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 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 \(\operatorname {Var}(f)\rho ^{I(f)/\operatorname {Var}(f)}\le \sum _{S\subseteq [n]}|S|\rho ^{|S|}\widehat f(S)^2\).

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.