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].
Categories: math.AP math.CA math.PR