Let \(n\ge 0\) and equip \(\{ 0,1\} ^n\) with its uniform probability measure; write \(\mathbb E\) for uniform expectation. Write \(x^{\{ i\} }\) for \(x\) with bit \(i\) flipped, \(I_i(h)=\mathbb E[(h(x)-h(x^{\{ i\} }))^2]/4\), and \(I(h)=\sum _i I_i(h)\). For every real function \(h\) on the cube and coordinate \(i\in [n]\), \(I_i(|h|)\le I_i(h)\).

Let \(n\ge 0\) and equip \(\{ 0,1\} ^n\) with its uniform probability measure; write \(\mathbb E\) for uniform expectation. For \(S\subseteq [n]\), put \(\chi _S(x)=\prod _{i\in S}(-1)^{x_i}\) and \(\widehat h(S)=\mathbb E[h\chi _S]\) for real cube functions \(h\). Write \(x^{\{ i\} }\) for \(x\) with bit \(i\) flipped, \(I_i(h)=\mathbb E[(h(x)-h(x^{\{ i\} }))^2]/4\), and \(I(h)=\sum _i I_i(h)\). For every real cube function \(g\), \(2(\mathbb E[g^2]-(\mathbb Eg)^2)\le I(g)+\sum _{|S|=1}\widehat g(S)^2\).

Let \(n\ge 0\) and equip \(\{ 0,1\} ^n\) with its uniform probability measure; write \(\mathbb E\) for uniform expectation. For \(S\subseteq [n]\), put \(\chi _S(x)=\prod _{i\in S}(-1)^{x_i}\) and \(\widehat h(S)=\mathbb E[h\chi _S]\) for real cube functions \(h\). Let \(g\) be a real cube function and \(a\in \mathbb R\). If \(|g(x)-g(\bar x)|\le 2|a|\) for every \(x\), where \(\bar x_i=1-x_i\), then \(\sum _{|S|=1}\widehat g(S)^2\le a^2\).

Let \(n\ge 0\) and equip \(\{ 0,1\} ^n\) with its uniform probability measure; write \(\mathbb E\) for uniform expectation. For \(S\subseteq [n]\), put \(\chi _S(x)=\prod _{i\in S}(-1)^{x_i}\) and \(\widehat h(S)=\mathbb E[h\chi _S]\) for real cube functions \(h\). Let \(P(x)=\sum _{S\subseteq [n]}a_S\chi _S(x)\) with real coefficients \(a_S\). Suppose \(a_S=0\) whenever \(|S|\gt 1\). For every \(x\in \{ 0,1\} ^n\), \(P(x)+P(\bar x)=2a_\varnothing \), where \(\bar x_i=1-x_i\).

Let \(n\ge 0\) and equip \(\{ 0,1\} ^n\) with its uniform probability measure; write \(\mathbb E\) for uniform expectation. For \(S\subseteq [n]\), put \(\chi _S(x)=\prod _{i\in S}(-1)^{x_i}\) and \(\widehat h(S)=\mathbb E[h\chi _S]\) for real cube functions \(h\). Let \(P(x)=\sum _{S\subseteq [n]}a_S\chi _S(x)\) with real coefficients \(a_S\). Write \(x^{\{ i\} }\) for \(x\) with bit \(i\) flipped, \(I_i(h)=\mathbb E[(h(x)-h(x^{\{ i\} }))^2]/4\), and \(I(h)=\sum _i I_i(h)\). If \(a_S=0\) for \(|S|\gt 1\), then \(I(P)+a_\varnothing ^2=\sum _S a_S^2\).