IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

Riesz transforms and fractional integrations for weighted orthogonal expansions on the unit sphere.

  • Nov. 8, 2016
  • 4:15 p.m.
  • LeConte 312

Abstract

I will report some recent progress on weighted orthogonal polynomial expansions (WOPEs) on the unit sphere of $\mathbb{R}^d$ with weights being invariant under a general finite reflection group introduced by C F Dunkl. For many problems in this area, one of the main difficulties comes from the fact that explicit closed formulas for reproducing kernels of the weighted orthogonal polynomial spaces are not available unless the finite reflection group is Abelian (in which case, a very useful formula was proved previously by Yuan Xu). In this talk, I will first show sharp estimates of Christoffel functions with doubling weights combined with a deep result of M. R\"osler on Dunkl's intertwining operators can be used to establish highly localized estimates for certain polynomial kernels in WOPEs. Secondly, I will show how these obtained estimates can be applied to determine the sharp power of the fractional Dunkl-Laplace-Beltrami operator for which the weighted Hardy-Littlewood-Sobolev inequality holds. Thirdly, I will show a new decomposition of the Dunkl-Laplace-Beltrami operator for the WOPEs using angular derivatives with respect to Euler angles and certain differential-difference operators which are easier to compute. Such a decomposition combined with the theory of fractional integration is used to introduce Riesz transforms on weighted spheres. Furhermore, the above mentioned localized polynomial kernel estimates allow us to show that these Riesz transforms enjoy properties similar to those of the classical Riesz transforms on the Euclidean space, including the $L^p$ boundedness. Finally, I will discuss briefly how these results can be used to establish similar results for WOPEs on the balls and simplexes. This is based on a method developed earlier by Yuan Xu. Our result on the ball extends a classical inequality of Muckenhoupt and Stein on conjugate ultraspherical polynomial expansions.

This is a joint work with Han Feng (University of Oregon).

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