Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(f:\Omega \to \mathbb R\), \(r\in \{ 0,1\} ^n\), and \(x\in \Omega \). Define \(R_rf(x)=\sum _{S\subseteq [n]}\chi _S(r)f_S(x)\), where \(f_S\) is the orthogonal component and \(\chi _S(r)=\prod _{i\in S}(-1)^{r_i}\).

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 \(f:\Omega \to \mathbb R\). For \(R_rf(x)=\sum _S\chi _S(r)f_S(x)\) and uniform independent \(r\in \{ 0,1\} ^n\), define \(\| Rf\| _q=(\mathbb E_{x\sim P}\mathbb E_r|R_rf(x)|^q)^{1/q}\) using real powers.

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 \(i\), \(\rho \in \mathbb R\), and \(f:\Omega \to \mathbb R\). Define \(N_{i,\rho }f=f-L_if+\rho L_if\), where \(L_i\) 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 \(\Omega =\prod _i\Omega _i\), rates \(r_i\in \mathbb R\), and \(f:\Omega \to \mathbb R\). Define \(T_{\boldsymbol r}f(x)=\sum _S(\prod _{i\in S}r_i)f_S(x)\) using the orthogonal components \(f_S\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(f:\Omega \to \mathbb R\), a coordinate \(i\), and \(x\in \Omega \). Define \(\mathcal I_i(f,x)=\sum _{S\ni i}f_S(x)^2\) using the orthogonal components of \(f\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(f:\Omega \to \mathbb R\), \(\tau \in \mathbb R\), and \(x\in \Omega \). Define \(J_\tau (f,x)=\{ i\in [n]:\tau \le \sum _{S\ni i}f_S(x)^2\} \).

For every \(J\subseteq [n]\) and \(k\in \mathbb R\), define \(\mathcal F(J,k)=\{ S\subseteq J:|S|\le k\} \), comparing the natural cardinality with the real cutoff.