Maximal operators in a fractal setting and geometric applications

Analysis Seminar
Wednesday, April 6, 2016 - 2:05pm for 1 hour (actually 50 minutes)
Skiles 005
Krystal Taylor – Ohio State University
Michael Lacey
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?