Regularity estimates of fractional heat semigroups related with uniformly elliptic operators

Abstract

Let $L=-\mathrm{div}(A(x)\cdot \nabla )+V(x)$ be a second-order uniformly elliptic operator on ${\mathbb{R}}^n$ $(n\ge 3)$, where $A(x)$ is a real symmetric matrix satisfying standard ellipticity conditions, and $V$ is a nonnegative potential belonging to the reverse Hölder class. For $\alpha \in (0,1)$, we study regularity estimates of the fractional heat semigroups $\{exp(-t{L}^{\alpha }){\}}_{t>0}$, via the subordination formula and the fundamental solution of the associated uniformly parabolic equation ${\partial }_{t}u+Lu=0$. This approach avoids the use of Fourier transforms and is applicable to second-order differential operators whose heat kernels satisfy Gaussian upper bounds. As an application, we characterize the Campanato-type space ${\Lambda }_{L,\gamma }\left({\mathbb{R}}^n\right)$ via the fractional heat semigroups $\{exp(-t{L}^{\alpha }){\}}_{t>0}$.