Strong Contraction and Influences in Tail Spaces

TL;DR

This paper investigates contraction properties and influence bounds for functions in tail spaces of the Boolean cube, proving new inequalities and providing counterexamples to extensions of classical theorems.

Contribution

It establishes an $L^{p}$ Poincaré inequality and moment decay estimates for tail space functions, and constructs counterexamples to certain influence bounds.

Findings

Proved an $L^{p}$ Poincaré inequality for tail space functions.

Established moment decay estimates for mean zero tail space functions.

Constructed functions with vanishing low-level Fourier coefficients and bounded influences, countering previous conjectures.

Abstract

We study contraction under a Markov semi-group and influence bounds for functions in $L^2$ tail spaces, i.e. functions all of whose low level Fourier coefficients vanish. It is natural to expect that certain analytic inequalities are stronger for such functions than for general functions in $L^2$. In the positive direction we prove an $L^{p}$ Poincar\'{e} inequality and moment decay estimates for mean $0$ functions and for all $1<p<\infty$, proving the degree one case of a conjecture of Mendel and Naor as well as the general degree case of the conjecture when restricted to Boolean functions. In the negative direction, we answer negatively two questions of Hatami and Kalai concerning extensions of the Kahn-Kalai-Linial and Harper Theorems to tail spaces. That is, we construct a function $f\colon\{-1,1\}^{n}\to\{-1,1\}$ whose Fourier coefficients vanish up to level $c \log n$, with all influences bounded by $C \log n/n$ for some constants $0<c,C< \infty$. We also construct a function $f\colon\{-1,1\}^{n}\to\{0,1\}$ with nonzero mean whose remaining Fourier coefficients vanish up to level $c' \log n$, with the sum of the influences bounded by $C'(\mathbb{E}f)\log(1/\mathbb{E}f)$ for some constants $0<c',C'<\infty$.