Square functions and the Hamming cube: duality

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 Poincaré-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.