Square functions and the Hamming cube: duality

A Volberg,F Nazarov,Paata Ivanisvili

doi:10.19086/da.3113

Abstract

Square functions and the Hamming cube: duality, Discrete Analysis 2018:1, 18 pp. This paper establishes a duality between, on the one hand, continuous objects such as square functions and stochastic processes in analysis, and on the other hand, discrete phenomena on the Hamming cube $\{−1, 1\}^n$ with large $n$. More specifically, there is a correspondence between estimates for square functions on the interval $[0,1]$ and discrete gradient estimates for functions on the Hamming cube. The rough idea of a square-function estimate is to cut a function $f$ up into pieces $f_i$ and consider instead the function $Sf=(\sum_i|f_i|^2)^{1/2}$, which turns out to be advantageous in many contexts. As for discrete gradient estimates, they are measures of how much the value of a function $f$ at an average point $x$ differs from the values at the neighbours of $x$. The paper demonstrates a strong connection between two different areas of harmonic analysis that could potentially have a further significant impact on both of them. The main result establishes a Poincare-type inequality for functions on the Hamming cube. Related estimates have been known previously (due to e.g. Naor and Schechtman), but the method used here is new, and in addition to offering a unified approach to a variety of related questions it also provides improved constants in important special cases. The authors show the versatility of their approach by applying it to several well known inequalities on both sides, such as the Sobolev, log-Sobolev, and Chang-Wilson-Wolff inequalities. They also improve a well-known inequality of Beckner for functions on the Hamming cube. The proofs, and the duality, are based on convexity in the guise of Bellman functions (a variant of the Legendre transform). Roughly speaking, one can start with a square function estimate, set up an appropriate Bellman function (the authors use a construction due to B. Davis and used also by G. Wang), dualize it using a minimax procedure to obtain a new “dual” Bellman function, and pass from there to an estimate on the Hamming cube. A similar argument works in the other direction.

Highlights

Our approach is based on a certain duality between the classical square function estimates on the Euclidean space and the gradient estimates on the Hamming cube
We think that the main contribution of the current paper is not just Theorem 1.1 that we obtain but rather a new duality approach that we develop between two different classes of extremal problems: square function estimates on the interval [0, 1] and gradient estimates on the Hamming cube, and Theorem 1.1 should be considered as an example
Speaking one can take a valid estimate for a square function, dualize it by a certain double Legendre transform, and one can write its corresponding dual estimate on the Hamming cube and vice versa

Summary

Main result

Consider the Hamming cube {−1, 1}n of an arbitrary dimension n ≥ 1. The estimate (1) improves the Naor–Schechtman bound [15] for the class of real valued functions for all 1 < p < 2 It follows from an application of Khinchin inequality with the sharp constant and (1) that we have the following corollary. Speaking one can take a valid estimate for a square function, dualize it by a certain double Legendre transform, and one can write its corresponding dual estimate on the Hamming cube and vice versa

An anonymous Bellman function

Going from U to M: from Square function to the Hamming cube

Going from M to U: from Hamming cube to square function

The dual to Log-Sobolev is Chang–Wilson–Wolff

Sobolev inequalities

Gaussian measure on Rn

Discrete surface measure

A Appendix

Heat inequality

Minimax theorem for noncompact sets