Understanding affine symmetry in geometric harmonic analysis

Abstract: Many geometrically-inspired problems of interest in harmonic analysis, including $L^p$-improving properties of Radon-like transforms and appropriate formulations of the Fourier restriction problem, exhibit natural affine invariance. In some cases, this can mean that the geometry of submanifolds viewed as subsets of Euclidean space fails to capture important subtleties in much the same way that understanding the size of the entries of a matrix fails to capture its nondegeneracy properties. In this talk, I will discuss how these ideas relate to recent work on characterizing the $L^p-L^q$ norm of Radon-like transforms for pairs $(1/p,1/q)$ on a natural scaling line. Along the way, natural connections to Geometric Invariant Theory will be described. This work is part of a broader project to understand other geometrically-flavored nonnegative operators related to fractional integration; some initial work on this broader project will be presented as time permits.

analysis of PDEsclassical analysis and ODEsfunctional analysis

Audience: researchers in the topic

OARS Online Analysis Research Seminar

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