Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(f:\Omega \to \mathbb R\), and \(x\in \Omega \). If \(R_rf(x)=\sum _S\chi _S(r)f_S(x)\) and \(r\) is uniform on the Boolean cube, then \(\mathbb E_r(R_rf(x))^2=\sum _Sf_S(x)^2\). 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); \(\chi _S(r)=\prod _{i\in S}(-1)^{r_i}\) for \(r\in \{ 0,1\} ^n\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) and \(f:\Omega \to \mathbb R\). Then \((\mathbb E_{x\sim P}\mathbb E_r|\sum _S\chi _S(r)f_S(x)|^2)^{1/2}=\| f\| _{2,P}\), where \(r\) is uniform on the Boolean cube. 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 \(p\gt 0\); \(\chi _S(r)=\prod _{i\in S}(-1)^{r_i}\) for \(r\in \{ 0,1\} ^n\).

Let \(f:\{ 0,1\} ^n\to \mathbb R\) be a function on the uniform Boolean cube, \(k\in \mathbb N\), and \(q\gt 1\). The Fourier projection \(f^{\le k}\) satisfies \(\| f^{\le k}\| _q\le (\sqrt{q-1})^k\| f\| _q\) when \(q\ge 2\), and \(\| f^{\le k}\| _q\le (1/\sqrt{q-1})^k\| f\| _q\) when \(q\le 2\). For a real function \(h\) on the cube, use \(\chi _S(x)=\prod _{i\in S}(-1)^{x_i}\), \(\widehat h(S)=\mathbb E_x[h(x)\chi _S(x)]\), \(h^{\le k}=\sum _{|S|\le k}\widehat h(S)\chi _S\), \(\| h\| _p=(\mathbb E_x|h(x)|^p)^{1/p}\), with uniform expectation \(\mathbb E_x\) and natural powers \(0^0=1\).

Let \(f:\{ 0,1\} ^n\to \mathbb R\) be a function on the uniform Boolean cube, \(k\in \mathbb N\), and \(0\lt \rho \le 1\). Then \(\| f^{\le k}\| _4\le (\sqrt3/\rho )^k\| T_\rho f\| _4\). For a real function \(h\) on the cube, use \(\chi _S(x)=\prod _{i\in S}(-1)^{x_i}\), \(\widehat h(S)=\mathbb E_x[h(x)\chi _S(x)]\), \(h^{\le k}=\sum _{|S|\le k}\widehat h(S)\chi _S\), \(\| h\| _p=(\mathbb E_x|h(x)|^p)^{1/p}\), \(T_a h=\sum _{S\subseteq [n]}a^{|S|}\widehat h(S)\chi _S\), with uniform expectation \(\mathbb E_x\) and natural powers \(0^0=1\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(f:\Omega \to \mathbb R\), and \(r\in \{ 0,1\} ^n\). For \(s_i=(-1)^{r_i}\), \(R_rf(x)=T_{\boldsymbol s}f(x)\) for every \(x\in \Omega \). 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); \(\chi _S(r)=\prod _{i\in S}(-1)^{r_i}\) for \(r\in \{ 0,1\} ^n\); \(R_rh=\sum _{S\subseteq [n]}\chi _S(r)h_S\); \(T_{\boldsymbol s}h=\sum _{S\subseteq [n]}(\prod _{i\in S}s_i)h_S\) for real coordinate rates \(s_i\).

For every \(S\subseteq [n]\) and uniform \(r\in \{ 0,1\} ^n\), let \(M(r)=\{ i:r_i=0\} \) and \(\chi _S(r)=\prod _{i\in S}(-1)^{r_i}\). Then \(\mathbb E_r[\mathbf1_{S\subseteq M(r)}\chi _S(r)]=2^{-|S|}\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(f:\Omega \to \mathbb R\), and \(x\in \Omega \). For uniform \(r\in \{ 0,1\} ^n\), \(R_rf=\sum _S\chi _S(r)f_S\), and \(M(r)=\{ i:r_i=0\} \), one has \(\mathbb E_r[E_{M(r)}(R_rf)(x)]=T_{1/2}f(x)\). 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 \(a\); \(\chi _S(r)=\prod _{i\in S}(-1)^{r_i}\) for \(r\in \{ 0,1\} ^n\); \(R_rh=\sum _{S\subseteq [n]}\chi _S(r)h_S\). 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 \(\Omega =\prod _i\Omega _i\), \(f:\Omega \to \mathbb R\), and \(q\ge 1\). Then \(\| T_{1/2}f\| _{q,P}\le (\mathbb E_{x\sim P}\mathbb E_r|\sum _S\chi _S(r)f_S(x)|^q)^{1/q}\), where \(r\) is uniform on the Boolean cube. 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 \(p\gt 0\); \(\chi _S(r)=\prod _{i\in S}(-1)^{r_i}\) for \(r\in \{ 0,1\} ^n\). The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).

For every \(q\ge 2\) there exists \(0\lt c\le 1\), with \(c=2/5\) if \(q=4\), such that every probability space \((\Omega ,\mu )\), every real \(X\in L^q(\mu )\) of mean zero, and every real \(a\) satisfy \(\| a-cX\| _q\le \| a+X\| _q\). Here \(\| Y\| _q=(\int _\Omega |Y|^q\, d\mu )^{1/q}\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(c\in \mathbb R\), \(f:\Omega \to \mathbb R\), and \(r\in \{ 0,1\} ^n\). Then \(R_r(T_cf)=T_{\boldsymbol s}f\) for \(s_i=c(-1)^{r_i}\). 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); \(T_a h=\sum _{S\subseteq [n]}a^{|S|}h_S\) for every real rate \(a\); \(\chi _S(r)=\prod _{i\in S}(-1)^{r_i}\) for \(r\in \{ 0,1\} ^n\); \(R_rh=\sum _{S\subseteq [n]}\chi _S(r)h_S\); \(T_{\boldsymbol s}h=\sum _{S\subseteq [n]}(\prod _{i\in S}s_i)h_S\) for real coordinate rates \(s_i\). 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 \(\Omega =\prod _i\Omega _i\), \(f,g:\Omega \to \mathbb R\), and \(q\gt 0\). If \(\| R_rf\| _{q,P}\le \| g\| _{q,P}\) for every sign vector \(r\), then \((\mathbb E_{x\sim P}\mathbb E_r|R_rf(x)|^q)^{1/q}\le \| g\| _{q,P}\), with uniform signs \(r\). 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); \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\); \(\chi _S(r)=\prod _{i\in S}(-1)^{r_i}\) for \(r\in \{ 0,1\} ^n\); \(R_rh=\sum _{S\subseteq [n]}\chi _S(r)h_S\).

For every \(q\gt 1\) there exists \(0\lt c\le 1\), with \(c=2/5\) if \(q\in \{ 4,4/3\} \), such that for every product \(P\) of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) and every real \(f\) on its outcome space, \((\mathbb E_{x\sim P}\mathbb E_r|R_r(T_cf)(x)|^q)^{1/q}\le \| f\| _{q,P}\), where \(r\) is uniform on the Boolean cube. 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 \(p\gt 0\); \(\chi _S(r)=\prod _{i\in S}(-1)^{r_i}\) for \(r\in \{ 0,1\} ^n\); \(R_rh=\sum _{S\subseteq [n]}\chi _S(r)h_S\). 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 \(\Omega =\prod _i\Omega _i\), \(\rho \ne 0\), and \(f:\Omega \to \mathbb R\). Then \(T_\rho (T_{1/\rho }f)=f\). 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); \(T_a h=\sum _{S\subseteq [n]}a^{|S|}h_S\) for every real rate \(a\). 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 \(\Omega =\prod _i\Omega _i\), \(\rho \in \mathbb R\), and \(f:\Omega \to \mathbb R\). Anisotropic noise with every coordinate rate equal to \(\rho \) satisfies \(T_{(\rho ,\ldots ,\rho )}f(x)=T_\rho f(x)\) for every \(x\in \Omega \). 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 \(a\); \(T_{\boldsymbol s}h=\sum _{S\subseteq [n]}(\prod _{i\in S}s_i)h_S\) for real coordinate rates \(s_i\). The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).

For every \(q\ge 2\) there exists \(C\ge 1\), with \(C=5\sqrt3\) if \(q=4\), such that for every product \(P\) of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), every \(k\in \mathbb N\), and every real \(f\) on its outcome space, \(\| f^{\le k}\| _{q,P}\le C^k\| f\| _{q,P}\). 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 \(p\gt 0\); \(h^{\le k}=\sum _{|S|\le k}h_S\).

For every \(q\gt 1\) there exists \(C\ge 1\), with \(C=5\sqrt3\) when \(q\in \{ 4,4/3\} \), such that for every product \(P\) of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), every \(k\in \mathbb N\), and every real \(f\) on its outcome space, \(\| f^{\le k}\| _{q,P}\le C^k\| f\| _{q,P}\). 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 \(p\gt 0\); \(h^{\le k}=\sum _{|S|\le k}h_S\).