Heat kernels of generalized degenerate Schrodinger operators and Hardy spaces

Let $displaystyle L = -frac{1}{w} , mathrm{div}(A , abla u) + mu$ be the generalized degenerate Schrodinger operator in $L^2_w(mathbb{R}^d)$ with $dge 3$ 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 Holder 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 $L$.