We study the Li--Yau inequality for the heat equation corresponding to the Dunkl harmonic oscillator, which is a non-local Schr\"{ o}dinger operator parameterized by reflections and multiplicity functions. In the particular case when the reflection group is isomorphic to $\mathbb{ Z}_2^d$, 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.
Categories: math.AP math.PR