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

Let $L=-\mathrm{\Delta }+V$ 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: $${\mathrm{\partial }}_{t}u+Lu={\mathrm{\partial }}_{t}u-\mathrm{\Delta }u+Vu=0,$$ we estimate the gradient and the time-fractional derivatives of the fractional heat kernel $K_{\alpha ,t}^L(\cdot ,\cdot )$, 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 $BM{O}_L^{\gamma }({\mathbb{R}}^n)$ via $\{e^{-t{L}^{\alpha }}{\}}_{t>0}$.