Refreshments will be served at 2:30 p.m. in 302 GP.

The presentation will begin at 3 p.m. in 155 Gordon Palmer Hall

Solyanik Estimates in Harmonic Analysis

Let $latex \mathcal{B}$ be a collection of open sets in $latex \mathbb{R}^n$.

Associated to $latex \mathcal{B}$ is the geometric maximal operator $latex M_{\mathcal{B}}$ defined by

$latex M_{\mathcal{B}}f(x) = \sup_{x \in R \in \mathcal{B}}\int_R|f|\;.$

For $latex 0 < \alpha < 1$, the associated Tauberian constant $latex C_{\mathcal{B}}(\alpha)$ is given by
$latex C_{\mathcal{B}}(\alpha) = \sup_{E \subset \mathbb{R}^n : 0 < |E| < \infty} \frac{1}{|E|}|\{x \in \mathbb{R}^n : M_{\mathcal{B}}\chi_E(x) > \alpha\}|\;.$

A maximal operator $latex M_\mathcal{B}$ such that $latex \lim_{\alpha \rightarrow 1^-}C_{\mathcal{B}}(\alpha) = 1$ is said to satisfy a 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.