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

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 $\{e^{-t{L}^\alpha}\}_{t>0}$.