Superexponential estimates and weighted lower bounds for the square function

TL;DR

This paper establishes a sharp superexponential distribution inequality for the classical dyadic square function in multiple dimensions, extending classical results and deriving new weighted lower bounds using good lambda techniques.

Contribution

It proves a sharp superexponential inequality for the classical dyadic square function in higher dimensions, improving upon previous work that used a different square function.

Findings

Superexponential tail estimate for the dyadic square function in any dimension.

Sharpness of the inequality as dimension tends to infinity.

Weighted and unweighted $L^p$ lower bounds derived from the distribution inequality.

Abstract

We prove the following superexponential distribution inequality: for any integrable $g$ on $[0,1)^{d}$ with zero average, and any $\lambda>0$ \[ |\{ x \in [0,1)^{d} \; :\; g \geq\lambda \}| \leq e^{- \lambda^{2}/(2^{d}\|S(g)\|_{\infty}^{2})}, \] where $S(g)$ denotes the classical dyadic square function in $[0,1)^{d}$. The estimate is sharp when dimension $d$ tends to infinity in the sense that the constant $2^{d}$ in the denominator cannot be replaced by $C2^{d}$ with $0<C<1$ independent of $d$ when $d \to \infty$. For $d=1$ this is a classical result of Chang--Wilson--Wolff [4]; however, in the case $d>1$ they work with a special square function $S_\infty$, and their result does not imply the estimates for the classical square function. Using good $\lambda$ inequalities technique we then obtain unweighted and weighted $L^p$ lower bounds for $S$; to get the corresponding good $\lambda$ inequalities we need to modify the classical construction. We also show how to obtain our superexponential distribution inequality (although with worse constants) from the weighted $L^2$ lower bounds for $S$, obtained in [5].