Heat kernels of generalized degenerate Schrödinger operators and Hardy spaces

Abstract

Let $L=-\frac{1}{w}\mathrm{div}(A\nabla u)+\mu$ be the generalized degenerate Schrödinger operator in $L_w^2({\mathbb{R}}^d)$ with $d\ge 3$ with suitable weight $w$ and measure $\mu$. The main aim of this paper is threefold. First, we obtain an upper bound for the fundamental solution of the operator $L$. Secondly, we prove some estimates for the heat kernel of $L$ including an upper bound, the Hölder continuity and a comparison estimate. Finally, we apply the results to study the maximal function characterization for the Hardy spaces associated to the critical function generated by the operator $L$.