Uniform Complex Time Heat Kernel Estimates Without Gaussian Bounds

TL;DR

This paper establishes uniform complex time heat kernel estimates for fractional Laplacians, especially when Gaussian bounds fail, and extends results to certain perturbed operators with potential V.

Contribution

It provides the first asymptotic and uniform estimates for complex time heat kernels of fractional Laplacians without relying on Gaussian bounds.

Findings

Asymptotic estimates near the imaginary axis

Uniform complex time heat kernel bounds established

Extensions to operators with higher order Kato class potentials

Abstract

In this paper, first we consider the uniform complex time heat kernel estimates of $e^{-z(-\Delta)^{\frac{\alpha}{2}}}$ for $\alpha>0, z\in \mathbb{C}^+$. When $\frac{\alpha}{2}$ is not an integer, generally the heat kernel doest not have the Gaussian upper bounds for real time. Thus the Phragm\'en-Lindel\"of methods fail to give the uniform complex time estimates. Instead, our first result gives the asymptotic estimates for $P(z, x)$ as $z$ tending to the imaginary axis. Then we prove the uniform complex time heat kernel estimates. Finally we also show the uniform estimates of analytic semigroup generated by $H=(-\Delta)^{\frac{\alpha}{2}}+V$ where $V$ belongs to higher order Kato class.