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

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Paata Ivanisvili, Princeton University
Fine Hall 314

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).