  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].
