Solyanik Estimates in Harmonic Analysis

02/19/2016 - 13:30
02/19/2016 - 14:30
Speaker: 
Paul Hagelstein, Baylor
Location: 
McGill, Burnside Hall, 805 Sherbrooke Str West, Room 920
Abstract: 

 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.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 - 12:29