Sharp Li–Yau inequalities for Dunkl harmonic oscillators

Abstract

Abstract We study the Li–Yau inequality for the heat equation corresponding to the Dunkl harmonic oscillator, which is a nonlocal Schrödinger operator parameterized by reflections and multiplicity functions. In the particular case when the reflection group is isomorphic to ℤ 2 d {\mathbb{Z}_{2}^{d}} , the result is sharp in the sense that equality is achieved by the heat kernel of the classic harmonic oscillator. We also provide the application on parabolic Harnack inequalities.