Maximal operators in a fractal setting and geometric applications

Analysis Seminar
Wednesday, April 6, 2016 - 2:05pm
1 hour (actually 50 minutes)
Skiles 005
Ohio State University
 We use Fourier analysis to establish $L^p$ bounds for Stein's  spherical    maximal theorem in the setting of compactly supported Borel measures $\mu, \nu$   satisfying natural local size assumptions $\mu(B(x,r)) \leq Cr^{s_{\mu}}, \nu(B(x,r)) \leq Cr^{s_{\nu}}$.  As an application, we address the following geometric problem: Suppose that $E\subset \mathbb{R}^d$ is a union of translations of the unit circle, $\{z \in \mathbb{R}^d: |z|=1\}$, by points in a set $U\subset \mathbb{R}^d$.  What are the minimal assumptions on the set $U$ which guarantee that the $d-$dimensional Lebesgue measure of $E$ is positive?