Solyanik Estimates in Harmonic Analysis

02/19/2016 - 13:30
02/19/2016 - 14:30
Paul Hagelstein (Baylor)
McGill University, 805 Shrebrooke St. West, Burnside Hall, Burn 920, 9th floor

Let $mathcal{B}$ be a collection of open sets in $mathbb{R}^n$. Associated to $mathcal{B}$ is the geometric maximal operator $M_{mathcal{B}}$ defined by $$M_{mathcal{B}}f(x) = sup_{x in R in mathcal{B}}int_R|f|;.$$ For $0 < alpha < 1$, the associated emph{Tauberian constant} $C_{mathcal{B}}(alpha)$ is given by $$C_{mathcal{B}}(alpha) = sup_{E subset mathbb{R}^n : 0 < |E| < infty} rac{1}{|E|}|{x in mathbb{R}^n : M_{mathcal{B}}chi_E(x) > alpha}|;.$$ A maximal operator $M_mathcal{B}$ such that $lim_{alpha ightarrow 1^-}C_{mathcal{B}}(alpha) = 1$ is said to satisfy a emph{Solyanik estimate}. In this talk we will prove that the uncentered Hardy-Littlewood maximal operator satisfies a Solyanik estimate. Moreover, we will indicate applications of Solyanik estimates to smoothness properties of Tauberian constants and to weighted norm inequalities. We will also discuss several fascinating open problems regarding Solyanik estimates.

This research is joint with Ioannis Parissis.

Last edited by on Fri, 02/12/2016 - 14:55