Compactness and Density Estimates for Weighted Fractional Heat Semigroups

We prove that the operator $L_0=-(1+\mathrm{\mid }x\mathrm{\mid }{)}^{\beta }(-\mathrm{\Delta }{)}^{\alpha \mathrm{/}2}$ with $\alpha \in (0,2)$, $d>\alpha$ and $\beta \ge 0$ generates a compact semigroup or resolvent on $L^2({\mathbb{R}}^d;(1+\mathrm{\mid }x\mathrm{\mid }{)}^{-\beta }dx)$, 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.