The Rate of Convergence on Fractional Power Dissipative Operator on Compact Manifolds

Abstract

Abstract On a compact connected manifold M $\mathbb{M}$ , we concern the fractional power dissipative operator e − t L α $e^{-t\left\vert \mathcal{L}\right\vert ^{\alpha}}$ , and obtain the almost-everywhere convergence rate (as t → 0+) of e − t L α f $e^{-t\left\vert \mathcal{L}\right\vert ^{\alpha}}\left( f\right)$ when f is in some Sobolev type Hardy spaces. The main result is a non-trivial extension of a recent result on ℝ n by Cao and Wang in 2.