Regularity of fractional heat semigroup associated with Schr\"odinger operators

TL;DR

This paper studies the fractional heat semigroup associated with Schr"odinger operators with potentials in the reverse H"older class, providing estimates for the heat kernel derivatives and characterizing certain function spaces.

Contribution

It introduces a novel approach to estimate fractional heat kernels without Fourier transform and applies it to characterize Campanato spaces related to Schr"odinger operators.

Findings

Derived gradient and fractional time-derivative estimates for the heat kernel.

Established a Carleson measure characterization of $BMO^{eta}_L$ spaces.

Method applicable to operators with Gaussian upper bound heat kernels.

Abstract

Let $L=-\Delta+V$ be a Schr\"odinger operator, where the potential $V$ belongs to the reverse H\"older class. By the subordinative formula, we introduce the fractional heat semigroup $\{e^{-t{L}^\alpha}\}_{t>0}, \alpha>0$, associated with ${L}$. By the aid of the fundamental solution of the heat equation: $$\partial_{t}u+L u=\partial_{t}u -\Delta u+Vu=0,$$ we estimate the gradient and the time-fractional derivatives of the fractional heat kernel $K^{L}_{\alpha,t}(\cdot, \cdot)$, respectively. This method is independent of the Fourier transform, and can be applied to the second order differential operators whose heat kernels satisfying Gaussian upper bounds. As an application, we establish a Carleson measure characterization of the Campanato type space $BMO^{\gamma}_{L}(\mathbb{R}^{n})$ via $\{e^{-t{L}^\alpha}\}_{t>0}$.