Superexponential estimates and weighted lower bounds for the square function

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, and Wolff; however, in the case of $d>1$ they work with special square function $S_\infty$, and their result does not imply the estimates for the classical square function.

Using a 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 by Domelevo et al.