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Seminar on Analysis, Geometry, and PDEs


Tid: 2021-04-13 15:15 till 16:15
Plats: This talk is given on Zoom. Email the organizer to receive an invitation.
Kontakt: eskil [dot] rydhe [at] math [dot] lu [dot] se
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The L^p-improving problem for Radon averages and Affine Invariant Measures: recent progress

Speaker: Marco Vitturi (University College Cork)

Abstract: The study of L^p-improving estimates for averages over surfaces (Radon averages) has witnessed some big advancements in recent times (due to P. Gressman): 

  • on the one hand we now have a general recipe for constructing weighted surface measures that are the ideal candidate for obtaining best-possible L^p-improving estimates (for the associated weighted Radon averages); 
  • on the other hand, we have methods for proving such estimates that are conditional on certain other weights being bounded away from zero.

The latter weight is generally unrelated to the former unfortunately, but the two happen to coincide in particular cases: a direct calculation shows that this is the case when the surface has codimension 1. In joint work with S. Dendrinos and A. Mustata we have established this is also the case for surfaces of codimension 2 in R^{2d}. The proof uses Geometric Invariant Theory techniques quite heavily. 

In the talk I will review the two weights above and explain how they can be related to each other in this special case of codimension 2 (in R^{2d}).