Regularity estimates of fractional heat semigroups related with uniformly elliptic operators

TL;DR

This paper establishes regularity estimates for fractional heat semigroups associated with uniformly elliptic operators, using a non-Fourier approach and characterizes related function spaces.

Contribution

It introduces a novel method to analyze fractional heat semigroups without Fourier transforms, applicable to operators with Gaussian heat kernel bounds.

Findings

Derived regularity estimates for fractional heat semigroups

Characterized Campanato-type spaces via these semigroups

Provided a new approach applicable to a broad class of elliptic operators

Abstract

Let $L = -{\rm div}( A(x) \cdot \nabla ) + V(x)$ be a second-order uniformly elliptic operator on $\mathbb{ R }^{n}$ $(n\geq 3)$, where $A(x)$ is a real symmetric matrix satisfying standard ellipticity conditions, and $V$ is a nonnegative potential belonging to the reverse H\"older class. For $ \alpha \in (0,1) $, we study regularity estimates of the fractional heat semigroups $ \{ exp (-tL^ {\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 } $.