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.