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Colloquium – Joris Roos, University of Wisconsin-Madison
January 24 @ 11:00 am - 12:00 pm
Topic: Maximal spherical averages and fractal dimensions
Abstract: Maximal spherical averages are a classical topic in real harmonic analysis arising from the study of differentiability properties of functions, going back to works of Stein and Bourgain.
In this talk we’ll consider maximal averages over spheres, where the radii are taken from a given fractal set. As is typical for averaging operators, these operators improve local integrability of input functions. We would like to know by exactly how much and how this depends on the set of radii. More precisely, we are interested in optimal $L^p$ improving properties, i.e. the sharp range of $(1/p, 1/q)$ such that $L^p\to L^q$ boundedness holds. It turns out that this range depends on various fractal dimensions of the set of radii, such as Minkowski and Assouad dimensions and the Assouad spectrum. One of our main results is a complete characterization of convex sets that arise as sharp $L^p$ improving region of such a spherical maximal operator, up to endpoints. A surprising feature is that these regions may be non-polygonal.
An application of the $L^p$ improving properties are sparse and weighted $L^p$ estimates for an associated global spherical maximal operator extending recent results of Lacey.
Based on joint works with A. Seeger and with T. Anderson, K. Hughes, A. Seeger.