Compactness and Density Estimates for Weighted Fractional Heat Semigroups

TL;DR

This paper investigates the spectral properties of a weighted fractional heat operator, establishing conditions for compactness, and deriving precise eigenvalue asymptotics and heat kernel bounds.

Contribution

It provides necessary and sufficient conditions for the operator to generate a compact semigroup and offers detailed asymptotic estimates and bounds for eigenvalues and heat kernels.

Findings

Operator generates compact semigroup if and only if β > α

Two-sided asymptotic estimates for high order eigenvalues

Sharp bounds for the heat kernel

Abstract

We prove that the operator $L_0=-(1+|x|)^\beta(-\Delta)^{\alpha/2}$ with $\alpha\in(0,2)$, $d>\alpha$ and $\beta\ge0$ generates a compact semigroup or resolvent on $L^2(\R^d;(1+|x|)^{-\beta}\,dx)$, if and only if $\beta>\alpha$. When $\beta>\alpha$, we obtain two-sided asymptotic estimates for high order eigenvalues, and sharp bounds for the corresponding heat kernel.