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Complex Analysis and Geometry Seminar

Spherical maximal functions and fractal dimensions of dilation sets

Joris Roos (U Mass-Lowell)

Location:  Hill Center 705
Date & time: Friday, 14 April 2023 at 10:30AM - 11:30PM

Abstract: This talk is about subtle properties of averages on spheres in two and higher-dimensional Euclidean space.Maximal spherical averages are a classic topic in harmonic analysis originating in questions on differentiability properties of functions. We consider maximal spherical averages with a supremum taken over a given dilation set. It turns out that the sharp Lp improving properties of such operators are closely related to fractal dimensions of the dilation set such as the Minkowski andAssouad dimensions. This leads to a surprising characterization of the closed convex sets which can occur as closure of the sharp Lp improving region of such a maximal operator. 

This is joint work with Andreas Seeger. If time allows we will also mention recent work on the Heisenberg group and work in progress on the variational setting.

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