The uncentered spherical maximal function and Nikodym sets

Abstract: Stein's spherical maximal function is an analogue of the Hardy-Littlewood maximal function, where the averages are taken over spheres instead of balls. While the uncentered Hardy-Littlewood maximal function is bounded on Lp for all p>1 and pointwise equivalent to its centered counterpart, the corresponding uncentered spherical maximal function is not as well-behaved.

We provide multidimensional versions of the Kakeya, Nikodym,and Besicovitch constructions associated with spheres. These yield counterexamples indicating that maximal operators given by translations of spherical averages are unbounded on Lp for all finite p.

However, for lower-dimensional sets of translations, we obtain Lp boundedness for the associated maximally translated spherical averages for a certain range of p that depends on the Minkowski dimension of the set of translations. This is joint work with A. Chang and J. Kim.

analysis of PDEsclassical analysis and ODEsfunctional analysismetric geometry

Audience: researchers in the topic

HA-GMT-PDE Seminar

Series comments: Description: A senior graduate student/postdoc series in harmonic analysis, geometric measure theory, and partial differential equations.

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