), and fourth moment at most \(9\). If \(\psi \in C^4(\mathbb R)\) and \(C\ge 0\) bounds \(|\psi ^{(4)}|\) everywhere, then \(|\mathbb E\psi (F(X))-\mathbb E\psi (F(Y))|\le (C/12)9^{k+1} I_i(F)^2\), where \(I_i(F)=\sum _{S\ni i}a_S^2\).

For integers \(n,r\ge 0\) and an index \(i\in \{ 0,\ldots ,n-1\} \), the hybrid selection chooses the right copy of \(i\) if \(i\lt r\) and the left copy otherwise, as an element of two disjoint copies of the coordinate set.

Let \(Z_{L,i},Z_{R,i}\colon \Omega \to \mathbb R\) be two indexed families, \(0\le i\lt n\), and let \(r\ge 0\) be an integer. The hybrid family \(H^{(r)}\) has \(H^{(r)}_i=Z_{R,i}\) for \(i\lt r\) and \(H^{(r)}_i=Z_{L,i}\) for \(i\ge r\).

For integers \(n,r\ge 0\), the map sending \(i\in \{ 0,\ldots ,n-1\} \) to its right copy when \(i\lt r\) and its left copy otherwise is injective.

Let \(Z_{L,i},Z_{R,i}\colon \Omega \to \mathbb R\) be two families indexed by \(0\le i\lt n\), and define \(H^{(r)}_i=Z_{R,i}\) if \(i\lt r\) and \(Z_{L,i}\) otherwise. Then \(H^{(0)}_i=Z_{L,i}\) for every \(i\).

Let \(Z_{L,i},Z_{R,i}\colon \Omega \to \mathbb R\) be two families indexed by \(0\le i\lt n\), and define \(H^{(r)}_i=Z_{R,i}\) if \(i\lt r\) and \(Z_{L,i}\) otherwise. Then \(H^{(n)}_i=Z_{R,i}\) for every \(i\), including the empty family when \(n=0\).

Let \(Z_{L,j},Z_{R,j}\colon \Omega \to \mathbb R\) be two families indexed by \(0\le j\lt n\), and put \(H^{(r)}_j=Z_{R,j}\) if \(j\lt r\) and \(Z_{L,j}\) otherwise. If \(i,j\in \{ 0,\ldots ,n-1\} \) and \(j\ne i\), then \(H^{(i)}_j=H^{(i+1)}_j\) as functions on \(\Omega \).

Let \(F(x)=\sum _{S\subseteq \{ 1,\ldots ,n\} }a_S\prod _{i\in S}x_i\) be a real multilinear polynomial of degree at most \(k\); let \(I_i(F)=\sum _{S\ni i}a_S^2\). Let \((Z_{L,i},Z_{R,i})_{i=1}^n\) be a mutually independent family of \(2n\) real inputs on a probability space, each measurable up to a null set with finite fourth absolute moment, first three moments \(0,1,0\), and fourth moment at most \(9\). If \(\psi \in C^4(\mathbb R)\) and \(C\ge 0\) bounds \(|\psi ^{(4)}|\) everywhere, then \(|\mathbb E\psi (F(Z_L))-\mathbb E\psi (F(Z_R))|\le (C/12)9^k\sum _i I_i(F)^2\).

Let \(I\) be a finite index set and let \((X_i)_{i\in I}\) and \((Y_i)_{i\in I}\) be real input families on probability spaces \((\Omega ,\mu )\) and \((\Xi ,\nu )\), respectively. If each family is independent and \(X_i\) and \(Y_i\) are measurable up to null sets with the same distribution for every \(i\), then the random vectors \((X_i)_{i\in I}\) and \((Y_i)_{i\in I}\) have the same distribution.

For real \(\eta ,x\) and any real function \(\psi \), the interval smoothing is \(S_\eta \psi (x)=\eta ^{-1}\int _0^\eta \psi (x+t)\, dt\), with value zero for \(\eta =0\). The integral is oriented when \(\eta \lt 0\) and is assigned zero when its integrand is not integrable.

For \(\eta \gt 0\) and a real function \(\psi \), write \(S_\eta \psi (x)=\eta ^{-1}\int _0^\eta \psi (x+t)\, dt\). If \(\psi \) is continuous, then \(S_\eta \psi \) is differentiable at every real \(x\) with derivative \((\psi (x+\eta )-\psi (x))/\eta \).

For \(\eta \gt 0\) and a real function \(\psi \), write \(S_\eta \psi (x)=\eta ^{-1}\int _0^\eta \psi (x+t)\, dt\). For each integer \(m\ge 0\), if \(\psi \) is \(C^m\) on \(\mathbb R\), then \(S_\eta \psi \) is \(C^{m+1}\) on \(\mathbb R\).

For \(\eta \gt 0\) and a real function \(\psi \), write \(S_\eta \psi (x)=\eta ^{-1}\int _0^\eta \psi (x+t)\, dt\). If \(m\ge 0\) is an integer and \(\psi \in C^m(\mathbb R)\), then \((S_\eta \psi )^{(m+1)}(x)=(\psi ^{(m)}(x+\eta )-\psi ^{(m)}(x))/\eta \) for every real \(x\).

For \(\eta \gt 0\) and a real function \(\psi \), write \(S_\eta \psi (x)=\eta ^{-1}\int _0^\eta \psi (x+t)\, dt\). If \(\psi \) is \(c\)-Lipschitz for \(c\ge 0\), then \(|(S_\eta \psi )\prime (x)|\le c\) for every real \(x\).

For \(\eta \gt 0\) and a real function \(\psi \), write \(S_\eta \psi (x)=\eta ^{-1}\int _0^\eta \psi (x+t)\, dt\). If \(\psi \colon \mathbb R\to \mathbb R\) is \(c\)-Lipschitz with \(c\ge 0\), then \(S_\eta \psi \) is also \(c\)-Lipschitz.

For \(\eta \gt 0\) and a real function \(\psi \), write \(S_\eta \psi (x)=\eta ^{-1}\int _0^\eta \psi (x+t)\, dt\). If \(\psi \) is \(c\)-Lipschitz for \(c\ge 0\), then \(|S_\eta \psi (x)-\psi (x)|\le c\eta \) for every real \(x\).

For \(\eta \gt 0\) and a real function \(\psi \), write \(S_\eta \psi (x)=\eta ^{-1}\int _0^\eta \psi (x+t)\, dt\). If \(\psi \) is \(c\)-Lipschitz with \(c\ge 0\) and \(m\ge 0\) is an integer, put \(\phi =S_\eta ^{m+1}\psi \), where the exponent denotes repeated application of \(S_\eta \). Then \(\phi \) is \(C^{m+1}\) and \(c\)-Lipschitz, and for every real \(x\), \(|\phi (x)-\psi (x)|\le (m+1)c\eta \) and \(|\phi ^{(m+1)}(x)|\le c(2/\eta )^m\).

For every \(\eta \gt 0\), \(c\ge 0\), and \(c\)-Lipschitz function \(\psi \colon \mathbb R\to \mathbb R\), there exists \(\phi \in C^4(\mathbb R)\) such that \(|\phi (x)-\psi (x)|\le 4c\eta \) and \(|\phi ^{(4)}(x)|\le 8c/\eta ^3\) for all real \(x\).

Let \(f,g\colon \Omega \to \mathbb R\) be functions on a probability space \((\Omega ,\mu )\), with \(f\) integrable and \(g\) measurable up to a null set. If a real constant \(D\) satisfies \(|f(\omega )-g(\omega )|\le D\) for every \(\omega \), then \(|\int f\, d\mu -\int g\, d\mu |\le D\).

Let \(U\in L^4(\mu )\) and \(V\in L^4(\nu )\) be real inputs on probability spaces, and let \(B\ge 0\). Suppose that for every \(\phi \in C^4(\mathbb R)\) and \(C\ge 0\) bounding \(|\phi ^{(4)}|\) everywhere, \(|\mathbb E_\mu \phi (U)-\mathbb E_\nu \phi (V)|\le CB\). For every \(c\ge 0\), \(c\)-Lipschitz function \(\psi \colon \mathbb R\to \mathbb R\), and \(\eta \gt 0\), \(|\mathbb E_\mu \psi (U)-\mathbb E_\nu \psi (V)|\le 8c\eta +(8c/\eta ^3)B\).

Let \(U\in L^4(\mu )\) and \(V\in L^4(\nu )\) be real inputs on probability spaces, and let \(B\ge 0\). Suppose that for every \(\phi \in C^4(\mathbb R)\) and \(C\ge 0\) bounding \(|\phi ^{(4)}|\) everywhere, \(|\mathbb E_\mu \phi (U)-\mathbb E_\nu \phi (V)|\le CB\). For every \(c\ge 0\) and \(c\)-Lipschitz function \(\psi \colon \mathbb R\to \mathbb R\), \(|\mathbb E_\mu \psi (U)-\mathbb E_\nu \psi (V)|\le 16cB^{1/4}\), including \(B=0\).

Let \(\psi \colon \mathbb R\to \mathbb R\) be \(C^4\) and let \(C\ge 0\) satisfy \(|\psi ^{(4)}(t)|\le C\) for all real \(t\). For all real \(a,b\), \(\left|\psi (a+b)-\sum _{j=0}^3\psi ^{(j)}(a)b^j/j!\right|\le Cb^4/24\).

Let \(Z\in L^4(\mu )\) be real-valued on a probability space \((\Omega ,\mu )\), and let \(\psi \colon \mathbb R\to \mathbb R\) be \(C^4\). If some \(C\ge 0\) bounds \(|\psi ^{(4)}(t)|\) for every real \(t\), then \(\psi (Z)\) is integrable under \(\mu \).

Let \(X\) be a real input on a probability space, measurable up to a null set with finite fourth absolute moment, first three moments \(0,1,0\), and fourth moment at most \(9\). For any real function \(\psi \) and real \(a,b\), \(\mathbb E[\sum _{j=0}^3\psi ^{(j)}(a)(bX)^j/j!]=\psi (a)+\tfrac 12\psi ^{(2)}(a)b^2\). Here iterated differentiation assigns zero at a point where the derivative does not exist, so the identity requires no smoothness of \(\psi \).

Let \(X\) be a real input on a probability space, measurable up to a null set with finite fourth absolute moment, first three moments \(0,1,0\), and fourth moment at most \(9\). Let \(\psi \colon \mathbb R\to \mathbb R\) be \(C^4\) and let \(C\ge 0\) satisfy \(|\psi ^{(4)}(t)|\le C\) for all real \(t\). For all real \(a,b\), \(|\mathbb E\psi (a+bX)-\psi (a)-\tfrac 12\psi ^{(2)}(a)b^2|\le (3C/8)b^4\).

Let \(X\) and \(Y\) be real inputs on possibly different probability spaces, each measurable up to a null set with finite fourth absolute moment, first three moments \(0,1,0\), and fourth moment at most \(9\). Let \(\psi \colon \mathbb R\to \mathbb R\) be \(C^4\) and let \(C\ge 0\) satisfy \(|\psi ^{(4)}(t)|\le C\) for all real \(t\). For every real \(a,b\), \(|\mathbb E\psi (a+bX)-\mathbb E\psi (a+bY)|\le (3C/4)b^4\).

Let \(\mu _m\) and \(\nu \) be probability measures on \(\mathbb R\). Assume that for every \(c\ge 0\) and every \(c\)-Lipschitz \(\psi :\mathbb R\to \mathbb R\), \(\int \psi \, d\mu _m\to \int \psi \, d\nu \). Then for each \(n\ge 0\) and every bounded continuous \(F:\mathbb R^n\to \mathbb R\), \(\int F\, d\mu _m^{\otimes n}\to \int F\, d\nu ^{\otimes n}\).

For \(m\ge 0\), let \(\nu _m\) be the probability law of \((m+1)^{-1/2}\sum _{j=1}^{m+1}\epsilon _j\), where the \(\epsilon _j\) are independent uniform signs in \(\{ -1,1\} \).

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). Let \(\nu _m\) be the law of the normalized sum of \(m+1\) independent uniform signs. For every \(n\ge 0\) and bounded continuous \(F:\mathbb R^n\to \mathbb R\), \(\int F\, d\nu _m^{\otimes n}\to \int F\, d\gamma _n\) as \(m\to \infty \).

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For independent uniform bits \(X_{ij}\), put \(S_r(X)_i=r^{-1/2}\sum _{j=1}^r(-1)^{X_{ij}}\). For every \(n\ge 0\) and bounded continuous \(F:\mathbb R^n\to \mathbb R\), \(\mathbb E F(S_{m+1}(X))\to \int F\, d\gamma _n\) as \(m\to \infty \).

For \(n\ge 0\), define a multilinear coefficient family \(P_n\) on subsets of \(\{ 1,\ldots ,n\} \) by \(P_n(S)=1/\sqrt n\) when \(|S|=1\) and \(P_n(S)=0\) otherwise; division by zero is zero. Its evaluation is \(n^{-1/2}\sum _i z_i\).

For every \(n\ge 0\), the multilinear polynomial with singleton coefficients \(1/\sqrt n\) and all other coefficients zero has degree at most one.

For \(n\ge 0\) and a coordinate \(i\in \{ 1,\ldots ,n\} \), let \(f(x)=n^{-1/2}\sum _j(-1)^{x_j}\) on the uniform Boolean cube. Its influence \(\mathbb E(D_i f)^2\), with \(D_i\) the coordinate half-difference, equals \(1/n\).

If \(n\gt 0\) and \(f(x)=n^{-1/2}\sum _{i=1}^n(-1)^{x_i}\) on the uniform Boolean cube, then \(\operatorname {Var}(f)=1\).

Let \(n\ge 0\), let \(\Omega \) be a measurable space, and let \(X_i:\Omega \to \mathbb R\) be arbitrary functions. The multilinear polynomial with singleton coefficients \(1/\sqrt n\) and all other coefficients zero evaluates at \((X_i(\omega ))_i\) to \(\sum _iX_i(\omega )/\sqrt n\) for every \(\omega \in \Omega \), with division by zero equal to zero.

There is a universal \(K\gt 0\) such that, for every \(n\gt 0\), probability space \((\Omega ,\mu )\), and independent real random variables \(X_1,\ldots ,X_n\) with finite fourth moments, \(\mathbb E X_i=\mathbb E X_i^3=0\), \(\mathbb E X_i^2=1\), and \(\mathbb E X_i^4\le 9\), every \(c\)-Lipschitz \(\psi :\mathbb R\to \mathbb R\) with \(c\ge 0\) satisfies \(|\mathbb E\psi (n^{-1/2}\sum _iX_i)-\mathbb E\psi (Z)|\le Kc\, n^{-1/4}\), where \(Z\) is standard Gaussian.

For every \(n\ge 0\) and \(f:\{ 0,1\} ^n\to \mathbb R\), integration against the product of \(n\) uniform bit measures equals the uniform cube expectation \(2^{-n}\sum _xf(x)\).

There is a universal \(K\gt 0\) such that, for \(n\gt 0\), independent uniform signs \(\epsilon _1,\ldots ,\epsilon _n\), a standard Gaussian \(Z\), and any \(c\)-Lipschitz \(\psi :\mathbb R\to \mathbb R\) with \(c\ge 0\), \(|\mathbb E\psi (n^{-1/2}\sum _i\epsilon _i)-\mathbb E\psi (Z)|\le Kc\, n^{-1/4}\).

For every \(c\ge 0\) and \(c\)-Lipschitz \(\psi :\mathbb R\to \mathbb R\), if \(\epsilon _1,\ldots ,\epsilon _{m+1}\) are independent uniform signs and \(Z\) is standard Gaussian, then \(\mathbb E\psi ((m+1)^{-1/2}\sum _i\epsilon _i)\to \mathbb E\psi (Z)\) as \(m\to \infty \), where the first expectation is integration under the product sign law.

For every \(c\ge 0\) and \(c\)-Lipschitz \(\psi :\mathbb R\to \mathbb R\), \(2^{-(m+1)}\sum _{x\in \{ 0,1\} ^{m+1}}\psi ((m+1)^{-1/2}\sum _i(-1)^{x_i})\to \mathbb E\psi (Z)\) as \(m\to \infty \), where \(Z\) is standard Gaussian.

Let \(I\) be a finite set, \(c\in \mathbb R^I\), \(v\gt 0\), and \(\mu _{c,v}=\bigotimes _{i\in I}N(c_i,v)\). If \(f:\mathbb R^I\to \mathbb R\) is measurable with \(|f|\le 1\) and \(i\in I\), then \(z\mapsto ((z_i-c_i)/v)f(z)\) is \(\mu _{c,v}\)-integrable and \(|\int ((z_i-c_i)/v)f(z)\, d\mu _{c,v}(z)|\le 1+1/v\).

Let \(I\) be a set, \(c,z\in \mathbb R^I\), \(v\gt 0\), \(i,j\in I\), and \(a\in \mathbb R\) with \(|a|\le 1\). Then \(|(((z_i-c_i)(z_j-c_j)/v^2-\delta _{ij}/v)a)|\le ((z_i-c_i)^2+(z_j-c_j)^2)/(2v^2)+1/v\), where \(\delta _{ij}=1\) if \(i=j\) and zero otherwise.

Let \(I\) be finite, \(c\in \mathbb R^I\), \(v\ge 0\), and \(i,j\in I\). The function \(z\mapsto ((z_i-c_i)^2+(z_j-c_j)^2)/(2v^2)+1/v\) is integrable under \(\bigotimes _kN(c_k,v)\), with division by zero interpreted as zero.

Let \(I\) be a finite set, \(c\in \mathbb R^I\), \(v\gt 0\), and \(\mu _{c,v}=\bigotimes _{i\in I}N(c_i,v)\). Put \(B_{ij}(z)=((z_i-c_i)^2+(z_j-c_j)^2)/(2v^2)+1/v\). For all \(i,j\in I\), \(\int B_{ij}\, d\mu _{c,v}=2/v\).

Let \(I\) be a finite set, \(c\in \mathbb R^I\), \(v\gt 0\), and \(\mu _{c,v}=\bigotimes _{i\in I}N(c_i,v)\). For \(i,j\in I\), put \(H_{ij}(z)=(z_i-c_i)(z_j-c_j)/v^2-\delta _{ij}/v\), where \(\delta _{ij}\) is one if \(i=j\) and zero otherwise. If \(f:\mathbb R^I\to \mathbb R\) is measurable and \(|f|\le 1\), then \(H_{ij}f\) is \(\mu _{c,v}\)-integrable for every \(i,j\in I\).

Let \(I\) be a finite set, \(c\in \mathbb R^I\), \(v\gt 0\), and \(\mu _{c,v}=\bigotimes _{i\in I}N(c_i,v)\). For \(i,j\in I\), put \(H_{ij}(z)=(z_i-c_i)(z_j-c_j)/v^2-\delta _{ij}/v\), where \(\delta _{ij}\) is one if \(i=j\) and zero otherwise. For any \(f:\mathbb R^I\to \mathbb R\) with \(|f|\le 1\) and \(i,j\in I\), \(|\int H_{ij}f\, d\mu _{c,v}|\le 2/v\), assigning zero to a nonintegrable integral.

For a finite set \(I\) and \(i\in I\), the coordinate projection \(\mathbb R^I\to \mathbb R\), \(x\mapsto x_i\), has operator norm at most one for the maximum norm.

Let \(I\) be a finite set, \(c\in \mathbb R^I\), \(v\gt 0\), and \(\mu _{c,v}=\bigotimes _{i\in I}N(c_i,v)\). The measure \(\mu _{c,v}\) has Lebesgue density \(z\mapsto \prod _i p_v(c_i,z_i)\), where \(p_v(c_i,z_i)\) is the scalar Gaussian density.

Let \(I\) be a finite set, \(c\in \mathbb R^I\), \(v\gt 0\), and \(\mu _{c,v}=\bigotimes _{i\in I}N(c_i,v)\). For a real normed vector space \(E\) and any \(f:\mathbb R^I\to E\), \(\int f\, d\mu _{c,v}=\int _{\mathbb R^I}(\prod _i p_v(c_i,z_i))f(z)\, dz\), where \(p_v\) is the scalar Gaussian density and nonintegrable integrals on either side are assigned zero.

Let \(I\) be finite, \(c\in \mathbb R^I\), and \(s\in \mathbb R\). The map \(y\mapsto (c_i+sy_i)_i\) pushes \(\bigotimes _iN(0,1)\) to \(\bigotimes _iN(c_i,s^2)\), including negative or zero \(s\).

Let \(I\) be finite, \(c\in \mathbb R^I\), \(s\ne 0\), and \(f:\mathbb R^I\to \mathbb R\) measurable. For a standard Gaussian vector \(Y\), \(\mathbb E f(c+sY)=\int _{\mathbb R^I}(\prod _i p_{s^2}(c_i,z_i))f(z)\, dz\), where \(p_{s^2}\) is the scalar Gaussian density and nonintegrable integrals are assigned zero.

For a finite set \(I\), \(c\in \mathbb R^I\), \(v\ge 0\), and \(i\in I\), the function \(z\mapsto (z_i-c_i)^2\) is integrable under \(\bigotimes _jN(c_j,v)\).

For a finite set \(I\), \(c\in \mathbb R^I\), \(v\ge 0\), and \(i\in I\), \(\int (z_i-c_i)^2\, d(\bigotimes _jN(c_j,v))(z)=v\).

For \(v\gt 0\) and \(M\ge 0\), let \(p_v(m,z)\) be the Gaussian density with mean \(m\) and variance \(v\). There exists a nonnegative Lebesgue-integrable \(b:\mathbb R\to \mathbb R\) such that \(p_v(m,z)\le b(z)\), \(|((z-m)/v)p_v(m,z)|\le b(z)\), and \(|((z-m)^2/v^2-1/v)p_v(m,z)|\le b(z)\) whenever \(|m|\le M\).

Let \(I\) be finite and \(v\ge 0\). Write \(p_v(c,z)\) for the \(N(c,v)\) density when \(v\gt 0\) and set \(p_0(c,z)=0\); define \(G_v(c,z)=\prod _{i\in I}p_v(c_i,z_i)\) and interpret division by zero as zero. Let \(A_v(c,z)\) be the linear functional \(h\mapsto \sum _i\bigl(\prod _{j\ne i}p_v(c_j,z_j)\bigr)((z_i-c_i)/v)p_v(c_i,z_i)h_i\). The product-density derivative is \(A_v(c,z)\) for mean and input vectors \(c,z\in \mathbb R^I\).

Let \(I\) be finite and \(v\ge 0\). Write \(p_v(c,z)\) for the \(N(c,v)\) density when \(v\gt 0\) and set \(p_0(c,z)=0\); define \(G_v(c,z)=\prod _{i\in I}p_v(c_i,z_i)\) and interpret division by zero as zero. Let \(A_v(c,z)\) be the linear functional \(h\mapsto \sum _i\bigl(\prod _{j\ne i}p_v(c_j,z_j)\bigr)((z_i-c_i)/v)p_v(c_i,z_i)h_i\). For \(c,z,h\in \mathbb R^I\), \(A_v(c,z)[h]=G_v(c,z)\sum _i((z_i-c_i)/v)h_i\).

Let \(I\) be finite and \(v\ge 0\). Write \(p_v(c,z)\) for the \(N(c,v)\) density when \(v\gt 0\) and set \(p_0(c,z)=0\); define \(G_v(c,z)=\prod _{i\in I}p_v(c_i,z_i)\) and interpret division by zero as zero. Let \(A_v(c,z)\) be the linear functional \(h\mapsto \sum _i\bigl(\prod _{j\ne i}p_v(c_j,z_j)\bigr)((z_i-c_i)/v)p_v(c_i,z_i)h_i\). For every \(c,z\in \mathbb R^I\), the derivative at \(c\) of \(m\mapsto G_v(m,z)\) is \(A_v(c,z)\).

Let \(I\) be finite and \(v\ge 0\). Write \(p_v(c,z)\) for the \(N(c,v)\) density when \(v\gt 0\) and set \(p_0(c,z)=0\); define \(G_v(c,z)=\prod _{i\in I}p_v(c_i,z_i)\) and interpret division by zero as zero. Let \(A_v(c,z)\) be the linear functional \(h\mapsto \sum _i\bigl(\prod _{j\ne i}p_v(c_j,z_j)\bigr)((z_i-c_i)/v)p_v(c_i,z_i)h_i\). If \(v\gt 0\) and \(M\ge 0\), there exists a nonnegative Lebesgue-integrable \(b:\mathbb R^I\to \mathbb R\) such that \(|G_v(m,z)|\le b(z)\) and \(\| A_v(m,z)\| \le b(z)\) for all \(z\) and all \(m\) with \(|m_i|\le M\) for every \(i\), with operator norm for the maximum norm.

Let \(I\) be finite and \(v\ge 0\). Write \(p_v(c,z)\) for the \(N(c,v)\) density when \(v\gt 0\) and set \(p_0(c,z)=0\); define \(G_v(c,z)=\prod _{i\in I}p_v(c_i,z_i)\) and interpret division by zero as zero. Let \(A_v(c,z)\) be the linear functional \(h\mapsto \sum _i\bigl(\prod _{j\ne i}p_v(c_j,z_j)\bigr)((z_i-c_i)/v)p_v(c_i,z_i)h_i\). For each fixed \(v\ge 0\), \((c,z)\mapsto A_v(c,z)\) is continuous in operator norm.

Let \(I\) be finite and \(v\ge 0\). Write \(p_v(c,z)\) for the \(N(c,v)\) density when \(v\gt 0\) and set \(p_0(c,z)=0\); define \(G_v(c,z)=\prod _{i\in I}p_v(c_i,z_i)\) and interpret division by zero as zero. Let \(A_v(c,z)\) be the linear functional \(h\mapsto \sum _i\bigl(\prod _{j\ne i}p_v(c_j,z_j)\bigr)((z_i-c_i)/v)p_v(c_i,z_i)h_i\). For fixed \(v\ge 0\) and \(c\in \mathbb R^I\), \(z\mapsto A_v(c,z)\) is continuous in operator norm.

Let \(I\) be finite and \(v\ge 0\). Write \(p_v(c,z)\) for the \(N(c,v)\) density when \(v\gt 0\) and set \(p_0(c,z)=0\); define \(G_v(c,z)=\prod _{i\in I}p_v(c_i,z_i)\) and interpret division by zero as zero. Let \(A_v(c,z)\) be the linear functional \(h\mapsto \sum _i\bigl(\prod _{j\ne i}p_v(c_j,z_j)\bigr)((z_i-c_i)/v)p_v(c_i,z_i)h_i\). Let \(f:\mathbb R^I\to \mathbb R\) be measurable with \(|f|\le 1\). If \(v\gt 0\) and \(c\in \mathbb R^I\), then \(z\mapsto f(z)A_v(c,z)\) is Lebesgue-integrable as a linear-functional-valued function.

Let \(I\) be finite and \(c,m\in \mathbb R^I\). If \(\| m-c\| _\infty \lt \varepsilon \) for a real \(\varepsilon \), then \(|m_i|\le \| c\| _\infty +\varepsilon \) for every \(i\in I\).

Let \(I\) be finite and \(v\ge 0\). Write \(p_v(c,z)\) for the \(N(c,v)\) density when \(v\gt 0\) and set \(p_0(c,z)=0\); define \(G_v(c,z)=\prod _{i\in I}p_v(c_i,z_i)\) and interpret division by zero as zero. Let \(A_v(c,z)\) be the linear functional \(h\mapsto \sum _i\bigl(\prod _{j\ne i}p_v(c_j,z_j)\bigr)((z_i-c_i)/v)p_v(c_i,z_i)h_i\). Let \(f:\mathbb R^I\to \mathbb R\) be measurable with \(|f|\le 1\). If \(v\gt 0\), the function \(m\mapsto \int _{\mathbb R^I}G_v(m,z)f(z)\, dz\) is differentiable at every \(c\), with derivative \(\int _{\mathbb R^I}f(z)A_v(c,z)\, dz\).

Let \(I\) be finite and \(v\ge 0\). Write \(p_v(c,z)\) for the \(N(c,v)\) density when \(v\gt 0\) and set \(p_0(c,z)=0\); define \(G_v(c,z)=\prod _{i\in I}p_v(c_i,z_i)\) and interpret division by zero as zero. Let \(A_v(c,z)\) be the linear functional \(h\mapsto \sum _i\bigl(\prod _{j\ne i}p_v(c_j,z_j)\bigr)((z_i-c_i)/v)p_v(c_i,z_i)h_i\). Let \(f:\mathbb R^I\to \mathbb R\) be measurable with \(|f|\le 1\). If \(v\gt 0\), the linear-functional-valued function \(c\mapsto \int _{\mathbb R^I}f(z)A_v(c,z)\, dz\) is continuous.

Let \(I\) be finite and \(v\ge 0\). Write \(p_v(c,z)\) for the \(N(c,v)\) density when \(v\gt 0\) and set \(p_0(c,z)=0\); define \(G_v(c,z)=\prod _{i\in I}p_v(c_i,z_i)\) and interpret division by zero as zero. Let \(A_v(c,z)\) be the linear functional \(h\mapsto \sum _i\bigl(\prod _{j\ne i}p_v(c_j,z_j)\bigr)((z_i-c_i)/v)p_v(c_i,z_i)h_i\). Let \(f:\mathbb R^I\to \mathbb R\) be measurable with \(|f|\le 1\). If \(v\gt 0\), the map \(m\mapsto \int _{\mathbb R^I}G_v(m,z)f(z)\, dz\) is continuously differentiable.

Let \(I\) be finite and \(v\ge 0\). Write \(p_v(c,z)\) for the \(N(c,v)\) density when \(v\gt 0\) and set \(p_0(c,z)=0\); define \(G_v(c,z)=\prod _{i\in I}p_v(c_i,z_i)\) and interpret division by zero as zero. Let \(A_v(c,z)\) be the linear functional \(h\mapsto \sum _i\bigl(\prod _{j\ne i}p_v(c_j,z_j)\bigr)((z_i-c_i)/v)p_v(c_i,z_i)h_i\). Let \(f:\mathbb R^I\to \mathbb R\) be measurable with \(|f|\le 1\). If \(v\gt 0\), \(c,h\in \mathbb R^I\), and \(Z\sim \bigotimes _iN(c_i,v)\), then \((\int f(z)A_v(c,z)\, dz)[h]=\sum _i\mathbb E[((Z_i-c_i)/v)f(Z)]h_i\).

Let \(I\) be finite and \(v\ge 0\). Write \(p_v(c,z)\) for the \(N(c,v)\) density when \(v\gt 0\) and set \(p_0(c,z)=0\); define \(G_v(c,z)=\prod _{i\in I}p_v(c_i,z_i)\) and interpret division by zero as zero. Let \(A_v(c,z)\) be the linear functional \(h\mapsto \sum _i\bigl(\prod _{j\ne i}p_v(c_j,z_j)\bigr)((z_i-c_i)/v)p_v(c_i,z_i)h_i\). Let \(f:\mathbb R^I\to \mathbb R\) be measurable with \(|f|\le 1\). If \(v\gt 0\) and \(c\in \mathbb R^I\), then \(\| \int f(z)A_v(c,z)\, dz\| \le |I|(1+1/v)\), in operator norm for the maximum norm.

Let \(I\) be finite and \(v\ge 0\). Write \(p_v(c,z)\) for the \(N(c,v)\) density when \(v\gt 0\) and set \(p_0(c,z)=0\); define \(G_v(c,z)=\prod _{i\in I}p_v(c_i,z_i)\) and interpret division by zero as zero. For \(c,z,h\in \mathbb R^I\), put \(S_v(c,z)[h]=\sum _i((z_i-c_i)/v)h_i\). The mean score is the linear functional \(S_v(c,z)\).

Let \(I\) be finite and \(v\ge 0\). Write \(p_v(c,z)\) for the \(N(c,v)\) density when \(v\gt 0\) and set \(p_0(c,z)=0\); define \(G_v(c,z)=\prod _{i\in I}p_v(c_i,z_i)\) and interpret division by zero as zero. For \(c,z,h\in \mathbb R^I\), put \(S_v(c,z)[h]=\sum _i((z_i-c_i)/v)h_i\). For every \(c,z\), the derivative in \(c\) of \(S_v(c,z)\) is the bilinear map \((h,k)\mapsto -\sum _i h_i k_i/v\).

Let \(I\) be finite and \(v\ge 0\). Put \(G_v(c,z)=\prod _i p_v(c_i,z_i)\), where \(p_v\) is the Gaussian density of variance \(v\) for \(v\gt 0\) and \(p_0=0\), and put \(S_v(c,z)[h]=\sum _i((z_i-c_i)/v)h_i\), with division by zero equal to zero. Define \(A_v(c,z)[h]=G_v(c,z)S_v(c,z)[h]\) and \(T_v(c,z)[h,k]=G_v(c,z)(S_v(c,z)[h]S_v(c,z)[k]-\sum _i h_i k_i/v)\). The product-density Hessian is the bilinear form \(T_v(c,z)\).

Let \(I\) be finite and \(v\ge 0\). Put \(G_v(c,z)=\prod _i p_v(c_i,z_i)\), where \(p_v\) is the Gaussian density of variance \(v\) for \(v\gt 0\) and \(p_0=0\), and put \(S_v(c,z)[h]=\sum _i((z_i-c_i)/v)h_i\), with division by zero equal to zero. Define \(A_v(c,z)[h]=G_v(c,z)S_v(c,z)[h]\) and \(T_v(c,z)[h,k]=G_v(c,z)(S_v(c,z)[h]S_v(c,z)[k]-\sum _i h_i k_i/v)\). For all \(c,z\in \mathbb R^I\), the derivative at \(c\) of \(m\mapsto A_v(m,z)\) is \(T_v(c,z)\).

Let \(I\) be finite and \(v\ge 0\). Put \(G_v(c,z)=\prod _i p_v(c_i,z_i)\), where \(p_v\) is the Gaussian density of variance \(v\) for \(v\gt 0\) and \(p_0=0\), and put \(S_v(c,z)[h]=\sum _i((z_i-c_i)/v)h_i\), with division by zero equal to zero. Define \(A_v(c,z)[h]=G_v(c,z)S_v(c,z)[h]\) and \(T_v(c,z)[h,k]=G_v(c,z)(S_v(c,z)[h]S_v(c,z)[k]-\sum _i h_i k_i/v)\). For all \(c,z,h,k\in \mathbb R^I\), \(T_v(c,z)[h,k]=G_v(c,z)\sum _{i,j}((z_i-c_i)(z_j-c_j)/v^2-\delta _{ij}/v)h_jk_i\), where \(\delta _{ij}\) is the Kronecker delta.

Let \(I\) be finite and \(v\ge 0\). Write \(p_v(c,z)\) for the \(N(c,v)\) density when \(v\gt 0\) and set \(p_0(c,z)=0\); define \(G_v(c,z)=\prod _{i\in I}p_v(c_i,z_i)\) and interpret division by zero as zero. Assume \(c,z,a\in \mathbb R^I\), \(a_i\ge 0\), \(p_v(c_i,z_i)\le a_i\), and \(|((z_i-c_i)^2/v^2-1/v)p_v(c_i,z_i)|\le a_i\) for every \(i\in I\). For every \(i\in I\), \(((z_i-c_i)^2/v^2)G_v(c,z)\le (1+1/v)\prod _j a_j\).

Let \(I\) be finite and \(v\ge 0\). Write \(p_v(c,z)\) for the \(N(c,v)\) density when \(v\gt 0\) and set \(p_0(c,z)=0\); define \(G_v(c,z)=\prod _{i\in I}p_v(c_i,z_i)\) and interpret division by zero as zero. Assume \(c,z,a\in \mathbb R^I\), \(a_i\ge 0\), \(p_v(c_i,z_i)\le a_i\), and \(|((z_i-c_i)^2/v^2-1/v)p_v(c_i,z_i)|\le a_i\) for every \(i\in I\). If \(v\gt 0\), then for every \(i,j\in I\), \(|((z_i-c_i)(z_j-c_j)/v^2-\delta _{ij}/v)G_v(c,z)|\le (1+2/v)\prod _k a_k\), where \(\delta _{ij}\) is the Kronecker delta.

Let \(I\) be finite and \(v\ge 0\). Put \(G_v(c,z)=\prod _i p_v(c_i,z_i)\), where \(p_v\) is the Gaussian density of variance \(v\) for \(v\gt 0\) and \(p_0=0\), and put \(S_v(c,z)[h]=\sum _i((z_i-c_i)/v)h_i\), with division by zero equal to zero. Define \(A_v(c,z)[h]=G_v(c,z)S_v(c,z)[h]\) and \(T_v(c,z)[h,k]=G_v(c,z)(S_v(c,z)[h]S_v(c,z)[k]-\sum _i h_i k_i/v)\). For fixed \(v\ge 0\), the map \((c,z)\mapsto T_v(c,z)\) is continuous in bilinear operator norm.

Let \(I\) be finite and \(a_{ij}\in \mathbb R\). For the bilinear form \(T(h,k)=\sum _{i,j}a_{ij}h_jk_i\) on maximum-norm vectors in \(\mathbb R^I\), \(\| T\| \le \sum _{i,j}|a_{ij}|\).

Let \(I\) be finite and \(v\ge 0\). Put \(G_v(c,z)=\prod _i p_v(c_i,z_i)\), where \(p_v\) is the Gaussian density of variance \(v\) for \(v\gt 0\) and \(p_0=0\), and put \(S_v(c,z)[h]=\sum _i((z_i-c_i)/v)h_i\), with division by zero equal to zero. Define \(A_v(c,z)[h]=G_v(c,z)S_v(c,z)[h]\) and \(T_v(c,z)[h,k]=G_v(c,z)(S_v(c,z)[h]S_v(c,z)[k]-\sum _i h_i k_i/v)\). If \(v\gt 0\) and \(M\ge 0\), there is a nonnegative Lebesgue-integrable \(b:\mathbb R^I\to \mathbb R\) such that \(\| T_v(m,z)\| \le b(z)\) for every \(z\) and every \(m\) satisfying \(|m_i|\le M\) for all \(i\); the bilinear operator norm uses maximum norms.

Let \(I\) be finite and \(v\ge 0\). Put \(G_v(c,z)=\prod _i p_v(c_i,z_i)\), where \(p_v\) is the Gaussian density of variance \(v\) for \(v\gt 0\) and \(p_0=0\), and put \(S_v(c,z)[h]=\sum _i((z_i-c_i)/v)h_i\), with division by zero equal to zero. Define \(A_v(c,z)[h]=G_v(c,z)S_v(c,z)[h]\) and \(T_v(c,z)[h,k]=G_v(c,z)(S_v(c,z)[h]S_v(c,z)[k]-\sum _i h_i k_i/v)\). Let \(f:\mathbb R^I\to \mathbb R\) be measurable with \(|f|\le 1\). If \(v\gt 0\) and \(c\in \mathbb R^I\), then \(z\mapsto f(z)T_v(c,z)\) is Lebesgue-integrable as a bilinear-form-valued function.

Let \(I\) be finite and \(v\ge 0\). Put \(G_v(c,z)=\prod _i p_v(c_i,z_i)\), where \(p_v\) is the Gaussian density of variance \(v\) for \(v\gt 0\) and \(p_0=0\), and put \(S_v(c,z)[h]=\sum _i((z_i-c_i)/v)h_i\), with division by zero equal to zero. Define \(A_v(c,z)[h]=G_v(c,z)S_v(c,z)[h]\) and \(T_v(c,z)[h,k]=G_v(c,z)(S_v(c,z)[h]S_v(c,z)[k]-\sum _i h_i k_i/v)\). Let \(f:\mathbb R^I\to \mathbb R\) be measurable with \(|f|\le 1\). If \(v\gt 0\), then \(m\mapsto \int f(z)A_v(m,z)\, dz\) is differentiable at every \(c\), with derivative \(\int f(z)T_v(c,z)\, dz\).

Let \(I\) be finite and \(v\ge 0\). Put \(G_v(c,z)=\prod _i p_v(c_i,z_i)\), where \(p_v\) is the Gaussian density of variance \(v\) for \(v\gt 0\) and \(p_0=0\), and put \(S_v(c,z)[h]=\sum _i((z_i-c_i)/v)h_i\), with division by zero equal to zero. Define \(A_v(c,z)[h]=G_v(c,z)S_v(c,z)[h]\) and \(T_v(c,z)[h,k]=G_v(c,z)(S_v(c,z)[h]S_v(c,z)[k]-\sum _i h_i k_i/v)\). Let \(f:\mathbb R^I\to \mathbb R\) be measurable with \(|f|\le 1\). If \(v\gt 0\), \(c,h,k\in \mathbb R^I\), and \(Z\sim \bigotimes _iN(c_i,v)\), then \((\int f(z)T_v(c,z)\, dz)[h,k]=\sum _{i,j}\mathbb E[((Z_i-c_i)(Z_j-c_j)/v^2-\delta _{ij}/v)f(Z)]h_jk_i\), where \(\delta _{ij}\) is the Kronecker delta.

Let \(I\) be finite and \(v\ge 0\). Put \(G_v(c,z)=\prod _i p_v(c_i,z_i)\), where \(p_v\) is the Gaussian density of variance \(v\) for \(v\gt 0\) and \(p_0=0\), and put \(S_v(c,z)[h]=\sum _i((z_i-c_i)/v)h_i\), with division by zero equal to zero. Define \(A_v(c,z)[h]=G_v(c,z)S_v(c,z)[h]\) and \(T_v(c,z)[h,k]=G_v(c,z)(S_v(c,z)[h]S_v(c,z)[k]-\sum _i h_i k_i/v)\). Let \(f:\mathbb R^I\to \mathbb R\) be measurable with \(|f|\le 1\). If \(v\gt 0\), then \(\| \int f(z)T_v(c,z)\, dz\| \le 2|I|^2/v\) for every \(c\in \mathbb R^I\), using maximum norms for the bilinear operator norm.

Let \(I\) be finite and \(v\ge 0\). Write \(p_v(c,z)\) for the \(N(c,v)\) density when \(v\gt 0\) and set \(p_0(c,z)=0\); define \(G_v(c,z)=\prod _{i\in I}p_v(c_i,z_i)\) and interpret division by zero as zero. Let \(A_v(c,z)\) be the linear functional \(h\mapsto \sum _i\bigl(\prod _{j\ne i}p_v(c_j,z_j)\bigr)((z_i-c_i)/v)p_v(c_i,z_i)h_i\). Let \(f:\mathbb R^I\to \mathbb R\) be measurable with \(|f|\le 1\). If \(v\gt 0\), the map \(c\mapsto \int f(z)A_v(c,z)\, dz\) is globally \(2|I|^2/v\)-Lipschitz, with maximum norm on mean vectors and operator norm on linear functionals.

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). The function \(Q\) is strictly increasing on \((0,1)\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). For every \(a\in (0,1)\), \(Q\) is continuous at \(a\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). For every \(a\in (0,1)\), \(Q'(a)=1/I(a)\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). For every \(a\in (0,1)\), \(I'(a)=-Q(a)\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). For every \(a\in (0,1)\), the derivative of \(p\mapsto -Q(p)\) at \(a\) is \(-1/I(a)\).

The standard Gaussian density satisfies \(\varphi (-x)=\varphi (x)\) for every \(x\in \mathbb R\).

For the standard Gaussian distribution function \(\Phi \), the function \(t\mapsto \Phi (-t)+\Phi (t)\) has derivative zero at every \(x\in \mathbb R\).

The standard Gaussian distribution function satisfies \(\Phi (-x)=1-\Phi (x)\) for every real \(x\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). For every \(a\in (0,1)\), \(Q(1-a)=-Q(a)\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). For every \(a\in (0,1)\), \(I(1-a)=I(a)\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). If \(a,b\in (0,1)\), \(a\le b\), and \(a+b\le 1\), then \(Q(b)^2\le Q(a)^2\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). If \(a,b\in (0,1)\), \(a\le b\), and \(a+b\le 1\), then \(I(a)\le I(b)\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). The function \(p\mapsto Q(p)^2\) is convex on \((0,1)\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). For \(a\in (0,1)\), the derivative of \(p\mapsto I(p)Q(p)\) at \(a\) is \(1-Q(a)^2\).

Let \(\Phi \) and \(\varphi \) be the standard Gaussian distribution function and density, and put \(Q=\Phi ^{-1}\) and \(I(p)=\varphi (Q(p))\) for \(0\lt p\lt 1\). For \(a\in (0,1)\), the derivative of \(p\mapsto I(p)^2\) at \(a\) is \(-2I(a)Q(a)\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. If \(a\in (0,1)\), then \(\overline I(a)=\varphi (\Phi ^{-1}(a))\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. Then \(\overline I(0)=0\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. Then \(\overline I(1)=0\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. For every real \(a\), \(\overline I(a)\ge 0\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. For every real \(a\), \(\overline I(1-a)=\overline I(a)\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. Then \(\lim _{p\to 0^+}\overline I(p)=0\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. The function \(\overline I\) is continuous at zero.

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. The function \(\overline I\) is continuous at one.

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. For \(a\in (0,1)\), \(\overline I'(a)=-\Phi ^{-1}(a)\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. The function \(\overline I\) is continuous on \([0,1]\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. The function \(\overline I\) is continuous on \(\mathbb R\).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. If \(\mu \) is a finite measure on a measurable space \(\Omega \) and \(f:\Omega \to \mathbb R\) is measurable with \(f(x)\in [0,1]\) for every \(x\), then \(\overline I\circ f\) is \(\mu \)-integrable.

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. For every \(\varepsilon \gt 0\) there exists \(C\gt 0\) such that \(\overline I(p)\le \varepsilon +C|p-b|\) for all \(p\in [0,1]\) and \(b\in \{ 0,1\} \).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For a measurable \(A\subseteq \mathbb R^n\), let \(u_\theta (x)=\mathbb E_Y1_A(\cos \theta \, x+\sin \theta \, Y)\) with \(Y\sim \gamma _n\), and put \(R_A(\theta )=\Pr (1_A(X)\ne 1_A(\cos \theta \, X+\sin \theta \, Y))\) for independent \(X,Y\sim \gamma _n\). For every \(\varepsilon \gt 0\) there exists \(C\gt 0\) such that \(\int \overline I(u_\theta (x))\, d\gamma _n(x)\le \varepsilon +CR_A(\theta )\) for every real \(\theta \).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For a measurable \(A\subseteq \mathbb R^n\), let \(u_\theta (x)=\mathbb E_Y1_A(\cos \theta \, x+\sin \theta \, Y)\) with \(Y\sim \gamma _n\), and put \(R_A(\theta )=\Pr (1_A(X)\ne 1_A(\cos \theta \, X+\sin \theta \, Y))\) for independent \(X,Y\sim \gamma _n\). Then \(\int \overline I(u_\theta (x))\, d\gamma _n(x)\to 0\) as \(\theta \to 0\) through real values.

Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). For any real \(t\), the map \((g,h)\mapsto (\cos t\, g+\sin t\, h,-\sin t\, g+\cos t\, h)\) preserves \(\gamma _n\otimes \gamma _n\).

Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). If \(A\subseteq \mathbb R^n\) is measurable, then \(s_A(s+t)\le s_A(s)+s_A(t)\) for all real \(s,t\).

Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). For any measurable \(A\subseteq \mathbb R^n\), integer \(\ell \ge 0\), and real angles \(\delta _1,\ldots ,\delta _\ell \), \(s_A(\sum _{j=1}^\ell \delta _j)\le \sum _{j=1}^\ell s_A(\delta _j)\).

Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). For any measurable \(A\subseteq \mathbb R^n\), real \(\delta \), and positive integer \(\ell \), \(s_A(\delta )\le \ell \, s_A(\delta /\ell )\).

Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). If \(A\subseteq \mathbb R^n\) is measurable and \(\alpha =\gamma _n(A)\), then \(s_A(\pi /2)=2\alpha (1-\alpha )\).

Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). If \(A\subseteq \mathbb R^n\) is measurable with \(\gamma _n(A)=1/2\), then \(s_A(\pi /(2\ell ))\ge 1/(2\ell )\) for every positive integer \(\ell \).

Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). For any measurable \(A\subseteq \mathbb R^n\), \(s_A(t)\to 0\) as the real angle \(t\to 0\).

Let \(f:\mathbb R^n\to \mathbb R\) have derivative \(L\) at \(x\in \mathbb R^n\), and let \(y\in \mathbb R^n\). Then \(\theta \mapsto f(\cos \theta \, x+\sin \theta \, y)\) has derivative \(L(y)\) at zero.

Let \(f:\mathbb R^n\to \mathbb R\) have derivative \(L\) at \(x\), and let \(y\in \mathbb R^n\). Then \(|f(\cos \theta \, x+\sin \theta \, y)-f(x)|/\theta \to |L(y)|\) as \(\theta \to 0^+\).

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\) and \(X,Y\) independent with that law. For a linear functional \(L:\mathbb R^n\to \mathbb R\), put \(|L|_2=\sqrt{\sum _i L(e_i)^2}\), where \(e_i\) is the canonical basis. Then \(\mathbb E|L(X)|=|L|_2\, 2\varphi (0)\), where \(\varphi \) is the one-dimensional standard Gaussian density.

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\) and \(X,Y\) independent with that law. Let \(f:\mathbb R^n\to \mathbb R\) be differentiable everywhere with derivative \(L(x)\), and put \(|L(x)|_2=\sqrt{\sum _i(L(x)e_i)^2}\), where \(e_i\) is the canonical basis. Let \(C\in [0,\infty ]\). If \(\mathbb E(|f(\cos \theta \, X+\sin \theta \, Y)-f(X)|/\theta )\le C\) for every \(\theta \gt 0\), then \(\mathbb E|L(X)Y|\le C\), with both expectations interpreted as nonnegative extended integrals.

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\) and \(X,Y\) independent with that law. Every continuous real linear functional \(L:\mathbb R^n\to \mathbb R\) is \(\gamma _n\)-integrable.

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\) and \(X,Y\) independent with that law. For a linear functional \(L:\mathbb R^n\to \mathbb R\), put \(|L|_2=\sqrt{\sum _i L(e_i)^2}\), where \(e_i\) is the canonical basis. The nonnegative extended expectation satisfies \(\mathbb E|L(X)|=|L|_2\, 2\varphi (0)\), with the finite expression on the right embedded into \([0,\infty ]\) and \(\varphi \) the standard Gaussian density.

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\) and \(X,Y\) independent with that law. If \(L:\mathbb R^n\to \mathcal L(\mathbb R^n,\mathbb R)\) is continuous, put \(|L(x)|_2=\sqrt{\sum _i(L(x)e_i)^2}\). Then \(\mathbb E|L(X)Y|=(\mathbb E|L(X)|_2)\, 2\varphi (0)\) as nonnegative extended integrals, where \(\varphi \) is the standard Gaussian density.

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\) and \(X,Y\) independent with that law. Let \(f:\mathbb R^n\to \mathbb R\) be differentiable everywhere with derivative \(L(x)\), and put \(|L(x)|_2=\sqrt{\sum _i(L(x)e_i)^2}\), where \(e_i\) is the canonical basis. Assume \(L\) is continuous and \(C\in [0,\infty ]\). If \(\mathbb E(\sqrt{\pi /2}\, |f(\cos \theta \, X+\sin \theta \, Y)-f(X)|/\theta )\le C\) for every \(\theta \gt 0\), then \(\mathbb E|L(X)|_2\le C\), with all expectations nonnegative extended integrals.

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\) and \(X,Y\) independent with that law. Let \(f:\mathbb R^n\to \mathbb R\) be differentiable everywhere with derivative \(L(x)\), and put \(|L(x)|_2=\sqrt{\sum _i(L(x)e_i)^2}\), where \(e_i\) is the canonical basis. Assume \(L\) is continuous, \(0\le f\le 1\), and \(C\in [0,\infty ]\). If \(\sqrt{\pi /2}\, \mathbb E|f(\cos \theta \, X+\sin \theta \, Y)-f(X)|/\theta \le C\) for every \(\theta \gt 0\), where the real expectation is embedded into \([0,\infty ]\), then the nonnegative extended expectation satisfies \(\mathbb E|L(X)|_2\le C\).

Let \(f\colon \mathbb R\to \mathbb R\) be continuous at zero with \(f(0)=0\) and \(f(s+t)\le f(s)+f(t)\) for all real \(s,t\). Then for every \(\delta \gt 0\), \([f(\delta )/\delta ]_+\le \liminf _{t\to 0^+}[f(t)/t]_+\) in \([0,\infty ]\), where \([u]_+=\max (u,0)\).

Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). Write \(P_\gamma (A)=\liminf _{t\to 0^+}\bigl[\sqrt{\pi /2}\, s_A(t)/t\bigr]_+\) in \([0,\infty ]\), where \([x]_+=\max (x,0)\). If \(A\subseteq \mathbb R^n\) is measurable and \(\delta \gt 0\), then \([\sqrt{\pi /2}\, s_A(\delta )/\delta ]_+\le P_\gamma (A)\).

Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). Write \(P_\gamma (A)=\liminf _{t\to 0^+}\bigl[\sqrt{\pi /2}\, s_A(t)/t\bigr]_+\) in \([0,\infty ]\), where \([x]_+=\max (x,0)\). For every measurable \(A\subseteq \mathbb R^n\), \([\sqrt{\pi /2}\, s_A(t)/t]_+\to P_\gamma (A)\) as \(t\to 0^+\), with convergence in \([0,\infty ]\).

Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). For any sets \(A,B\subseteq \mathbb R^n\) with \(\mathbf1_A=\mathbf1_B\) almost everywhere under \(\gamma _n\), \(s_A(t)=s_B(t)\) for every real \(t\).

Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). Write \(P_\gamma (A)=\liminf _{t\to 0^+}\bigl[\sqrt{\pi /2}\, s_A(t)/t\bigr]_+\) in \([0,\infty ]\), where \([x]_+=\max (x,0)\). If measurable sets \(A,B\subseteq \mathbb R^n\) agree almost everywhere under \(\gamma _n\), then \(P_\gamma (A)=P_\gamma (B)\).

Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). Write \(P_\gamma (A)=\liminf _{t\to 0^+}\bigl[\sqrt{\pi /2}\, s_A(t)/t\bigr]_+\) in \([0,\infty ]\), where \([x]_+=\max (x,0)\). For a vector \(v\in \mathbb R^n\) with \(\sum _i v_i^2=1\) and real \(t\), the halfspace \(H=\{ z:\sum _i v_i z_i\le t\} \) satisfies \(P_\gamma (H)=\varphi (t)\), where \(\varphi (t)=(2\pi )^{-1/2}e^{-t^2/2}\).

Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). Write \(P_\gamma (A)=\liminf _{t\to 0^+}\bigl[\sqrt{\pi /2}\, s_A(t)/t\bigr]_+\) in \([0,\infty ]\), where \([x]_+=\max (x,0)\). For every measurable \(A\subseteq \mathbb R^n\) and \(\delta \gt 0\), \([\sqrt{\pi /2}\, s_A(\delta )/\delta ]_+\le P_\gamma (A)\).

Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). Write \(P_\gamma (A)=\liminf _{t\to 0^+}\bigl[\sqrt{\pi /2}\, s_A(t)/t\bigr]_+\) in \([0,\infty ]\), where \([x]_+=\max (x,0)\). If \(A\subseteq \mathbb R^n\) is measurable with \(\alpha =\gamma _n(A)\), then \(2\sqrt{2/\pi }\, \alpha (1-\alpha )\le P_\gamma (A)\).

Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). Write \(P_\gamma (A)=\liminf _{t\to 0^+}\bigl[\sqrt{\pi /2}\, s_A(t)/t\bigr]_+\) in \([0,\infty ]\), where \([x]_+=\max (x,0)\). Let \(A\subseteq \mathbb R^n\) be measurable and \(f\colon \mathbb R^n\to [0,1]\) be differentiable with continuous derivative \(D(x)\). If \(\mathbb E|f(G)-f(\cos \theta \, G+\sin \theta \, G\prime )|\le s_A(\theta )\) for every real \(\theta \), then \(\int \| \nabla f(x)\| _2\, d\gamma _n(x)\le P_\gamma (A)\) as a nonnegative extended integral.

Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). Write \(P_\gamma (A)=\liminf _{t\to 0^+}\bigl[\sqrt{\pi /2}\, s_A(t)/t\bigr]_+\) in \([0,\infty ]\), where \([x]_+=\max (x,0)\). Let \(A\subseteq \mathbb R^n\) be measurable, \(0\lt \alpha \lt 1\), and let \(U\) be the standard Gaussian isoperimetric profile extended by zero outside \((0,1)\). Set \(f_u(x)=\mathbb E[\mathbf1_A(\cos u\, x+\sin u\, G)]\). If \(U(\alpha )\le [\int U(f_u)\, d\gamma _n]_++P_\gamma (A)\) for every \(0\lt u\lt \pi /2\), then \(U(\alpha )\le P_\gamma (A)\).

Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). Write \(P_\gamma (A)=\liminf _{t\to 0^+}\bigl[\sqrt{\pi /2}\, s_A(t)/t\bigr]_+\) in \([0,\infty ]\), where \([x]_+=\max (x,0)\). Let \(A\subseteq \mathbb R^n\) be measurable with \(\gamma _n(A)=\alpha \in (0,1)\) and put \(f_u(x)=\mathbb E[\mathbf1_A(\cos u\, x+\sin u\, G)]\). Suppose that for each \(0\lt u\lt \pi /2\), \(f_u\) has a derivative family \(D_u\) and there exist constants \(C_u\ge 0\) and \(M_u\in \mathbb R\) such that \(D_u\) is \(C_u\)-Lipschitz and \(\| D_u(x)\| \le M_u\) for all \(x\). Then \(\varphi (\Phi ^{-1}(\alpha ))\le P_\gamma (A)\), where \(\varphi \) and \(\Phi \) are the standard Gaussian density and distribution function.

Let \(n\ge 0\), let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\), and let \(G,G\prime \) be independent vectors with law \(\gamma _n\). Write \(s_A(t)\) for the real value of the product-measure mass \((\gamma _n\otimes \gamma _n)\{ (g,h):\mathbf1_A(g)\ne \mathbf1_A(\cos t\, g+\sin t\, h)\} \). Write \(P_\gamma (A)=\liminf _{t\to 0^+}\bigl[\sqrt{\pi /2}\, s_A(t)/t\bigr]_+\) in \([0,\infty ]\), where \([x]_+=\max (x,0)\). For every measurable \(A\subseteq \mathbb R^n\) of Gaussian volume \(\alpha \in (0,1)\), \(\varphi (\Phi ^{-1}(\alpha ))\le P_\gamma (A)\), where \(\varphi \) and \(\Phi \) are the standard Gaussian density and distribution function.

Let \(f:\{ 0,1\} ^n\to \{ 0,1\} \) and \(0\lt \rho \le 1\). Then under the uniform cube law \(\mathbb E[fT_\rho f]\le (\mathbb Ef)^{2/(1+\rho )}\).

Let \(f:\{ 0,1\} ^n\to \{ 0,1\} \) have uniform mean \(\alpha \gt 0\), and let \(a_i\in \mathbb R\) satisfy \(\sum _ia_i^2=1\). Then \(\big(\mathbb E[f(x)\sum _i a_i(-1)^{x_i}]\big)^2\le 2\alpha ^2\log (1/\alpha )\).

Let \(f:\{ 0,1\} ^n\to \mathbb R\) be a function on the uniform Boolean cube, \(i\in [n]\), and \(\rho \ge 0\). Then \(I_i^{(\rho )}(f)=\| T_{\sqrt\rho }D_if\| _2^2\), where \(D_if(x)=(f(x^{i\leftarrow 0})-f(x^{i\leftarrow 1}))/2\).

Let \(f:\{ 0,1\} ^n\to \{ -1,1\} \) be a function on the uniform Boolean cube, \(i\in [n]\), and \(p\gt 0\). Then \(\mathbb E|D_if|^p=I_i(f)\), where \(D_if(x)=(f(x^{i\leftarrow 0})-f(x^{i\leftarrow 1}))/2\). For a real cube function \(h\), define \(I_i(h)=\tfrac 14\mathbb E_x(h(x)-h(x^{(i)}))^2\), where \(x^{(i)}\) flips coordinate \(i\) and \(\mathbb E_x\) is uniform expectation.

Let \(f:\{ 0,1\} ^n\to \{ -1,1\} \) be a function on the uniform Boolean cube, \(i\in [n]\), and \(0\le \rho \le 1\). Then \(I_i^{(\rho )}(f)\le I_i(f)^{2/(1+\rho )}\). For a real cube function \(h\), define \(I_i(h)=\tfrac 14\mathbb E_x(h(x)-h(x^{(i)}))^2\), where \(x^{(i)}\) flips coordinate \(i\) and \(\mathbb E_x\) is uniform expectation.

Let \(f:\{ 0,1\} ^n\to \{ -1,1\} \) be a function on the uniform Boolean cube and \(i\in [n]\). Then \(I_i^{(1/3)}(f)\le I_i(f)^{3/2}\). For a real cube function \(h\), define \(I_i(h)=\tfrac 14\mathbb E_x(h(x)-h(x^{(i)}))^2\), where \(x^{(i)}\) flips coordinate \(i\) and \(\mathbb E_x\) is uniform expectation.

Let \(f:\{ 0,1\} ^n\to \mathbb R\) be a function on the uniform Boolean cube, \(k\in \mathbb N\), and \(0\le \rho \le 1\). Then \(\rho ^k\sum _{|S|\le k}\widehat f(S)^2\le \mathbb E[fT_\rho f]\).

For every \(0\lt \alpha \le 1\) and \(\rho \ge 0\), \(\alpha ^{2/(1+\rho )}\le \alpha ^2\exp (2\rho \log (1/\alpha ))\).

Let \(0\lt \alpha \le 1\), \(W\in \mathbb R\), and a positive integer \(k\) satisfy \(k\le 2\log (1/\alpha )\). If \(\rho ^kW\le \alpha ^{2/(1+\rho )}\) for every \(0\lt \rho \le 1\), then \(W\le ((2e/k)\log (1/\alpha ))^k\alpha ^2\).

Let \(f:\{ 0,1\} ^n\to \{ 0,1\} \) have uniform mean \(\alpha \gt 0\), and let \(k\) be a positive integer with \(k\le 2\log (1/\alpha )\). Then \(\sum _{|S|\le k}\widehat f(S)^2\le ((2e/k)\log (1/\alpha ))^k\alpha ^2\).

Let \(f:\{ 0,1\} ^n\to \{ 0,1\} \) have uniform mean \(\alpha \gt 0\). Then \(\sum _{|S|=1}\widehat f(S)^2\le 2\alpha ^2\log (1/\alpha )\).

Let \(f:\{ 0,1\} ^n\to \{ -1,1\} \) be a function on the uniform Boolean cube with variance \(V\gt 0\), and put \(a=I(f)/V\). Then \((9/a^2)9^{-a}\le \max _{i\in [n]}|I_i(f)|\). For a real cube function \(h\), define \(I_i(h)=\tfrac 14\mathbb E_x(h(x)-h(x^{(i)}))^2\), where \(x^{(i)}\) flips coordinate \(i\) and \(\mathbb E_x\) is uniform expectation, and \(I(h)=\sum _{i=0}^{n-1}I_i(h)\).

Let \(f:\{ 0,1\} ^n\to \{ -1,1\} \) be a function on the uniform Boolean cube in dimension \(n\ge 2\). Some coordinate \(i\) satisfies \(\operatorname {Var}(f)\log n/(30n)\le I_i(f)\). For a real cube function \(h\), define \(I_i(h)=\tfrac 14\mathbb E_x(h(x)-h(x^{(i)}))^2\), where \(x^{(i)}\) flips coordinate \(i\) and \(\mathbb E_x\) is uniform expectation.

There exists an absolute \(c\gt 0\) such that for every natural dimension \(n\) and every \(f:\{ 0,1\} ^n\to \{ -1,1\} \) under the uniform law, \(c\operatorname {Var}(f)(\log n/n)\le \sup _{i\in [n]}|I_i(f)|\), with the empty supremum and division by zero interpreted as zero. For a real cube function \(h\), define \(I_i(h)=\tfrac 14\mathbb E_x(h(x)-h(x^{(i)}))^2\), where \(x^{(i)}\) flips coordinate \(i\) and \(\mathbb E_x\) is uniform expectation.

There exists an absolute \(C\gt 0\) such that for every \(f:\{ 0,1\} ^n\to \{ -1,1\} \) under the uniform law with \(\operatorname {Var}(f)\ge 1/2\), there is a nonempty \(S\subseteq [n]\) with \(|S|\le CI(f)\) and \(\widehat f(S)^2\ge \exp (-CI(f)^2)\). For a real cube function \(h\), define \(I_i(h)=\tfrac 14\mathbb E_x(h(x)-h(x^{(i)}))^2\), where \(x^{(i)}\) flips coordinate \(i\) and \(\mathbb E_x\) is uniform expectation, and \(I(h)=\sum _{i=0}^{n-1}I_i(h)\).

For any finite set \(s\) and \(k\in \mathbb N\), the number of subsets \(t\subseteq s\) with \(|t|\le k\) is at most \((|s|+1)^k\).

Let \(f:\{ 0,1\} ^n\to \{ -1,1\} \) be a function on the uniform Boolean cube, \(J\subseteq [n]\), \(k\in \mathbb N\), \(0\le \rho \le 1\), and \(\tau \ge 0\). If \(I_i(f)\le \tau \) for every \(i\notin J\), then \(\rho ^k\sum _{|S|\le k,\ S\nsubseteq J}\widehat f(S)^2\le \tau ^{(1-\rho )/(1+\rho )}I(f)\), with \(0^0=1\). For a real cube function \(h\), define \(I_i(h)=\tfrac 14\mathbb E_x(h(x)-h(x^{(i)}))^2\), where \(x^{(i)}\) flips coordinate \(i\) and \(\mathbb E_x\) is uniform expectation, and \(I(h)=\sum _{i=0}^{n-1}I_i(h)\).

Let \(d\) be a DNF on \(n\) variables of width at most a positive integer \(w\). Suppose every term has no repeated literals and any two literals in a term sharing a variable are equal. For every \(t\in \mathbb N\), its sign encoding \(f(x)=(-1)^{d(x)}\) satisfies \(\sum _{|S|\gt 120w(t+1)}\widehat f(S)^2\le 8\, 2^{-t}\) under the uniform cube law.

Let \(d\) be a DNF on \(n\) variables of width at most a positive integer \(w\), and let \(t\in \mathbb N\). Its sign encoding \(f(x)=(-1)^{d(x)}\) satisfies \(\sum _{|S|\gt 120w(t+1)}\widehat f(S)^2\le 8\, 2^{-t}\) under the uniform cube law.

Let \(d\) be a DNF on \(n\) variables and let \(x\in \{ 0,1\} ^n\) satisfy \(d(x)=1\). The number of coordinates whose individual flip changes the sign encoding \((-1)^{d(x)}\) is at most the width of \(d\).

For every DNF \(d\) on \(n\) variables, the total influence under the uniform cube law of \(x\mapsto (-1)^{d(x)}\) is at most twice the width of \(d\). For a real cube function \(h\), define \(I_i(h)=\tfrac 14\mathbb E_x(h(x)-h(x^{(i)}))^2\), where \(x^{(i)}\) flips coordinate \(i\) and \(\mathbb E_x\) is uniform expectation, and \(I(h)=\sum _{i=0}^{n-1}I_i(h)\).

There exists an absolute \(C\gt 0\) such that for every positive integer \(w\) and \(0\lt \varepsilon \le 1/2\), there exists \(t\in \mathbb N\) satisfying \(8\, 2^{-t}\le \varepsilon \) and \(((2w)^3/\varepsilon ^2)9^{120w(t+1)}\le (1/\varepsilon )^{Cw}\).

There exists an absolute \(C\gt 0\) such that for every DNF \(d\) on \(n\) variables of width at most \(w\in \mathbb N\) and every \(0\lt \varepsilon \le 1\), there exist \(J\subseteq [n]\) and \(h:\{ 0,1\} ^n\to \{ -1,1\} \) depending only on \(J\), with \(|J|\le (1/\varepsilon )^{Cw}\) and \(\Pr _x((-1)^{d(x)}\ne h(x))\le \varepsilon \) under the uniform cube law.

Let \(f:\{ 0,1\} ^n\to \{ -1,1\} \) be a function on the uniform Boolean cube, \(0\lt \varepsilon \le 1\), and \(k\in \mathbb N\). Write \(I=I(f)\), put \(\tau =1\) if \(I=0\) and \(\tau =\varepsilon ^2/(I^2 9^k)\) otherwise, and set \(J=\{ i:I_i(f)\ge \tau \} \). Then \(|J|\le I^3 9^k/\varepsilon ^2\) and \(\sum _{|S|\le k,\ S\nsubseteq J}\widehat f(S)^2\le \varepsilon \); if \(\sum _{|S|\gt k}\widehat f(S)^2\le \varepsilon \), then \(\sum _{\neg (S\subseteq J\ \mathrm{and}\ |S|\le k)}\widehat f(S)^2\le 2\varepsilon \) and there exists \(h:\{ 0,1\} ^n\to \{ -1,1\} \) depending only on \(J\) with \(\Pr (f\ne h)\le \varepsilon \). For a real cube function \(h\), define \(I_i(h)=\tfrac 14\mathbb E_x(h(x)-h(x^{(i)}))^2\), where \(x^{(i)}\) flips coordinate \(i\) and \(\mathbb E_x\) is uniform expectation, and \(I(h)=\sum _{i=0}^{n-1}I_i(h)\).

There exists an absolute \(C\gt 0\) such that for every \(f:\{ 0,1\} ^n\to \{ -1,1\} \) under the uniform law and \(0\lt \varepsilon \le 1\), there exist \(J\subseteq [n]\) and \(h:\{ 0,1\} ^n\to \{ -1,1\} \) depending only on \(J\) with \(|J|\le \exp (CI(f)/\varepsilon )\), \(\sum _{\neg (S\subseteq J\ \mathrm{and}\ |S|\le I(f)/\varepsilon )}\widehat f(S)^2\le 2\varepsilon \), and \(\Pr (f\ne h)\le \varepsilon \). For a real cube function \(h\), define \(I_i(h)=\tfrac 14\mathbb E_x(h(x)-h(x^{(i)}))^2\), where \(x^{(i)}\) flips coordinate \(i\) and \(\mathbb E_x\) is uniform expectation, and \(I(h)=\sum _{i=0}^{n-1}I_i(h)\).

Let \(f:\{ 0,1\} ^n\to \{ -1,1\} \) be a function on the uniform Boolean cube with \(I=I(f)\gt 0\). For \(\varepsilon ,\eta \gt 0\) and \(k\in \mathbb N\) satisfying \(\sum _{|S|\gt k}\widehat f(S)^2\le \varepsilon \), put \(\rho =\eta /(2+\eta )\). There exist \(J\subseteq [n]\) and \(h:\{ 0,1\} ^n\to \{ -1,1\} \) depending only on \(J\), with \(|J|\le I^{2+\eta }/(\varepsilon \rho ^k)^{1+\eta }\) and \(\Pr (f\ne h)\le \varepsilon \). For a real cube function \(h\), define \(I_i(h)=\tfrac 14\mathbb E_x(h(x)-h(x^{(i)}))^2\), where \(x^{(i)}\) flips coordinate \(i\) and \(\mathbb E_x\) is uniform expectation, and \(I(h)=\sum _{i=0}^{n-1}I_i(h)\).

Let \(S,J\subseteq [n]\) and \(x\in \{ 0,1\} ^n\), and let \(x\star _J y\) equal \(x\) on \(J\) and \(y\) elsewhere. Then for uniform \(y\), \(\mathbb E_y\chi _S(x\star _J y)=\chi _S(x)\) if \(S\subseteq J\), and zero otherwise.

Let \(f:\{ 0,1\} ^n\to \mathbb R\) be a function on the uniform Boolean cube, \(J\subseteq [n]\), and \(x\in \{ 0,1\} ^n\). If \(x\star _Jy\) keeps \(x\) on \(J\) and \(y\) elsewhere, then \(\mathbb E_y f(x\star _Jy)=\sum _{S\subseteq J}\widehat f(S)\chi _S(x)\).

Let \(f:\{ 0,1\} ^n\to \{ -1,1\} \) be a function on the uniform Boolean cube and \(J\subseteq [n]\). For uniform independent \(z\) and \(x\), and \(x\star _Jz\) keeping \(x\) on \(J\) and \(z\) elsewhere, \(\mathbb E_z\Pr _x(f(x)\ne f(x\star _Jz))=\tfrac 12\sum _{S\nsubseteq J}\widehat f(S)^2\).

Let \(f:\{ 0,1\} ^n\to \{ -1,1\} \) be a function on the uniform Boolean cube and \(J\subseteq [n]\). There exists \(h:\{ 0,1\} ^n\to \{ -1,1\} \) depending only on \(J\) such that \(\Pr (f\ne h)\le \tfrac 12\sum _{S\nsubseteq J}\widehat f(S)^2\).

Let \(f:\{ 0,1\} ^n\to \{ -1,1\} \) be a function on the uniform Boolean cube. There exists a constant \(h:\{ 0,1\} ^n\to \{ -1,1\} \) such that \(\Pr (f\ne h)\le \operatorname {Var}(f)/2\).

There exists an absolute \(B\ge 1\) such that for every linear threshold function \(f:\{ 0,1\} ^n\to \{ -1,1\} \) and \(0\lt \varepsilon \le 1/2\), there is \(k\in \mathbb N\) with \(k\le B/\varepsilon ^2\) and \(\sum _{|S|\gt k}\widehat f(S)^2\le \varepsilon \) under the uniform cube law.

Let \(B\ge 1\), \(0\lt \varepsilon ,\eta \le 1/2\), and \(k\in \mathbb N\) with \(k\le B/\varepsilon ^2\). Then \(1/(\varepsilon (\eta /(2+\eta ))^k)^{1+\eta }\le (1/\eta )^{200B/\varepsilon ^2}\).

There exists an absolute \(C\gt 0\) such that for every linear threshold function \(f:\{ 0,1\} ^n\to \{ -1,1\} \) under the uniform law and \(0\lt \varepsilon ,\eta \le 1/2\), there exist \(J\subseteq [n]\) and \(h:\{ 0,1\} ^n\to \{ -1,1\} \) depending only on \(J\) with \(|J|\le I(f)^{2+\eta }(1/\eta )^{C/\varepsilon ^2}\) and \(\Pr (f\ne h)\le \varepsilon \). For a real cube function \(h\), define \(I_i(h)=\tfrac 14\mathbb E_x(h(x)-h(x^{(i)}))^2\), where \(x^{(i)}\) flips coordinate \(i\) and \(\mathbb E_x\) is uniform expectation, and \(I(h)=\sum _{i=0}^{n-1}I_i(h)\).

Let \(f:\{ 0,1\} ^n\to \mathbb R\) be a function on the uniform Boolean cube of Fourier degree at most \(k\in \mathbb N\). Then \(e^{-2k}\mathbb Ef^2\le (\mathbb E|f|)^2\).

Let \(n\in \mathbb N\) and \(-1\le \rho \le 1\), and put \(W(x,y)=2^{-n}\prod _{i=0}^{n-1}[(1+\rho (-1)^{x_i+y_i})/2]\) for cube points \(x,y\). Then \(W(x,y)\ge 0\), \(W(x,y)=W(y,x)\), \(\sum _yW(x,y)=2^{-n}\) for every \(x\), and \(\sum _{x,y}W(x,y)=1\).

Let \(f:\{ 0,1\} ^n\to \mathbb R\) be a function on the uniform Boolean cube of Fourier degree at most \(k\in \mathbb N\) with variance \(V\gt 0\). Then \(9/(16\cdot 9^k)\le \Pr (|f-\mathbb Ef|\gt \sqrt V/2)\).

Let \(f:\{ 0,1\} ^n\to \mathbb R\) be a function on the uniform Boolean cube of Fourier degree at most \(k\in \mathbb N\), and let \(q\ge 2\). Then \(\| f\| _q\le (\sqrt{q-1})^k\| f\| _2\).

Let \(f:\{ 0,1\} ^n\to \mathbb R\) be a function on the uniform Boolean cube of Fourier degree at most \(k\in \mathbb N\), and let \(0\lt p\lt 2\lt q\). Then \(\| f\| _2\le \exp (kq(2-p)/(2p))\| f\| _p\).

Let \(f:\{ 0,1\} ^n\to \mathbb R\) be a function on the uniform Boolean cube of Fourier degree at most \(k\in \mathbb N\), and let \(1\le p\le 2\). Then \(\| f\| _2\le \exp ((2/p-1)k)\| f\| _p\).

Let \(k\gt 0\) be an integer and \(t\ge (\sqrt{2e})^k\). Then \(t\gt 0\) and \(t^{2/k}/e\ge 2\).

Let \(k\gt 0\) be an integer and \(t\gt 0\), and put \(q=t^{2/k}/e\). Then \(((\sqrt q)^k/t)^q=\exp (-kt^{2/k}/(2e))\).

Let \(f:\{ 0,1\} ^n\to \mathbb R\) be a function on the uniform Boolean cube of Fourier degree at most a positive integer \(k\), with \(\| f\| _2\gt 0\). For \(t\ge (\sqrt{2e})^k\), \(\Pr (|f|\ge t\| f\| _2)\le \exp (-kt^{2/k}/(2e))\).

Let \(f:\{ 0,1\} ^n\to \mathbb R\) be a function on the uniform Boolean cube of Fourier degree at most \(k\in \mathbb N\), with positive variance. Then \(e^{-2k}/4\le \Pr (\mathbb Ef\lt f)\).

For a set \(A\subseteq \{ 0,1\} ^n\), define \(\mathbf1_A(x)=1\) if \(x\in A\) and \(\mathbf1_A(x)=0\) otherwise.

For a set \(A\subseteq \{ 0,1\} ^n\), define its volume \(\mu (A)=\mathbb E_x\mathbf1_A(x)=|A|/2^n\) under the uniform cube law.

For every \(A\subseteq \{ 0,1\} ^n\) and \(x\in \{ 0,1\} ^n\), \(\mathbf1_A(x)\ge 0\).

For every \(A\subseteq \{ 0,1\} ^n\) and \(x\in \{ 0,1\} ^n\), \(\mathbf1_A(x)\le 1\).

For every \(A\subseteq \{ 0,1\} ^n\) and \(x\in \{ 0,1\} ^n\), \(|\mathbf1_A(x)|=\mathbf1_A(x)\).

For every \(A\subseteq \{ 0,1\} ^n\) and \(x\in \{ 0,1\} ^n\), \(\mathbf1_A(x)^2=\mathbf1_A(x)\).

For every \(A\subseteq \{ 0,1\} ^n\), \(x\in \{ 0,1\} ^n\), and \(r\gt 0\), \(\mathbf1_A(x)^r=\mathbf1_A(x)\).

For every \(A\subseteq \{ 0,1\} ^n\), \(x\in \{ 0,1\} ^n\), and \(r\gt 0\), \(|\mathbf1_A(x)|^r=\mathbf1_A(x)\).

For every \(A\subseteq \{ 0,1\} ^n\) and \(r\gt 0\), the uniform expectation satisfies \(\mathbb E|\mathbf1_A|^r=|A|/2^n\).

For every \(A\subseteq \{ 0,1\} ^n\) and \(r\gt 0\), its moment norm under the uniform cube law satisfies \(\| \mathbf1_A\| _r=(|A|/2^n)^{1/r}\).

For every \(A\subseteq \{ 0,1\} ^n\), its uniform volume satisfies \(\mu (A)=|A|/2^n\ge 0\).

For every \(A\subseteq \{ 0,1\} ^n\), its uniform volume satisfies \(\mu (A)=|A|/2^n\le 1\).

Let \(A,B\subseteq \{ 0,1\} ^n\) have uniform volumes \(\alpha ,\beta \), and let real \(p,u,\rho \) satisfy \(1\le p\le u\), \(u\ge 2\), \(0\le \rho \le 1\), and \(\rho \le \sqrt{(p-1)/(u-1)}\). Then \(\mathbb E[\mathbf1_AT_\rho \mathbf1_B]\le \alpha ^{(u-1)/u}\beta ^{1/p}\).

For every \(A\subseteq \{ 0,1\} ^n\) of uniform volume \(\alpha \) and \(0\lt \rho \le 1\), \(\mathbb E[\mathbf1_AT_\rho \mathbf1_A]\le \alpha ^{2/(1+\rho )}\).

Let \(k\gt 0\) be an integer and \(t\ge (\sqrt{2e})^k\). Then \(t\gt 0\) and \(t^{2/k}/e\ge 2\).

Let \(k\gt 0\) be an integer and \(t\gt 0\), and put \(q=t^{2/k}/e\). Then \(((\sqrt q)^k/t)^q=\exp (-kt^{2/k}/(2e))\).

For an integer \(k\gt 0\) and reals \(\lambda ,t\gt 0\), \((t(\sqrt\lambda )^k)^{2/k}=\lambda t^{2/k}\).

Let \(k\gt 0\) be an integer, \(\lambda \gt 0\), and \(t\ge (\sqrt{2e/\lambda })^k\). Then \(t\gt 0\) and \(\lambda t^{2/k}/e\ge 2\).

Let \(k\gt 0\) be an integer and \(\lambda ,t\gt 0\), and put \(q=\lambda t^{2/k}/e\). Then \(((\sqrt q\lambda ^{1/q-1/2})^k/t)^q=\lambda ^k\exp (-k\lambda t^{2/k}/(2e))\).

Let \(f:\{ 0,1\} ^n\to \mathbb {R}\), \(i\in [n]\), and \(S\subseteq [n]\). For the signed coordinate derivative \(D_i f(x)=\tfrac 12(-1)^{x_i}(f(x)-f(x^{\oplus i}))\), where \(x^{\oplus i}\) flips coordinate \(i\), one has \(\widehat{D_i f}(S)=0\) if \(i\in S\), and \(\widehat{D_i f}(S)=\widehat f(S\cup \{ i\} )\) otherwise.

Let \(f:\{ 0,1\} ^n\to \mathbb {R}\) have Fourier degree at most \(k\in \mathbb N\), and let \(c\in \mathbb {R}\). The function \(x\mapsto f(x)-c\) also has Fourier degree at most \(k\).

For an event \(A\subseteq \{ 0,1\} ^n\), its indicator \(\mathbf1_A\) is the real-valued function equal to \(1\) on \(A\) and \(0\) outside \(A\).

For an event \(A\subseteq \{ 0,1\} ^n\), define its uniform probability by \(\Pr [A]=\mathbb {E}[\mathbf1_A]=2^{-n}|A|\).

Let \(f:\{ 0,1\} ^n\to \mathbb {R}\), define \(\operatorname {Var}(f)=\mathbb {E}[(f-\mathbb {E}f)^2]\), with uniform expectation on the cube.

For every \(f:\{ 0,1\} ^n\to \mathbb {R}\), uniform variance satisfies \(\operatorname {Var}(f)=\sum _{\varnothing \ne S\subseteq [n]}\widehat f(S)^2\).

Let \(f:\{ 0,1\} ^n\to \mathbb {R}\) have Fourier degree at most \(k\in \mathbb N\). Its total influence satisfies \(I(f)\le k\operatorname {Var}(f)\), where \(I(f)=\sum _{S\subseteq [n]}|S|\widehat f(S)^2\).

Let \(f:\{ 0,1\} ^n\to \mathbb {R}\) have Fourier degree at most \(k\), variance at most \(1\), and coordinate influences \(\operatorname {Inf}_i(f)\le \varepsilon \) for every \(i\), where \(\varepsilon \ge 0\). Then \(\sum _i\operatorname {Inf}_i(f)^2\le k\varepsilon \).

Let \(f:\{ 0,1\} ^n\to \mathbb {R}\) have positive uniform variance, and let \(\rho \gt 0\). Writing \(I(f)=\sum _S|S|\widehat f(S)^2\), one has \(\operatorname {Var}(f)\rho ^{I(f)/\operatorname {Var}(f)}\le \sum _{S\subseteq [n]}|S|\rho ^{|S|}\widehat f(S)^2\).

Let \(p\in \mathbb {R}\) and \(f:\{ 0,1\} ^n\to \mathbb {R}\), define \(\| f\| _p=(\mathbb {E}|f|^p)^{1/p}\) using real powers and uniform expectation, with \(1/0=0\), \(a^0=1\) for \(a\ge 0\), and \(0^r=0\) for \(r\ne 0\).

For every \(f:\{ 0,1\} ^n\to \mathbb {R}\), the power mean \((\mathbb {E}|f|^2)^{1/2}\) equals the second norm \(\sqrt{\mathbb {E}[f^2]}\).

Let \(f:\{ 0,1\} ^n\to \mathbb {R}\) and \(q\gt 0\), the power mean \(\| f\| _q=(\mathbb {E}|f|^q)^{1/q}\) satisfies \(\| f\| _q^q=\mathbb {E}|f|^q\).

Let \(f:\{ 0,1\} ^n\to \mathbb {R}\) and \(q,c\gt 0\), uniform probability satisfies \(\Pr [|f|\ge c]\le (\| f\| _q/c)^q\), where \(\| f\| _q=(\mathbb {E}|f|^q)^{1/q}\).

Let \(f:\{ 0,1\} ^n\to \mathbb {R}\) and \(0\lt p\lt 2\lt q\). With uniform expectations, \(\mathbb {E}[f^2]\le (\mathbb {E}|f|^p)^{(q-2)/(q-p)}(\mathbb {E}|f|^q)^{(2-p)/(q-p)}\).

Let \(A,M,u,v,c\in \mathbb {R}\) satisfy \(A\ge 0\), \(M\gt 0\), \(u\gt 0\), and \(u+v=2\). If \(M^2\le A^u(e^cM)^v\), then \(M\le e^{cv/u}A\).

Let \(f:\{ 0,1\} ^n\to \mathbb {R}\), \(0\lt p\lt 2\lt q\), and \(c\in \mathbb {R}\), with \(\| f\| _r=(\mathbb {E}|f|^r)^{1/r}\). If \(\| f\| _q\le e^c\| f\| _2\), then \(\| f\| _2\le \exp \! \left(\frac{cq(2-p)}{p(q-2)}\right)\| f\| _p\).

Let \(g:\{ 0,1\} ^n\to \mathbb {R}\) have uniform mean zero. Then \((\mathbb {E}|g|)^2\le 4\mathbb {E}[g^2]\Pr [g\gt 0]\).

Let \(f:\{ 0,1\} ^n\to \{ 0,1\} \) have uniform mean \(\alpha \gt 0\), let \(g:\{ 0,1\} ^n\to \mathbb {R}\), and let \(t,b\in \mathbb {R}\). If \(\mathbb {E}[e^{tg}]\le e^b\), then \(t\mathbb {E}[fg]\le \alpha (b-\log \alpha )\).

Let \(f:\{ 0,1\} ^n\to \mathbb {R}\) have Fourier degree at most \(k\in \mathbb N\), and let \(\rho \ge 1\). For the Fourier multiplier \(\widehat{T_\rho f}(S)=\rho ^{|S|}\widehat f(S)\), one has \(\| T_\rho f\| _2\le \rho ^k\| f\| _2\) under the uniform measure. For \(d\in \mathbb N\), \(h:\{ 0,1\} ^d\to \mathbb {R}\), and \(\tau \in \mathbb {R}\), the spectral noise operator is \(T_\tau h(x)=\sum _{S\subseteq [d]}\tau ^{|S|}\widehat h(S)\chi _S(x)\), where \(\chi _S(x)=(-1)^{\sum _{i\in S}x_i}\) and \(\widehat h(S)=\mathbb {E}[h\chi _S]\) under the uniform measure.

For every \(f:\{ 0,1\} ^n\to \mathbb {R}\) and \(k\in \mathbb N\), \(\sum _{|S|\gt k}\widehat f(S)^2\le I(f)/(k+1)\), where \(I(f)=\sum _S|S|\widehat f(S)^2\) is total influence.

A weighted directed graph on a set \(V\) consists of a directed graph on \(V\), with loops permitted, together with a real weight \(w(x,y)\) for every ordered pair \(x,y\in V\); no positivity or support condition is imposed.

For \(n\in \mathbb N\) and \(\rho \in \mathbb {R}\), the \(\rho \)-stable cube is the complete directed graph, including loops, on \(\{ 0,1\} ^n\), with weight \(w(x,y)=2^{-n}\prod _{i=1}^n\frac{1+\rho (-1)^{x_i+y_i}}{2}\). For \(-1\le \rho \le 1\), these weights are the joint probabilities of a uniform correlated pair.

For \(n\in \mathbb N\) and \(\delta \in \mathbb {R}\), the \(\delta \)-noisy cube is the \((1-2\delta )\)-stable cube: the complete directed graph on \(\{ 0,1\} ^n\) with weights \(2^{-n}\prod _{i=1}^n\frac{1+(1-2\delta )(-1)^{x_i+y_i}}{2}\).

Let \(q\ge 2\) and \(a,t\in \mathbb {R}\). Then \((|a+t|^q+|a-t|^q)/2\le (a^2+(q-1)t^2)^{q/2}\).

A real-valued function \(f\) on \(\{ 0,1\} ^n\) is nonnegative when \(f(x)\ge 0\) for every cube point \(x\).

Let \(p\in \mathbb {R}\) and \(f:\{ 0,1\} ^n\to \mathbb {R}\), define \(M_p(f)=0\) when \(p\le 0\) and \(f\) has a zero; otherwise set \(M_0(f)=\exp (\mathbb {E}\log |f|)\) and \(M_p(f)=(\mathbb {E}|f|^p)^{1/p}\) for \(p\ne 0\), with uniform expectation.

For \(p\gt 0\) and \(f:\{ 0,1\} ^n\to \mathbb {R}\), the extended power mean agrees with the ordinary formula \(M_p(f)=(\mathbb {E}|f|^p)^{1/p}\) under the uniform measure.

For \(p\gt 0\) and every \(f:\{ 0,1\} ^n\to \mathbb {R}\), the uniform power mean \(M_p(f)=(\mathbb {E}|f|^p)^{1/p}\) is nonnegative.

For \(p\gt 0\) and functions \(f,g:\{ 0,1\} ^n\to \mathbb {R}\) satisfying \(0\le f(x)\le g(x)\) for every \(x\), one has \(M_p(f)\le M_p(g)\), where \(M_p(u)=(\mathbb {E}|u|^p)^{1/p}\).

Let \(p,c\gt 0\) and \(f:\{ 0,1\} ^n\to \mathbb {R}\). With \(M_p(u)=(\mathbb {E}|u|^p)^{1/p}\), one has \(M_p(cf)=cM_p(f)\).

Let \(p\gt 0\) and let \(f:\{ 0,1\} ^0\to [0,\infty )\). Its uniform power mean \(M_p(f)=(\mathbb {E}f^p)^{1/p}\) equals the value of \(f\) at the unique empty string.

Let \(p\gt 0\) and \(f:\{ 0,1\} ^{n+1}\to \mathbb {R}\), and write \(M_{p,z}\) for the uniform power mean \((\mathbb {E}_z|\cdot |^p)^{1/p}\). Then \(M_{p,(x,b)}f(x,b)=M_{p,x}\bigl(M_{p,b}f(x,b)\bigr)\), where \(x\in \{ 0,1\} ^n\) and \(b\in \{ 0,1\} \).

Let \(p\gt 0\) and \(F:\{ 0,1\} ^n\times \{ 0,1\} \to \mathbb {R}\). For uniform power means \(M_{p,z}u=(\mathbb {E}_z|u|^p)^{1/p}\), one has \(M_{p,x}(M_{p,b}F(x,b))=M_{p,b}(M_{p,x}F(x,b))\).

Let \(f:\{ 0,1\} ^n\to \mathbb {R}\), \(c\in \mathbb {R}\), and \(S\subseteq [n]\), uniform Fourier–Walsh coefficients satisfy \(\widehat{cf}(S)=c\widehat f(S)\).

Let \(f:\{ 0,1\} ^n\to \mathbb {R}\) and \(\rho ,c\in \mathbb {R}\), the Fourier noise operator satisfies \(T_\rho (cf)=cT_\rho f\). For \(d\in \mathbb N\), \(h:\{ 0,1\} ^d\to \mathbb {R}\), and \(\tau \in \mathbb {R}\), the spectral noise operator is \(T_\tau h(x)=\sum _{S\subseteq [d]}\tau ^{|S|}\widehat h(S)\chi _S(x)\), where \(\chi _S(x)=(-1)^{\sum _{i\in S}x_i}\) and \(\widehat h(S)=\mathbb {E}[h\chi _S]\) under the uniform measure.

Let \(\rho \in \mathbb {R}\) and \(f:\{ 0,1\} ^0\to \mathbb {R}\), the Fourier noise operator satisfies \(T_\rho f=f\). For \(d\in \mathbb N\), \(h:\{ 0,1\} ^d\to \mathbb {R}\), and \(\tau \in \mathbb {R}\), the spectral noise operator is \(T_\tau h(x)=\sum _{S\subseteq [d]}\tau ^{|S|}\widehat h(S)\chi _S(x)\), where \(\chi _S(x)=(-1)^{\sum _{i\in S}x_i}\) and \(\widehat h(S)=\mathbb {E}[h\chi _S]\) under the uniform measure.

Let \(0\le \rho \le 1\) and \(f:\{ 0,1\} ^n\to [0,\infty )\). Then \(T_\rho f(x)\ge 0\) at every cube point \(x\). For \(d\in \mathbb N\), \(h:\{ 0,1\} ^d\to \mathbb {R}\), and \(\tau \in \mathbb {R}\), the spectral noise operator is \(T_\tau h(x)=\sum _{S\subseteq [d]}\tau ^{|S|}\widehat h(S)\chi _S(x)\), where \(\chi _S(x)=(-1)^{\sum _{i\in S}x_i}\) and \(\widehat h(S)=\mathbb {E}[h\chi _S]\) under the uniform measure.

For \(\rho ,a\in \mathbb {R}\), let \(f(b)=1+a(-1)^b\) on \(\{ 0,1\} \). Then \(T_\rho f(b)=1+\rho a(-1)^b\) for both bits \(b\). For \(d\in \mathbb N\), \(h:\{ 0,1\} ^d\to \mathbb {R}\), and \(\tau \in \mathbb {R}\), the spectral noise operator is \(T_\tau h(x)=\sum _{S\subseteq [d]}\tau ^{|S|}\widehat h(S)\chi _S(x)\), where \(\chi _S(x)=(-1)^{\sum _{i\in S}x_i}\) and \(\widehat h(S)=\mathbb {E}[h\chi _S]\) under the uniform measure.

Let \(f:\{ 0,1\} ^{n+1}\to \mathbb {R}\), \(f_b(x)=f(x,b)\), and \(g=(f_0+f_1)/2\). For every \(S\subseteq [n]\), \(\widehat g(S)=(\widehat{f_0}(S)+\widehat{f_1}(S))/2\).

Let \(f:\{ 0,1\} ^{n+1}\to \mathbb {R}\), \(f_b(x)=f(x,b)\), and \(h=(f_0-f_1)/2\). For every \(S\subseteq [n]\), \(\widehat h(S)=(\widehat{f_0}(S)-\widehat{f_1}(S))/2\).

Let \(f:\{ 0,1\} ^{n+1}\to \mathbb {R}\), \(\rho \in \mathbb {R}\), and \(f_b(z)=f(z,b)\). For \(x\in \{ 0,1\} ^n\) and \(b\in \{ 0,1\} \), \(T_\rho f(x,b)=T_\rho \bigl(t\mapsto T_\rho f_t(x)\bigr)(b)\); the outer operator acts on one bit and the inner operator on \(n\) bits. For \(d\in \mathbb N\), \(h:\{ 0,1\} ^d\to \mathbb {R}\), and \(\tau \in \mathbb {R}\), the spectral noise operator is \(T_\tau h(x)=\sum _{S\subseteq [d]}\tau ^{|S|}\widehat h(S)\chi _S(x)\), where \(\chi _S(x)=(-1)^{\sum _{i\in S}x_i}\) and \(\widehat h(S)=\mathbb {E}[h\chi _S]\) under the uniform measure.

Let \(q\le p\le 1\), \(q\lt 1\), \(0\le \rho \le 1\), and \(\rho ^2\le (1-p)/(1-q)\). Every \(f:\{ 0,1\} ^n\to [0,\infty )\) satisfies \(M_q(T_\rho f)\ge M_p(f)\). For nonnegative functions \(u\), the extended mean is \(M_r(u)=(\mathbb {E}u^r)^{1/r}\) when \(r\gt 0\); the same formula applies for \(r\lt 0\) when \(u\) is positive everywhere. For positive \(u\), \(M_0(u)=\exp (\mathbb {E}\log u)\); if \(r\le 0\) and \(u\) has a zero, \(M_r(u)=0\). All expectations are uniform. For \(d\in \mathbb N\), \(h:\{ 0,1\} ^d\to \mathbb {R}\), and \(\tau \in \mathbb {R}\), the spectral noise operator is \(T_\tau h(x)=\sum _{S\subseteq [d]}\tau ^{|S|}\widehat h(S)\chi _S(x)\), where \(\chi _S(x)=(-1)^{\sum _{i\in S}x_i}\) and \(\widehat h(S)=\mathbb {E}[h\chi _S]\) under the uniform measure.

Let \(q\le p\le 1\), \(q\lt 1\), \(0\le \rho \le 1\), and \(\rho ^2\le (1-p)/(1-q)\). Every \(f:\{ 0,1\} ^n\to [0,\infty )\) satisfies \(M_q(T_\rho f)\ge M_p(f)\). For nonnegative functions \(u\), the extended mean is \(M_r(u)=(\mathbb {E}u^r)^{1/r}\) when \(r\gt 0\); the same formula applies for \(r\lt 0\) when \(u\) is positive everywhere. For positive \(u\), \(M_0(u)=\exp (\mathbb {E}\log u)\); if \(r\le 0\) and \(u\) has a zero, \(M_r(u)=0\). All expectations are uniform. For \(d\in \mathbb N\), \(h:\{ 0,1\} ^d\to \mathbb {R}\), and \(\tau \in \mathbb {R}\), the spectral noise operator is \(T_\tau h(x)=\sum _{S\subseteq [d]}\tau ^{|S|}\widehat h(S)\chi _S(x)\), where \(\chi _S(x)=(-1)^{\sum _{i\in S}x_i}\) and \(\widehat h(S)=\mathbb {E}[h\chi _S]\) under the uniform measure.

Let \(0\lt r\lt 1\), \(a\ge 0\), and \(b\gt 0\). Writing \(s=r/(r-1)\lt 0\), one has \(ab\ge a^r/r+b^s/s\).

Let \(r\in \mathbb {R}\) and \(u:\{ 0,1\} ^n\to [0,\infty )\) have at least one positive value. Then \(\mathbb {E}[u^r]\gt 0\), with the convention \(0^0=1\) and \(0^r=0\) for \(r\ne 0\).

Let \(r\ne 0\), let \(u:\{ 0,1\} ^n\to [0,\infty )\), and assume \(A=\mathbb {E}[u^r]\gt 0\) under the uniform measure. Then \(\mathbb {E}[(u/A^{1/r})^r]=1\).

Let \(u,v:\{ 0,1\} ^n\to \mathbb {R}\) and nonzero \(A,B\in \mathbb {R}\), uniform expectation satisfies \(\mathbb {E}[uv]=AB\mathbb {E}[(u/A)(v/B)]\).

Let \(0\lt r\lt 1\) and \(u,v:\{ 0,1\} ^n\to [0,\infty )\). Then \(\mathbb {E}[uv]\ge M_r(u)M_{r/(r-1)}(v)\). For nonnegative functions \(u\), the extended mean is \(M_r(u)=(\mathbb {E}u^r)^{1/r}\) when \(r\gt 0\); the same formula applies for \(r\lt 0\) when \(u\) is positive everywhere. For positive \(u\), \(M_0(u)=\exp (\mathbb {E}\log u)\); if \(r\le 0\) and \(u\) has a zero, \(M_r(u)=0\). All expectations are uniform.

Let \(p\lt 1\), \(p\ne 0\), and \(f,g:\{ 0,1\} ^n\to [0,\infty )\). Then \(\mathbb {E}[fg]\ge M_p(f)M_{p/(p-1)}(g)\). For nonnegative functions \(u\), the extended mean is \(M_r(u)=(\mathbb {E}u^r)^{1/r}\) when \(r\gt 0\); the same formula applies for \(r\lt 0\) when \(u\) is positive everywhere. For positive \(u\), \(M_0(u)=\exp (\mathbb {E}\log u)\); if \(r\le 0\) and \(u\) has a zero, \(M_r(u)=0\). All expectations are uniform.

Let \(s\subseteq I\) and \(t\subseteq J\) be finite sets, let \(r\ge 1\), and let \(a:I\times J\to \mathbb {R}\) satisfy \(a(i,j)\ge 0\) for \(i\in s\), \(j\in t\). Then \((\sum _{j\in t}(\sum _{i\in s}a(i,j))^r)^{1/r}\le \sum _{i\in s}(\sum _{j\in t}a(i,j)^r)^{1/r}\).

Let \(0\lt q\le p\) and \(F:\{ 0,1\} ^n\times \{ 0,1\} \to [0,\infty )\). Writing \(M_{r,z}u=(\mathbb {E}_z u^r)^{1/r}\) for uniform means, \(M_{q,x}(M_{p,b}F(x,b))\ge M_{p,b}(M_{q,x}F(x,b))\).

Let \(0\lt q\lt p\) and \(0\le \rho \le 1\). Suppose \(M_q(T_\rho u)\ge M_p(u)\) for every nonnegative function \(u\) on one bit. Then \(M_q(T_\rho f)\ge M_p(f)\) for every \(n\in \mathbb N\) and \(f:\{ 0,1\} ^n\to [0,\infty )\), where \(M_r(u)=(\mathbb {E}|u|^r)^{1/r}\). For \(d\in \mathbb N\), \(h:\{ 0,1\} ^d\to \mathbb {R}\), and \(\tau \in \mathbb {R}\), the spectral noise operator is \(T_\tau h(x)=\sum _{S\subseteq [d]}\tau ^{|S|}\widehat h(S)\chi _S(x)\), where \(\chi _S(x)=(-1)^{\sum _{i\in S}x_i}\) and \(\widehat h(S)=\mathbb {E}[h\chi _S]\) under the uniform measure.

Let \(0\lt q\lt p\lt 1\), \(0\le \rho \le 1\), and \(\rho ^2=(1-p)/(1-q)\). Every \(f:\{ 0,1\} ^n\to [0,\infty )\) satisfies \(M_q(T_\rho f)\ge M_p(f)\), where \(M_r(u)=(\mathbb {E}|u|^r)^{1/r}\) under the uniform measure. For \(d\in \mathbb N\), \(h:\{ 0,1\} ^d\to \mathbb {R}\), and \(\tau \in \mathbb {R}\), the spectral noise operator is \(T_\tau h(x)=\sum _{S\subseteq [d]}\tau ^{|S|}\widehat h(S)\chi _S(x)\), where \(\chi _S(x)=(-1)^{\sum _{i\in S}x_i}\) and \(\widehat h(S)=\mathbb {E}[h\chi _S]\) under the uniform measure.

Let \(q\lt 1\), \(0\le \rho \le \sigma \le 1\), and \(f:\{ 0,1\} ^n\to [0,\infty )\). Then \(M_q(T_\rho f)\ge M_q(T_\sigma f)\). For nonnegative functions \(u\), the extended mean is \(M_r(u)=(\mathbb {E}u^r)^{1/r}\) when \(r\gt 0\); the same formula applies for \(r\lt 0\) when \(u\) is positive everywhere. For positive \(u\), \(M_0(u)=\exp (\mathbb {E}\log u)\); if \(r\le 0\) and \(u\) has a zero, \(M_r(u)=0\). All expectations are uniform. For \(d\in \mathbb N\), \(h:\{ 0,1\} ^d\to \mathbb {R}\), and \(\tau \in \mathbb {R}\), the spectral noise operator is \(T_\tau h(x)=\sum _{S\subseteq [d]}\tau ^{|S|}\widehat h(S)\chi _S(x)\), where \(\chi _S(x)=(-1)^{\sum _{i\in S}x_i}\) and \(\widehat h(S)=\mathbb {E}[h\chi _S]\) under the uniform measure.

Let \(a\le b\le 1\) and \(f:\{ 0,1\} ^n\to [0,\infty )\). Then \(M_a(f)\le M_b(f)\). For nonnegative functions \(u\), the extended mean is \(M_r(u)=(\mathbb {E}u^r)^{1/r}\) when \(r\gt 0\); the same formula applies for \(r\lt 0\) when \(u\) is positive everywhere. For positive \(u\), \(M_0(u)=\exp (\mathbb {E}\log u)\); if \(r\le 0\) and \(u\) has a zero, \(M_r(u)=0\). All expectations are uniform.

Let \(p,q\lt 1\), \(0\le \rho \le 1\), and \(\rho ^2\le (1-p)(1-q)\). For all \(f,g:\{ 0,1\} ^n\to [0,\infty )\), uniform expectation satisfies \(\mathbb {E}[fT_\rho g]\ge M_p(f)M_q(g)\). For nonnegative functions \(u\), the extended mean is \(M_r(u)=(\mathbb {E}u^r)^{1/r}\) when \(r\gt 0\); the same formula applies for \(r\lt 0\) when \(u\) is positive everywhere. For positive \(u\), \(M_0(u)=\exp (\mathbb {E}\log u)\); if \(r\le 0\) and \(u\) has a zero, \(M_r(u)=0\). All expectations are uniform. For \(d\in \mathbb N\), \(h:\{ 0,1\} ^d\to \mathbb {R}\), and \(\tau \in \mathbb {R}\), the spectral noise operator is \(T_\tau h(x)=\sum _{S\subseteq [d]}\tau ^{|S|}\widehat h(S)\chi _S(x)\), where \(\chi _S(x)=(-1)^{\sum _{i\in S}x_i}\) and \(\widehat h(S)=\mathbb {E}[h\chi _S]\) under the uniform measure.

Let \(s,x\in \mathbb {R}\) with \(|x|\lt 1\), the generalized binomial series converges and \(\sum _{k=0}^{\infty }\binom {s}{k}x^k=(1+x)^s\), where \(\binom {s}{k}=s(s-1)\cdots (s-k+1)/k!\) and \(\binom {s}{0}=1\).

For every \(s,x\in \mathbb {R}\) with \(|x|\lt 1\), the series \(\sum _{k=0}^{\infty }\binom {s}{2k}x^{2k}\) is summable, with generalized binomial coefficients.

Let \(s,x\in \mathbb {R}\) with \(|x|\lt 1\), \(((1+x)^s+(1-x)^s)/2=\sum _{k=0}^{\infty }\binom {s}{2k}x^{2k}\), using generalized binomial coefficients.

Let \(s,b\in \mathbb {R}\) with \(|b|\lt 1\), uniform one-bit expectation satisfies \(\mathbb {E}_{t\in \{ 0,1\} }|1+b(-1)^t|^s=((1+b)^s+(1-b)^s)/2\).

For every \(s\in \mathbb {R}\), the generalized binomial coefficient satisfies \(\binom {s}{2}=s(s-1)/2\).

Let \(s\in \mathbb {R}\) and \(k\in \mathbb N\), the generalized binomial coefficients satisfy \(\binom {s}{2k+2}=\binom {s}{2k}(s-2k)(s-2k-1)/((2k+1)(2k+2))\).

Let \(0\lt s\lt 1\). For every integer \(k\ge 1\), the generalized binomial coefficient satisfies \(\binom {s}{2k}\le 0\).

Let \(0\lt q\lt p\lt 1\), \(\rho \ge 0\), \(\rho ^2=(1-p)/(1-q)\), and \(m\in \mathbb N\) with \(m\ge 2\). Then \(\rho (m-q)\le m-p\).

Let \(0\lt q\lt p\lt 1\), \(\rho \ge 0\), \(\rho ^2=(1-p)/(1-q)\), and integer \(K\ge 1\). If \(\binom p{2K}\le (p/q)\binom q{2K}\rho ^{2K}\), then \(\binom p{2K+2}\le (p/q)\binom q{2K+2}\rho ^{2K+2}\), with generalized binomial coefficients.

Let \(0\lt q\lt p\lt 1\), \(\rho \ge 0\), and \(\rho ^2=(1-p)/(1-q)\). For every integer \(k\ge 1\), \(\binom p{2k}\le (p/q)\binom q{2k}\rho ^{2k}\), with generalized binomial coefficients.

Let \(0\lt q\lt p\lt 1\), \(\rho \ge 0\), \(\rho ^2=(1-p)/(1-q)\), and \(|a|\lt 1\). For \(f(t)=1+a(-1)^t\) on \(\{ 0,1\} \), \(M_q(T_\rho f)\ge M_p(f)\), where \(M_r(u)=(\mathbb {E}|u|^r)^{1/r}\) under the uniform measure. For \(d\in \mathbb N\), \(h:\{ 0,1\} ^d\to \mathbb {R}\), and \(\tau \in \mathbb {R}\), the spectral noise operator is \(T_\tau h(x)=\sum _{S\subseteq [d]}\tau ^{|S|}\widehat h(S)\chi _S(x)\), where \(\chi _S(x)=(-1)^{\sum _{i\in S}x_i}\) and \(\widehat h(S)=\mathbb {E}[h\chi _S]\) under the uniform measure.

Every point of \(\{ 0,1\} ^1\) is either the string whose only coordinate is \(0\) or the string whose only coordinate is \(1\).

For every nonnegative function \(f:\{ 0,1\} \to [0,\infty )\) that is not identically zero, there exist \(c\gt 0\) and \(a\in [-1,1]\) such that \(f(t)=c(1+a(-1)^t)\) for both bits \(t\).

For every \(s\gt 0\), the map \(a\mapsto (\mathbb {E}_{t\in \{ 0,1\} }|1+a(-1)^t|^s)^{1/s}\) is continuous on \(\mathbb {R}\).

Let \(0\lt q\lt p\lt 1\), \(\rho \ge 0\), and \(\rho ^2=(1-p)/(1-q)\). Every \(f:\{ 0,1\} \to [0,\infty )\) satisfies \(M_q(T_\rho f)\ge M_p(f)\), where \(M_r(u)=(\mathbb {E}|u|^r)^{1/r}\) under the uniform measure. For \(d\in \mathbb N\), \(h:\{ 0,1\} ^d\to \mathbb {R}\), and \(\tau \in \mathbb {R}\), the spectral noise operator is \(T_\tau h(x)=\sum _{S\subseteq [d]}\tau ^{|S|}\widehat h(S)\chi _S(x)\), where \(\chi _S(x)=(-1)^{\sum _{i\in S}x_i}\) and \(\widehat h(S)=\mathbb {E}[h\chi _S]\) under the uniform measure.

For \(q,\lambda \in \mathbb {R}\), set \(u=\log ((1-\lambda )/\lambda )\) and define \(\rho (q,\lambda )=\sqrt{\sinh (u/q)/\sinh (u/(q/(q-1)))}\), where division by zero is zero, \(\log 0=0\), \(\log x=\log |x|\) for \(x\lt 0\), and \(\sqrt x=0\) for \(x\le 0\).

For \(\lambda \in \mathbb {R}\), define \(C(\lambda )=e\) if \(\lambda =1/2\), and \(C(\lambda )=((1-\lambda )/\lambda )^{1/(2(1-2\lambda ))}\) otherwise, where division by zero is zero; for \(x\lt 0\), \(x^r=|x|^r\cos (\pi r)\); and \(0^r=0\) for \(r\ne 0\), while \(0^0=1\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(c\in \mathbb R\), and \(f:\Omega \to \mathbb R\). Then \(\operatorname {Stab}_{c^2}(f)=\| T_cf\| _{2,P}^2\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(T_a h=\sum _{S\subseteq [n]}a^{|S|}h_S\) for every real rate \(a\); \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\); \(\operatorname {Stab}_a(h)=\mathbb E_P[hT_a h]\). The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), an event \(A\subseteq \Omega \), and \(p\gt 0\). Then \(\| \mathbf1_A\| _{p,P}^2=\Pr _P(A)^{2/p}\). For a real function \(h\) on \(\Omega \), use \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\); \(\Pr _P(A)=\mathbb E_P\mathbf1_A\), where \(\mathbf1_A\) is \(1\) on \(A\) and \(0\) elsewhere.

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with every coordinate atom at least \(0\lt \lambda \le 1\), and let \(q\ge 2\) and \(0\le \rho \le \lambda ^{1-2/q}/(q-1)\). For every event \(A\subseteq \Omega \), \(\operatorname {Stab}_\rho (\mathbf1_A)\le \Pr _P(A)^{2-2/q}\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(T_a h=\sum _{S\subseteq [n]}a^{|S|}h_S\) for every real rate \(a\); \(\operatorname {Stab}_a(h)=\mathbb E_P[hT_a h]\); \(\Pr _P(A)=\mathbb E_P\mathbf1_A\), where \(\mathbf1_A\) is \(1\) on \(A\) and \(0\) elsewhere. The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(\rho \ge 0\), a coordinate \(i\), and \(f:\Omega \to \mathbb R\). Then \(\rho I_i^{(\rho )}(f)=\| T_{\sqrt\rho }L_if\| _{2,P}^2\), where \(I_i^{(\rho )}\) is stable influence and \(L_i\) is the coordinate Laplacian. For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(T_a h=\sum _{S\subseteq [n]}a^{|S|}h_S\) for every real rate \(a\); \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\); \(L_i h=h-E_{[n]\setminus \{ i\} }h\); \(I_i^{(a)}(h)=\sum _{S\ni i}a^{|S|-1}\mathbb E_Ph_S^2\) for \(a\in \mathbb R\). The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(f:\Omega \to \{ -1,1\} \), and \(S\subseteq [n]\). For conditional expectation \(E_S\) onto \(S\), \(\mathbb E_P|f-E_Sf|=\mathbb E_P(f-E_Sf)^2\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(f:\Omega \to \{ -1,1\} \), a coordinate \(i\), and \(1\le p\le 2\). Then \(\mathbb E_P|L_if|^p\le I_i(f)\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(L_i h=h-E_{[n]\setminus \{ i\} }h\); \(I_i(h)=\mathbb E_P(h-E_{[n]\setminus \{ i\} }h)^2\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(q\ge 2\), \(\rho \ge 0\), \(f:\Omega \to \{ -1,1\} \), and a coordinate \(i\). If \(\| T_{\sqrt\rho }L_if\| _{2,P}\le \| L_if\| _{q/(q-1),P}\), then \(\rho I_i^{(\rho )}(f)\le I_i(f)^{2-2/q}\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(T_a h=\sum _{S\subseteq [n]}a^{|S|}h_S\) for every real rate \(a\); \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\); \(L_i h=h-E_{[n]\setminus \{ i\} }h\); \(I_i(h)=\mathbb E_P(h-E_{[n]\setminus \{ i\} }h)^2\); \(I_i^{(a)}(h)=\sum _{S\ni i}a^{|S|-1}\mathbb E_Ph_S^2\) for \(a\in \mathbb R\). The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with every coordinate atom at least \(0\lt \lambda \le 1\), and let \(q\ge 2\) and \(0\le \rho \le \lambda ^{1-2/q}/(q-1)\). For every \(f:\Omega \to \{ -1,1\} \) and coordinate \(i\), \(\rho I_i^{(\rho )}(f)\le I_i(f)^{2-2/q}\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(I_i(h)=\mathbb E_P(h-E_{[n]\setminus \{ i\} }h)^2\); \(I_i^{(a)}(h)=\sum _{S\ni i}a^{|S|-1}\mathbb E_Ph_S^2\) for \(a\in \mathbb R\). The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with every coordinate atom at least \(0\lt \lambda \le 1\). For every \(f:\Omega \to \{ -1,1\} \) and coordinate \(i\), \(\sum _{S\ni i}(\sqrt\lambda /3)^{|S|}\mathbb E_Pf_S^2\le I_i(f)^{3/2}\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(I_i(h)=\mathbb E_P(h-E_{[n]\setminus \{ i\} }h)^2\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with every coordinate atom at least \(0\lt \lambda \le 1/2\). If \(f:\Omega \to \{ -1,1\} \) has positive variance and \(a=I(f)/\operatorname {Var}_P(f)\), there is a coordinate \(i\) such that \((a^{-1})^2(9/\lambda )^{-a}\le I_i(f)\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(I_i(h)=\mathbb E_P(h-E_{[n]\setminus \{ i\} }h)^2\); \(I(h)=\sum _{i=0}^{n-1}I_i(h)=\sum _{S\subseteq [n]}|S|\mathbb E_Ph_S^2\); \(\operatorname {Var}_P(h)=\mathbb E_P(h-\mathbb E_Ph)^2\).

There is an absolute \(c\gt 0\) such that for every \(n\ge 2\), every product \(P\) of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), with every coordinate atom at least \(0\lt \lambda \le 1/2\), and every \(f:\Omega \to \{ -1,1\} \), some coordinate \(i\) satisfies \(c\operatorname {Var}_P(f)\log n/(n\log (1/\lambda ))\le I_i(f)\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(I_i(h)=\mathbb E_P(h-E_{[n]\setminus \{ i\} }h)^2\); \(\operatorname {Var}_P(h)=\mathbb E_P(h-\mathbb E_Ph)^2\).

There is an absolute \(C\gt 0\) such that for every product \(P\) of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), with every coordinate atom at least \(0\lt \lambda \le 1/2\), every \(f:\Omega \to \{ -1,1\} \), and every \(0\lt \varepsilon \le 1\), there exist \(J\subseteq [n]\) and \(g:\Omega \to \{ -1,1\} \) depending only on \(J\) with \(|J|\le (1/\lambda )^{CI(f)/\varepsilon }\) and \(\Pr _P(f\ne g)\le \varepsilon \). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(I_i(h)=\mathbb E_P(h-E_{[n]\setminus \{ i\} }h)^2\); \(I(h)=\sum _{i=0}^{n-1}I_i(h)=\sum _{S\subseteq [n]}|S|\mathbb E_Ph_S^2\); \(\Pr _P(A)=\mathbb E_P\mathbf1_A\), where \(\mathbf1_A\) is \(1\) on \(A\) and \(0\) elsewhere.

For every real \(\lambda \), define \(C_\lambda =\begin{cases} e,& \lambda =1/2,\\ ((1-\lambda )/\lambda )^{1/(2(1-2\lambda ))},& \lambda \ne 1/2.\end{cases}\). Then \(\lim _{\lambda \to 0^+}C_\lambda \sqrt\lambda =1\).

For every real \(\lambda \), define \(C_\lambda =\begin{cases} e,& \lambda =1/2,\\ ((1-\lambda )/\lambda )^{1/(2(1-2\lambda ))},& \lambda \ne 1/2.\end{cases}\). Then \(\lim _{\lambda \to (1/2)^-,\ 0\lt \lambda \lt 1/2}C_\lambda =e\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with every coordinate atom at least \(0\lt \lambda \lt 1/2\), and let \(q\gt 2\). Define \(r_q(\lambda )=\sqrt{\frac{\sinh (\log ((1-\lambda )/\lambda )/q)}{\sinh (\log ((1-\lambda )/\lambda )/(q/(q-1)))}}\); if \(0\le \rho \le r_q(\lambda )^2\), then \(\operatorname {Stab}_\rho (\mathbf1_A)\le \Pr _P(A)^{2-2/q}\) for every event \(A\subseteq \Omega \). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(T_a h=\sum _{S\subseteq [n]}a^{|S|}h_S\) for every real rate \(a\); \(\operatorname {Stab}_a(h)=\mathbb E_P[hT_a h]\); \(\Pr _P(A)=\mathbb E_P\mathbf1_A\), where \(\mathbf1_A\) is \(1\) on \(A\) and \(0\) elsewhere. The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with every coordinate atom at least \(0\lt \lambda \lt 1/2\), and let \(q\gt 2\). Define \(r_q(\lambda )=\sqrt{\frac{\sinh (\log ((1-\lambda )/\lambda )/q)}{\sinh (\log ((1-\lambda )/\lambda )/(q/(q-1)))}}\); if \(0\le \rho \le r_q(\lambda )^2\), then \(\rho I_i^{(\rho )}(f)\le I_i(f)^{2-2/q}\) for every \(f:\Omega \to \{ -1,1\} \) and coordinate \(i\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(I_i(h)=\mathbb E_P(h-E_{[n]\setminus \{ i\} }h)^2\); \(I_i^{(a)}(h)=\sum _{S\ni i}a^{|S|-1}\mathbb E_Ph_S^2\) for \(a\in \mathbb R\). The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(0\le c\le 1\), and \(f:\Omega \to \mathbb R\). Then \(\| T_cf\| _{2,P}\le \| f\| _{2,P}\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(T_a h=\sum _{S\subseteq [n]}a^{|S|}h_S\) for every real rate \(a\); \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\). The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(c\ge 1\), \(k\in \mathbb N\), and \(f:\Omega \to \mathbb R\) of degree at most \(k\). Then \(\| T_cf\| _{2,P}\le c^k\| f\| _{2,P}\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(T_a h=\sum _{S\subseteq [n]}a^{|S|}h_S\) for every real rate \(a\); \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\); degree at most \(k\) means \(h_S=0\) for every \(|S|\gt k\). The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with every coordinate atom at least \(\lambda \), where \(0\lt \lambda \le 1\), and let \(q\gt 2\). For \(k\in \mathbb N\) and \(f:\Omega \to \mathbb R\) of degree at most \(k\), \(\| f\| _{q,P}\le (\sqrt{q-1}\lambda ^{1/q-1/2})^k\| f\| _{2,P}\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\); degree at most \(k\) means \(h_S=0\) for every \(|S|\gt k\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with every coordinate atom at least \(0\lt \lambda \lt 1/2\). For \(f:\Omega \to \mathbb R\) of degree at most \(k\in \mathbb N\), \(\| f\| _{2,P}\le C_\lambda ^k\| f\| _{1,P}\), where \(C_\lambda =((1-\lambda )/\lambda )^{1/(2(1-2\lambda ))}\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\); degree at most \(k\) means \(h_S=0\) for every \(|S|\gt k\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with every coordinate atom at least \(\lambda \), where \(0\lt \lambda \le 1/2\). Define \(C_\lambda =\begin{cases} e,& \lambda =1/2,\\ ((1-\lambda )/\lambda )^{1/(2(1-2\lambda ))},& \lambda \ne 1/2.\end{cases}\) For \(f:\Omega \to \mathbb R\) of degree at most \(k\in \mathbb N\), \(\| f\| _{2,P}\le C_\lambda ^k\| f\| _{1,P}\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\); degree at most \(k\) means \(h_S=0\) for every \(|S|\gt k\).

For every \(0\lt \lambda \le 1/2\), define \(C_\lambda =\begin{cases} e,& \lambda =1/2,\\ ((1-\lambda )/\lambda )^{1/(2(1-2\lambda ))},& \lambda \ne 1/2.\end{cases}\). Then \(C_\lambda \le e/\sqrt{2\lambda }\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with every coordinate atom at least \(0\lt \lambda \le 1/2\). If \(f:\Omega \to \mathbb R\) has degree at most \(k\in \mathbb N\) and positive variance, then \(\tfrac 14(e^2/(2\lambda ))^{-k}\le \Pr _P(\mathbb E_Pf\lt f)\) and \((15/\lambda )^{-k}\le \tfrac 14(e^2/(2\lambda ))^{-k}\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(\operatorname {Var}_P(h)=\mathbb E_P(h-\mathbb E_Ph)^2\); \(\Pr _P(A)=\mathbb E_P\mathbf1_A\), where \(\mathbf1_A\) is \(1\) on \(A\) and \(0\) elsewhere; degree at most \(k\) means \(h_S=0\) for every \(|S|\gt k\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with every coordinate atom at least \(0\lt \lambda \le 1\). For \(q\ge 2\) and \(f:\Omega \to \mathbb R\) of degree at most \(k\in \mathbb N\), \(\| f\| _{q,P}\le (\sqrt q\lambda ^{1/q-1/2})^k\| f\| _{2,P}\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\); degree at most \(k\) means \(h_S=0\) for every \(|S|\gt k\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with every coordinate atom at least \(0\lt \lambda \le 1\). If \(k\gt 0\) is an integer, \(f:\Omega \to \mathbb R\) has degree at most \(k\), \(\| f\| _{2,P}\gt 0\), and \(t\ge (\sqrt{2e/\lambda })^k\), then \(\Pr _P(|f|\ge t\| f\| _{2,P})\le \lambda ^k\exp (-k\lambda t^{2/k}/(2e))\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\); \(\Pr _P(A)=\mathbb E_P\mathbf1_A\), where \(\mathbf1_A\) is \(1\) on \(A\) and \(0\) elsewhere; degree at most \(k\) means \(h_S=0\) for every \(|S|\gt k\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(0\le c\le 1\), and \(f:\Omega \to \mathbb R\). Then \(\| T_cf\| _{2,P}\le \| f\| _{2,P}\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(T_a h=\sum _{S\subseteq [n]}a^{|S|}h_S\) for every real rate \(a\); \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\). The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).

For a finite set \(S\) and a real function \(g\) on finite subsets of an ambient set, define \(E_S(g)=\sum _{A,B\subseteq S}(-1)^{|S|-|A|}(-1)^{|S|-|B|}g(A\cap B)\).

For a finite set \(S\), an element \(i\notin S\), and a real function \(g\) on finite subsets, put \(E_T(g)=\sum _{A,B\subseteq T}(-1)^{|T|-|A|}(-1)^{|T|-|B|}g(A\cap B)\). Then \(E_{S\cup \{ i\} }(g)=E_S(T\mapsto g(T\cup \{ i\} ))-E_S(g)\).

For a finite set \(U\) and a real function \(g\) on finite subsets, put \(E_S(g)=\sum _{A,B\subseteq S}(-1)^{|S|-|A|}(-1)^{|S|-|B|}g(A\cap B)\). Then \(\sum _{S\subseteq U}E_S(g)=g(U)\).

For every \(n\in \mathbb N\), a finite probability product consists of finite sets \(\Omega _i\) for \(i\in \{ 0,\ldots ,n-1\} \) and real weights \(w_i(a)\gt 0\) satisfying \(\sum _{a\in \Omega _i}w_i(a)=1\). The coordinate sets and weights may depend on \(i\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\). Its outcome space is \(\Omega =\prod _{i=0}^{n-1}\Omega _i\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), with coordinate weights \(w_i\). The mass of \(x\in \prod _i\Omega _i\) is \(m(x)=\prod _iw_i(x_i)\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), with product masses \(m(x)\). For \(f:\prod _i\Omega _i\to \mathbb R\), define \(\mathbb E_Pf=\sum _xm(x)f(x)\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) and let \(A\) be an event in its outcome space. Its indicator \(\mathbf1_A\) equals \(1\) on \(A\) and \(0\) elsewhere.

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) and let \(A\) be an event in its outcome space. Define \(\Pr _P(A)=\mathbb E_P\mathbf1_A\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(q\in \mathbb R\), and \(f:\prod _i\Omega _i\to \mathbb R\). Define \(\| f\| _{q,P}=(\mathbb E_P|f|^q)^{1/q}\) using real powers and division; for \(q\gt 0\) this is the usual moment expression.

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) and \(f:\prod _i\Omega _i\to \mathbb R\). Define \(\operatorname {Var}_P(f)=\mathbb E_P(f-\mathbb E_Pf)^2\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(S\subseteq \{ 0,\ldots ,n-1\} \), and \(f:\prod _i\Omega _i\to \mathbb R\). Say that \(f\) depends only on \(S\) if \(f(x)=f(y)\) whenever \(x_i=y_i\) for every \(i\in S\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(S\subseteq \{ 0,\ldots ,n-1\} \), and \(f:\prod _i\Omega _i\to \mathbb R\). Define \(E_Sf(x)=\mathbb E_{Y\sim P}f(z)\), where \(z_i=x_i\) for \(i\in S\) and \(z_i=Y_i\) otherwise.

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(S\subseteq \{ 0,\ldots ,n-1\} \), and \(f:\prod _i\Omega _i\to \mathbb R\). Its \(S\)-component is \(f_S=\sum _{T\subseteq S}(-1)^{|S|-|T|}E_Tf\), where \(E_T\) conditions on the coordinates in \(T\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(k\in \mathbb N\), and \(f:\prod _i\Omega _i\to \mathbb R\). The predicate that \(f\) has degree at most \(k\) means that every orthogonal component \(f_S\) with \(|S|\gt k\) vanishes identically.

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(k\in \mathbb N\), and \(f:\prod _i\Omega _i\to \mathbb R\). Define \(f^{\le k}(x)=\sum _{|S|\le k}f_S(x)\), where \(f_S\) is the orthogonal component indexed by \(S\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(\rho \in \mathbb R\), and \(f:\prod _i\Omega _i\to \mathbb R\). Define \(T_\rho f(x)=\sum _S\rho ^{|S|}f_S(x)\), where \(f_S\) are the orthogonal components and the powers have natural exponents.

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), a coordinate \(i\), and \(f:\prod _j\Omega _j\to \mathbb R\). Define \(L_if=f-E_{[n]\setminus \{ i\} }f\), where \(E_S\) conditions on the coordinates in \(S\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), a coordinate \(i\), and \(f:\prod _j\Omega _j\to \mathbb R\). Define \(I_i(f)=\mathbb E_P(L_if)^2\), where \(L_if\) subtracts the conditional expectation over coordinate \(i\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) and \(f:\prod _i\Omega _i\to \mathbb R\). Define \(I(f)=\sum _{i=0}^{n-1}I_i(f)\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(\rho \in \mathbb R\), a coordinate \(i\), and \(f:\prod _j\Omega _j\to \mathbb R\). Define \(I_i^{(\rho )}(f)=\sum _{S\ni i}\rho ^{|S|-1}\mathbb E_Pf_S^2\), with natural exponents, including \(\rho =0\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(\rho \in \mathbb R\), and \(f:\prod _i\Omega _i\to \mathbb R\). Define \(\operatorname {Stab}_\rho (f)=\mathbb E_P[fT_\rho f]\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with coordinate weights \(w_i\) and let \(\lambda \in \mathbb R\). The atom lower bound \(\lambda \) means that \(\lambda \le w_i(a)\) for every coordinate \(i\) and outcome \(a\in \Omega _i\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) and \(f:\prod _i\Omega _i\to \mathbb R\). The Boolean-valued predicate means \(f(x)\in \{ -1,1\} \) at every outcome \(x\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), with product masses \(m(x)\). Then \(\sum _xm(x)=1\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(S\subseteq [n]\), and \(F:\Omega \times \Omega \to \mathbb R\), where \(\Omega =\prod _i\Omega _i\). For independent \(X,Y\sim P\), let \(X\mathbin {\star _S}Y\) keep \(X_i\) on \(S\) and \(Y_i\) elsewhere; then \(\mathbb E F(X\mathbin {\star _S}Y,Y\mathbin {\star _S}X)=\mathbb EF(X,Y)\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) and \(c\in \mathbb R\). Then \(\mathbb E_P[c]=c\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), let \(s\) be a finite index set, and let \(a_j\in \mathbb R\) and \(F_j:\Omega \to \mathbb R\). Then \(\mathbb E_P\sum _{j\in s}a_jF_j=\sum _{j\in s}a_j\mathbb E_PF_j\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(S,T\subseteq [n]\), \(f:\Omega \to \mathbb R\), and \(x\in \Omega \). For coordinate conditional expectations, \(E_S(E_Tf)(x)=E_{S\cap T}f(x)\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(S\subseteq [n]\), and \(f,g:\Omega \to \mathbb R\). For conditional expectation \(E_S\) onto \(S\), \(\mathbb E_P[fE_Sg]=\mathbb E_P[(E_Sf)g]\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), a finite index set \(s\), and functions \(F_j:\Omega \to \mathbb R\). Then \(\mathbb E_P\sum _{j\in s}F_j=\sum _{j\in s}\mathbb E_PF_j\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(a\in \mathbb R\), and \(f:\Omega \to \mathbb R\). Then \(\mathbb E_P[af]=a\mathbb E_Pf\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) and \(f:\Omega \to \mathbb R\) with orthogonal components \(f_S\). Then \(\mathbb E_Pf^2=\sum _{S\subseteq [n]}\mathbb E_Pf_S^2\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(J\subseteq [n]\), and functions \(f,h:\Omega \to \mathbb R\). If \(f\) takes values in \(\{ -1,1\} \) and \(h\) depends only on \(J\), there exists \(g:\Omega \to \{ -1,1\} \) depending only on \(J\) such that \(\Pr _P(f\ne g)\le \mathbb E_P(f-h)^2\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(f:\Omega \to \mathbb R\) with variance \(V\gt 0\), and \(\rho \gt 0\). Writing \(I=\sum _iI_i(f)\) and \(f_S\) for the orthogonal components, \(V\rho ^{I/V}\le \sum _S|S|\rho ^{|S|}\mathbb E_Pf_S^2\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(q\gt 2\), \(c\in \mathbb R\), and \(f:\Omega \to \mathbb R\). If \(\| f\| _{q,P}\le e^c\| f\| _{2,P}\), then \(\| f\| _{2,P}\le e^{cq/(q-2)}\| f\| _{1,P}\).

For every \(u\gt 0\), the real function \(q\mapsto \tfrac 12[\log \sinh (u/q)-\log \sinh (u-u/q)]\) has derivative \(-u\cosh (u/2)/(4\sinh (u/2))\) at \(q=2\).

For every \(0\lt \lambda \lt 1/2\), \(\cosh (\tfrac 12\log ((1-\lambda )/\lambda ))/\sinh (\tfrac 12\log ((1-\lambda )/\lambda ))=1/(1-2\lambda )\).

Let \(0\lt \lambda \lt 1/2\), define \(r_q(\lambda )=\sqrt{\frac{\sinh (\log ((1-\lambda )/\lambda )/q)}{\sinh (\log ((1-\lambda )/\lambda )/(q/(q-1)))}}\), and put \(C_\lambda =((1-\lambda )/\lambda )^{1/(2(1-2\lambda ))}\). Then \(\lim _{q\to 2^+}\exp (-q\log r_q(\lambda )/(q-2))=C_\lambda \).

For every real \(\lambda \), define \(C_\lambda =\begin{cases} e,& \lambda =1/2,\\ ((1-\lambda )/\lambda )^{1/(2(1-2\lambda ))},& \lambda \ne 1/2.\end{cases}\). Then \(\lim _{\lambda \to (1/2)^-,\ 0\lt \lambda \lt 1/2}C_\lambda =e\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) and let \(g:\Omega \to \mathbb R\) have mean zero and positive second moment. If \(C\gt 0\) and \(\| g\| _{2,P}\le C\| g\| _{1,P}\), then \(1/(4C^2)\le \Pr _P(g\gt 0)\).

For every \(\lambda \gt 0\) and a positive integer \(k\), \((15/\lambda )^{-k}\le \tfrac 14(e^2/(2\lambda ))^{-k}\), where negative powers have real exponents.

For every \(\lambda ,V\gt 0\) and \(I\in \mathbb R\), \(((V(\sqrt\lambda /3)^{I/V})/I)^2=((I/V)^{-1})^2(9/\lambda )^{-I/V}\), with division and inversion taking value zero at zero denominators.

For every \(0\lt \lambda \le 1/2\), \(1/2\le \log (1/\lambda )\le \log (9/\lambda )\le 5\log (1/\lambda )\).

Let \(t\) be a positive real number. Then \(e^{-2t}\le (t^{-1})^2\).

Let \(L,D,x,t\in \mathbb R\) satisfy \(2\le 100L\), \(2(D+2)\le 100L\), \(x\ge 0\), and \(t\gt 0\). Then either \(x/(100Le^x)\le t/e^x\) or \(x/(100Le^x)\le t^{-2}e^{-Dt}\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(f:\Omega \to \mathbb R\), \(J\subseteq [n]\), \(k\in \mathbb N\), \(0\le \rho \le 1\), and \(\tau \ge 0\). If \(I_i(f)\le \tau \) for \(i\notin J\) and \(\sum _{S\ni i}\rho ^{|S|}\mathbb E_Pf_S^2\le I_i(f)^{3/2}\) for every \(i\), then \(\rho ^k\sum _{|S|\le k,\ S\nsubseteq J}\mathbb E_Pf_S^2\le \sqrt\tau \, I(f)\).

Let \(0\lt \lambda \le 1/2\), \(I,\varepsilon \gt 0\), and \(k\in \mathbb N\). Put \(\rho =\sqrt\lambda /3\) and \(\tau =(\varepsilon /(2I))^2\rho ^{2k}\); then \(0\lt \rho \le 1\), \(\tau \gt 0\), \(\sqrt\tau I=(\varepsilon /2)\rho ^k\), and \(I/\tau =(4I^3/\varepsilon ^2)(9/\lambda )^k\).

Let \(0\lt \lambda \le 1/2\), \(0\lt \varepsilon \le 1\), \(\varepsilon \le I\), and \(k\in \mathbb N\) with \(k\le 3I/\varepsilon \). Then \((4I^3/\varepsilon ^2)(9/\lambda )^k\le (1/\lambda )^{100I/\varepsilon }\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(p\ge 1\), \(q\gt 1\), and \(0\le \rho \le 1\). If every coordinate probability law satisfies \(\| \rho g+(1-\rho )\mathbb E_i g\| _{q,i}\le \| g\| _{p,i}\) for every real \(g\) on that coordinate, then \(\| T_\rho f\| _{q,P}\le \| f\| _{p,P}\) for every \(f:\Omega \to \mathbb R\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(T_a h=\sum _{S\subseteq [n]}a^{|S|}h_S\) for every real rate \(a\); \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\); \(\| g\| _{p,i}=(\sum _{a\in \Omega _i}w_i(a)|g(a)|^p)^{1/p}\) and \(\mathbb E_i g=\sum _{a\in \Omega _i}w_i(a)g(a)\), where \(w_i\) gives the probability masses in coordinate \(i\). The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(1\le p\le q\), and \(0\le \rho \le 1\). If every coordinate satisfies \(\| \rho g+(1-\rho )\mathbb E_i g\| _{q,i}\le \| g\| _{p,i}\) for every real coordinate function \(g\), then \(\| T_\rho f\| _{q,P}\le \| f\| _{p,P}\) for every \(f:\Omega \to \mathbb R\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(T_a h=\sum _{S\subseteq [n]}a^{|S|}h_S\) for every real rate \(a\); \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\); \(\| g\| _{p,i}=(\sum _{a\in \Omega _i}w_i(a)|g(a)|^p)^{1/p}\) and \(\mathbb E_i g=\sum _{a\in \Omega _i}w_i(a)g(a)\), where \(w_i\) gives the probability masses in coordinate \(i\). The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).

For every \(A,B\ge 0\) and \(p,q\in \mathbb R\), the inequality \(A^{2/q}\le B^{2/p}\) implies \(A^{1/q}\le B^{1/p}\).

Let \(I\) be a finite set with nonnegative weights \(w_i\) satisfying \(\sum _iw_i=1\), and let \(g:I\to \mathbb R\) and \(0\le \rho \le \sigma \). Then \(\rho ^2\sum _iw_ig_i^2+(1-\rho ^2)(\sum _iw_ig_i)^2\le \sigma ^2\sum _iw_ig_i^2+(1-\sigma ^2)(\sum _iw_ig_i)^2\).

Let \(I\) be finite with weights \(w_i\ge \lambda \), \(\sum _iw_i=1\), \(0\lt \lambda \lt 1/2\), and \(q\gt 2\). Put \(r=\sqrt{\sinh (\log ((1-\lambda )/\lambda )/q)/\sinh (\log ((1-\lambda )/\lambda )/(q/(q-1)))}\). For \(0\le \rho \le r\), the operator \(Ng=\rho g+(1-\rho )\sum _iw_ig_i\) satisfies \(\| Ng\| _{q,w}\le \| g\| _{2,w}\) and \(\| Ng\| _{2,w}\le \| g\| _{q/(q-1),w}\). Here \(\| h\| _{p,w}=(\sum _{i\in I}w_i|h_i|^p)^{1/p}\) for \(p\gt 0\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with every coordinate atom at least \(\lambda \), where \(0\lt \lambda \lt 1/2\), and let \(q\gt 2\). Put \(r=\sqrt{\sinh (\log ((1-\lambda )/\lambda )/q)/\sinh (\log ((1-\lambda )/\lambda )/(q/(q-1)))}\). For \(0\le \rho \le r\) and every \(f:\Omega \to \mathbb R\), \(\| T_\rho f\| _{q,P}\le \| f\| _{2,P}\) and \(\| T_\rho f\| _{2,P}\le \| f\| _{q/(q-1),P}\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(T_a h=\sum _{S\subseteq [n]}a^{|S|}h_S\) for every real rate \(a\); \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\). The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).

Let \(a,b\) be real numbers with \(0\lt a\le b\). Then \((a/b)e^{a-b}\le \sinh (a)/\sinh (b)\).

Let \(q\gt 2\) and \(0\lt \lambda \lt 1\). For \(r_q(t)=\sqrt{\sinh (\log ((1-t)/t)/q)/\sinh (\log ((1-t)/t)/(q/(q-1)))}\), one has \(\lambda ^{1/2-1/q}/\sqrt{q-1}\le r_q(\lambda /(1+\lambda ))\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with every coordinate atom at least \(1\), and let \(q\gt 2\) and \(\rho \in \mathbb R\). For every \(f:\Omega \to \mathbb R\), \(\| T_\rho f\| _{q,P}\le \| f\| _{2,P}\) and \(\| T_\rho f\| _{2,P}\le \| f\| _{q/(q-1),P}\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(T_a h=\sum _{S\subseteq [n]}a^{|S|}h_S\) for every real rate \(a\); \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\). The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with every coordinate atom at least \(\lambda \), where \(0\lt \lambda \le 1\), and let \(q\gt 2\). For \(0\le \rho \le \lambda ^{1/2-1/q}/\sqrt{q-1}\) and every \(f:\Omega \to \mathbb R\), \(\| T_\rho f\| _{q,P}\le \| f\| _{2,P}\) and \(\| T_\rho f\| _{2,P}\le \| f\| _{q/(q-1),P}\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(T_a h=\sum _{S\subseteq [n]}a^{|S|}h_S\) for every real rate \(a\); \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\). The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).

Let \(P\) be a probability law on a finite outcome space \(\Omega \), with every atom of mass at least \(\lambda \gt 0\), where \(0\lt \lambda \le 1\), and let \(q\gt 2\). For \(0\le \rho \le \lambda ^{1/2-1/q}/\sqrt{q-1}\), the operator \(T_\rho f=\rho f+(1-\rho )\mathbb E_Pf\) satisfies \(\| T_\rho f\| _{q,P}\le \| f\| _{2,P}\) and \(\| T_\rho f\| _{2,P}\le \| f\| _{q/(q-1),P}\) for every real \(f\). For a real function \(h\) on \(\Omega \), use \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with coordinate weights \(w_i\), \(S\subseteq [n]\), and \(f:\Omega \to \mathbb R\). Then \(E_Sf(x)=\sum _y[\prod _{i\in S}\mathbf1_{x_i=y_i}\prod _{i\notin S}w_i(y_i)]f(y)\) for every \(x\in \Omega \).

For a finite set \(I\), \(S\subseteq I\), and real functions \(a,b\) on \(I\), \(\sum _{T\subseteq S}(-1)^{|S|-|T|}\prod _{i\in T}a_i\prod _{i\notin T}b_i=\prod _{i\in S}(a_i-b_i)\prod _{i\notin S}b_i\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with coordinate weights \(w_i\), \(S\subseteq [n]\), and \(f:\Omega \to \mathbb R\). Its orthogonal component satisfies \(f_S(x)=\sum _y[\prod _{i\in S}(\mathbf1_{x_i=y_i}-w_i(y_i))\prod _{i\notin S}w_i(y_i)]f(y)\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with coordinate weights \(w_i\), \(\rho \in \mathbb R\), and \(f:\Omega \to \mathbb R\). Then \(T_\rho f(x)=\sum _y[\prod _i(\rho \mathbf1_{x_i=y_i}+(1-\rho )w_i(y_i))]f(y)\) for every \(x\in \Omega \).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(0\le \rho \le 1\), and \(f:\Omega \to \mathbb R\). Then \(\| T_\rho f\| _{1,P}\le \| f\| _{1,P}\).

For every \(q\gt 1\) and \(z\in \mathbb R\), set \(g=z|z|^{q-2}\). Then \(zg=|z|^q\) and \(|g|^{q/(q-1)}=|z|^q\).

Let \(I\) be a finite set with nonnegative weights \(w_i\), and let \(p,q\gt 1\) satisfy \(1/p+1/q=1\). For \(f,g:I\to \mathbb R\), \(\sum _iw_if_ig_i\le (\sum _iw_i|f_i|^p)^{1/p}(\sum _iw_i|g_i|^q)^{1/q}\).

Let \(I\) be a finite set with nonnegative weights \(w_i\), \(q\gt 1\), \(z:I\to \mathbb R\), and \(N\ge 0\). If \(\sum _iw_iz_ig_i\le N(\sum _iw_i|g_i|^{q/(q-1)})^{(q-1)/q}\) for every \(g:I\to \mathbb R\), then \((\sum _iw_i|z_i|^q)^{1/q}\le N\).

Let \(I\) be a finite set with nonnegative weights \(w_i\) and define \(\| f\| _{r,w}=(\sum _iw_i|f_i|^r)^{1/r}\). If an operator \(T\) satisfies \(\sum _iw_i(Tf)_ig_i=\sum _iw_if_i(Tg)_i\) for all \(f,g\) and \(\| Tg\| _{q/(q-1),w}\le \| g\| _{q/(q-1),w}\) for every \(g\), where \(q\gt 1\), then \(\| Tf\| _{q,w}\le \| f\| _{q,w}\) for every \(f\).

Let \(I\) be a finite set with nonnegative weights \(w_i\), \(f:I\to \mathbb R\), and \(q,c\gt 0\). Then \(\sum _{i:c\le |f_i|}w_i\le ((\sum _iw_i|f_i|^q)^{1/q}/c)^q\).

Let \(I\) be a finite set with nonnegative weights \(w_i\) and let \(g:I\to \mathbb R\) satisfy \(\sum _iw_ig_i=0\). Then \((\sum _iw_i|g_i|)^2\le 4(\sum _iw_ig_i^2)(\sum _{i:g_i\gt 0}w_i)\).

Let \(I\) be a finite set with nonnegative weights \(w_i\) with \(\sum _iw_i\gt 0\), let \(d_i\in \mathbb R\), and let \(\rho \gt 0\). Then \((\sum _iw_i)\rho ^{(\sum _iw_id_i)/(\sum _iw_i)}\le \sum _iw_i\rho ^{d_i}\).

Let \(I\) be finite and let \(a_i\ge 0\) have positive sum. If \(A\ge 0\) and \(A\le \sum _i a_i^{3/2}\), then some \(i\in I\) satisfies \((A/\sum _ja_j)^2\le a_i\).

Let \(\alpha \) be finite, let \(E(S)\ge 0\) for every \(S\subseteq \alpha \), let \(J\subseteq \alpha \), \(k\in \mathbb N\), and \(0\le \rho \le 1\). Then \(\rho ^k\sum _{|S|\le k,\ S\nsubseteq J}E(S)\le \sum _{i\notin J}\sum _{S\ni i}\rho ^{|S|}E(S)\).

Let \(\Omega _i\) be finite sets, \(w_i(a)\ge 0\) weights, and \(k_i(a,b)\ge 0\) kernels for \(i\in [n]\). For \(p,r\ge 1\), suppose \(\sum _{a,b}k_i(a,b)u(a)v(b)\le \| u\| _{p,w_i}\| v\| _{r,w_i}\) for every coordinate \(i\) and real \(u,v\). Then \(\sum _{x,y}(\prod _i k_i(x_i,y_i))f(x)g(y)\le \| f\| _{p,w}\| g\| _{r,w}\) for all real \(f,g\) on \(\prod _i\Omega _i\), where \(w(x)=\prod _iw_i(x_i)\) and \(\| h\| _{s,w}=(\sum _xw(x)|h(x)|^s)^{1/s}\).

Let \(\Omega _i\) be finite sets, \(w_i(a)\ge 0\) weights, and \(k_i(y,x)\ge 0\) kernels for \(i\in [n]\). For \(p\ge 1\) and \(q\gt 1\), suppose \(\| \sum _xk_i(\, \cdot \, ,x)h(x)\| _{q,w_i}\le \| h\| _{p,w_i}\) for every coordinate \(i\) and real \(h\). Then the product operator \(Tf(y)=\sum _x\prod _i k_i(y_i,x_i)f(x)\) satisfies \(\| Tf\| _{q,w}\le \| f\| _{p,w}\) for every real \(f\) on \(\prod _i\Omega _i\), with \(w(x)=\prod _iw_i(x_i)\) and \(\| h\| _{s,w}=(\sum _xw(x)|h(x)|^s)^{1/s}\).

For a real function \(X\) on a measure space \((\Omega ,\mu )\) and extended exponents \(p,q\in [0,\infty ]\), \((p,q,\rho )\)-hypercontractivity means \(1\le p\le q\), \(0\le \rho \lt 1\), \(X\in L^q(\mu )\), and \(\| a+\rho bX\| _q\le \| a+bX\| _p\) for every \(a,b\in \mathbb {R}\), with extended-valued norms and the essential supremum at infinity.

For a real function \(X\) on a measure space \((\Omega ,\mu )\) and \(q\in \mathbb {R}\), define \(N_q(X)\) by taking the extended norm at exponent \(\max (q,0)\) and converting it to a real number; a finite value is retained and an infinite value is assigned \(0\).

A real function \(X\) on a measure space \((\Omega ,\mu )\) is symmetric when \(X\) and \(-X\) are almost everywhere measurable and have the same distribution under \(\mu \).

A Rademacher variable on a measure space \((\Omega ,\mu )\) is a measurable \(r:\Omega \to \mathbb {R}\) satisfying \(r\in \{ -1,1\} \) almost everywhere and \(\mu (r=1)=\mu (r=-1)=1/2\).

Let \(F\) assign a real coefficient \(F(S)\) to every \(S\subseteq [n]\), and let \(X_i:\Omega \to \mathbb {R}\). Define \(F(X)(\omega )=\sum _{S\subseteq [n]}F(S)\prod _{i\in S}X_i(\omega )\).

A real function \(X\) on \((\Omega ,\mu )\) has atom lower bound \(\lambda \in \mathbb {R}\) when there exists a finite \(s\subseteq \mathbb {R}\) supporting \(X\) almost everywhere and every \(x\in s\) satisfies \(\mu (X=x)\ge \max (\lambda ,0)\).

Let \(q\gt 0\) and \(X\in L^q(\mu )\) on an arbitrary measure space. Its finite real norm satisfies \(\| X\| _q^q=\int |X|^q\, d\mu \).

Let \((\Omega ,\mu )\) be a probability space, let \(X\in L^2(\mu )\) satisfy \(\mathbb {E}X=0\), and let \(a,b\in \mathbb {R}\). Then \(\| a+bX\| _2^2=a^2+b^2\| X\| _2^2\).

Let \(r\) be a measurable random variable on a probability space, with \(r\in \{ -1,1\} \) almost surely and \(\Pr [r=1]=\Pr [r=-1]=1/2\). Its law is \(\tfrac 12\delta _1+\tfrac 12\delta _{-1}\).

Let \(X\) be centered on a probability space, let \(q\gt 2\), and suppose \(H_{2,q,\rho }(X)\). Then \(H_{q/(q-1),2,\rho }(X)\). Write \(H_{p,q,\rho }(X)\) for \(1\le p\le q\le \infty \), \(0\le \rho \lt 1\), \(X\in L^q(\mu )\), and \(\| a+\rho bX\| _q\le \| a+bX\| _p\) for all \(a,b\in \mathbb {R}\).

Let \(X\) be a real function on an arbitrary measure space, suppose \(H_{p,q,\rho }(X)\), and let \(c\in \mathbb {R}\). Then \(H_{p,q,\rho }(cX)\). Write \(H_{p,q,\rho }(X)\) for \(1\le p\le q\le \infty \), \(0\le \rho \lt 1\), \(X\in L^q(\mu )\), and \(\| a+\rho bX\| _q\le \| a+bX\| _p\) for all \(a,b\in \mathbb {R}\).

Let \(X\) be centered on a probability space, suppose \(H_{p,q,\rho }(X)\), and let \(0\le \sigma \le \rho \). Then \(H_{p,q,\sigma }(X)\), including extended exponents equal to infinity. Write \(H_{p,q,\rho }(X)\) for \(1\le p\le q\le \infty \), \(0\le \rho \lt 1\), \(X\in L^q(\mu )\), and \(\| a+\rho bX\| _q\le \| a+bX\| _p\) for all \(a,b\in \mathbb {R}\).

Let \(\mu \) be a finite measure and \(X:\Omega \to \mathbb {R}\) measurable. If a finite set supports \(X\) almost everywhere and each listed value has mass at least \(\max (\lambda ,0)\) for some real \(\lambda \), then \(X\in L^p(\mu )\) for every \(p\in [0,\infty ]\), including essential boundedness.

Let \(X_1,\ldots ,X_n\) be mutually independent real random variables on a probability space, each belonging to \(L^4\). For arbitrary real coefficients \(F(S)\), the multilinear evaluation \(\sum _{S\subseteq [n]}F(S)\prod _{i\in S}X_i\) belongs to \(L^4\).

Let \(R,a,b,A,D,C\in \mathbb {R}\) satisfy \(R\ge 9\), \(a,b,C,D\ge 0\), \(A\le Ra^2\), \(D\le b^2\), and \(C^2\le AD\). Then \(A+6C+RD\le R(a+b)^2\).

Let \(s\) be a finite subset of an index set \(I\), let \(i\notin s\), let \(F\) assign real coefficients to finite subsets of \(I\), and let \(X_j:\Omega \to \mathbb {R}\). For every \(\omega \in \Omega \), \(\sum _{T\subseteq s\cup \{ i\} }F(T)\prod _{j\in T}X_j(\omega )=\sum _{T\subseteq s}F(T)\prod _{j\in T}X_j(\omega )+X_i(\omega )\sum _{T\subseteq s}F(T\cup \{ i\} )\prod _{j\in T}X_j(\omega )\).

Let \(\mu \) be a finite measure, let \(p,a,b\in \mathbb N\) satisfy \(a+b\le p\), and let real functions \(f,g\in L^p(\mu )\). Then \(f^ag^b\) is integrable.

Let \(q\gt 2\) and \(\lambda \gt 0\). Let \(X\) be a measurable real random variable on a probability space, supported almost surely on a finite set whose listed atoms all have probability at least \(\lambda \). Then \(\| X\| _q\le (1/\lambda )^{1/2-1/q}\| X\| _2\).

Let \(q\gt 2\), \(0\lt \lambda \le 1\), and let \(X\) be measurable and centered on a probability space, supported almost surely on a finite set whose atoms each have mass at least \(\lambda \). Setting \(\rho =\lambda ^{1/2-1/q}/(2\sqrt{q-1})\), both \(H_{2,q,\rho }(X)\) and \(H_{q/(q-1),2,\rho }(X)\) hold. Write \(H_{p,q,\rho }(X)\) for \(1\le p\le q\le \infty \), \(0\le \rho \lt 1\), \(X\in L^q(\mu )\), and \(\| a+\rho bX\| _q\le \| a+bX\| _p\) for all \(a,b\in \mathbb {R}\).

Let \(q\gt 2\), \(0\lt \lambda \le 1\), and let measurable \(X\) have a symmetric law on a probability space, supported almost surely on a finite set whose atoms each have mass at least \(\lambda \). Setting \(\rho =\lambda ^{1/2-1/q}/\sqrt{q-1}\), both \(H_{2,q,\rho }(X)\) and \(H_{q/(q-1),2,\rho }(X)\) hold. Write \(H_{p,q,\rho }(X)\) for \(1\le p\le q\le \infty \), \(0\le \rho \lt 1\), \(X\in L^q(\mu )\), and \(\| a+\rho bX\| _q\le \| a+bX\| _p\) for all \(a,b\in \mathbb {R}\).