# Superexponential estimates for the dyadic square function and lower bounds for maximal functions.

# Superexponential estimates for the dyadic square function and lower bounds for maximal functions.

I will speak about two topics. We will find the bounds on the measure of upper-level sets of an integrable function on the n-dimensional unit cube [0,1]^n under the assumption that its classical dyadic square function is uniformly bounded by 1. For n=1 this is the classical result of Chang--Wilson--Wolff. For n>1, our estimate is new, and it depends on the dimension n in a sharp way as n goes to infinity (joint work with S. Treil).

In the second part I will speak about lower bounds for maximal functions in any dimension. Is it true that for any finite p>1 there exists a constant C(p)>1 such that ||Mf||_p > C(p) ||f||_p for any f from Lp space where Mf denotes the centered maximal function defined over translations and dilations of a fixed centrally symmetric convex body? Uncentered maximal function Mf? ``Almost centered'' maximal function? Does it depend on dimension? We will answer to some of these questions (joint work with B. Jaye and F. Nazarov).