Heat kernels of generalized degenerate Schr\"odinger operators and Hardy spaces

TL;DR

This paper investigates heat kernels of a generalized degenerate Schrödinger operator, providing bounds, regularity estimates, and applications to Hardy space characterizations in weighted $L^2$ spaces.

Contribution

It establishes new upper bounds and regularity estimates for heat kernels of degenerate Schrödinger operators and applies these to Hardy space theory.

Findings

Derived an upper bound for the fundamental solution.

Proved Hölder continuity and comparison estimates for the heat kernel.

Characterized Hardy spaces via maximal functions associated with the operator.

Abstract

Let $\displaystyle L = -\frac{1}{w} \, \mathrm{div}(A \, \nabla u) + \mu$ be the generalized degenerate Schr\"odinger operator in $L^2_w(\mathbb{R}^d)$ 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 H\"older 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 $L$.