Yao Liu

First Position

Dissertation

Riesz Distributions Assiciated to Dunkl Operators

Advisor

Yuri Berest

Abstract

Dunkl operators are a deformation of the usual derivatives by certain non-local operators that involve reflections from a given root system, and remarkably they (for different directions) commute with each other. For the last 25 years or so, they have been studied extensively, on both the analytical and algebraic sides, and old connections and new applications have been found in many disparate fields of mathematics. One of the highlights was a Huygens Principle for the Dunkl wave equation, a phenomenon with a long history in the differential case. The theoryof Riesz distributions associated to Dunkl operators is developed here in order to shed new light on the Dunkl wave equation. The techniques are also applied to a higher order hyperbolic equation. These equations are very special for they have to satisfy a Bernstein-Sato type identity with Dunkl operators, and one may attribute the phenomenon of Huygens Principle to the fact that Dunkl operators shift the roots of the Bernstein-Sato polynomial.