Uniform Complex Time Heat Kernel Estimates Without Gaussian Bounds

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 Phragmén-Lindelöf 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.