288 Boolean Analysis — Comparison
288.1 Overview
Weighted interpolation and the sharp noise radius give quantitative comparison of the first and second norms. Scalar estimates isolate the explicit constants used in finite-product anticoncentration, KKL bounds, and junta approximation.
288.2 Declarations
Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(f:\Omega \to \mathbb R\) with variance \(V\gt 0\), and \(\rho \gt 0\). Writing \(I=\sum _iI_i(f)\) and \(f_S\) for the orthogonal components, \(V\rho ^{I/V}\le \sum _S|S|\rho ^{|S|}\mathbb E_Pf_S^2\).
Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(q\gt 2\), \(c\in \mathbb R\), and \(f:\Omega \to \mathbb R\). If \(\| f\| _{q,P}\le e^c\| f\| _{2,P}\), then \(\| f\| _{2,P}\le e^{cq/(q-2)}\| f\| _{1,P}\).
For every \(u\gt 0\), the real function \(q\mapsto \tfrac 12[\log \sinh (u/q)-\log \sinh (u-u/q)]\) has derivative \(-u\cosh (u/2)/(4\sinh (u/2))\) at \(q=2\).
For every \(0\lt \lambda \lt 1/2\), \(\cosh (\tfrac 12\log ((1-\lambda )/\lambda ))/\sinh (\tfrac 12\log ((1-\lambda )/\lambda ))=1/(1-2\lambda )\).
Let \(0\lt \lambda \lt 1/2\), define \(r_q(\lambda )=\sqrt{\frac{\sinh (\log ((1-\lambda )/\lambda )/q)}{\sinh (\log ((1-\lambda )/\lambda )/(q/(q-1)))}}\), and put \(C_\lambda =((1-\lambda )/\lambda )^{1/(2(1-2\lambda ))}\). Then \(\lim _{q\to 2^+}\exp (-q\log r_q(\lambda )/(q-2))=C_\lambda \).
For every real \(\lambda \), define \(C_\lambda =\begin{cases} e,& \lambda =1/2,\\ ((1-\lambda )/\lambda )^{1/(2(1-2\lambda ))},& \lambda \ne 1/2.\end{cases}\). Then \(\lim _{\lambda \to (1/2)^-,\ 0\lt \lambda \lt 1/2}C_\lambda =e\).
Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) and let \(g:\Omega \to \mathbb R\) have mean zero and positive second moment. If \(C\gt 0\) and \(\| g\| _{2,P}\le C\| g\| _{1,P}\), then \(1/(4C^2)\le \Pr _P(g\gt 0)\).
For every \(\lambda \gt 0\) and a positive integer \(k\), \((15/\lambda )^{-k}\le \tfrac 14(e^2/(2\lambda ))^{-k}\), where negative powers have real exponents.
For every \(\lambda ,V\gt 0\) and \(I\in \mathbb R\), \(((V(\sqrt\lambda /3)^{I/V})/I)^2=((I/V)^{-1})^2(9/\lambda )^{-I/V}\), with division and inversion taking value zero at zero denominators.
For every \(0\lt \lambda \le 1/2\), \(1/2\le \log (1/\lambda )\le \log (9/\lambda )\le 5\log (1/\lambda )\).
Let \(t\) be a positive real number. Then \(e^{-2t}\le (t^{-1})^2\).
Let \(L,D,x,t\in \mathbb R\) satisfy \(2\le 100L\), \(2(D+2)\le 100L\), \(x\ge 0\), and \(t\gt 0\). Then either \(x/(100Le^x)\le t/e^x\) or \(x/(100Le^x)\le t^{-2}e^{-Dt}\).
Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(f:\Omega \to \mathbb R\), \(J\subseteq [n]\), \(k\in \mathbb N\), \(0\le \rho \le 1\), and \(\tau \ge 0\). If \(I_i(f)\le \tau \) for \(i\notin J\) and \(\sum _{S\ni i}\rho ^{|S|}\mathbb E_Pf_S^2\le I_i(f)^{3/2}\) for every \(i\), then \(\rho ^k\sum _{|S|\le k,\ S\nsubseteq J}\mathbb E_Pf_S^2\le \sqrt\tau \, I(f)\).
Let \(0\lt \lambda \le 1/2\), \(I,\varepsilon \gt 0\), and \(k\in \mathbb N\). Put \(\rho =\sqrt\lambda /3\) and \(\tau =(\varepsilon /(2I))^2\rho ^{2k}\); then \(0\lt \rho \le 1\), \(\tau \gt 0\), \(\sqrt\tau I=(\varepsilon /2)\rho ^k\), and \(I/\tau =(4I^3/\varepsilon ^2)(9/\lambda )^k\).
Let \(0\lt \lambda \le 1/2\), \(0\lt \varepsilon \le 1\), \(\varepsilon \le I\), and \(k\in \mathbb N\) with \(k\le 3I/\varepsilon \). Then \((4I^3/\varepsilon ^2)(9/\lambda )^k\le (1/\lambda )^{100I/\varepsilon }\).