Gradient Estimates for the Heat Kernel of Schrödinger Operators on Stratified Lie Groups and Their Applications

Abstract

In this paper, we investigate the pointwise \(L^\infty \) estimates and Hölder continuity of the difference between the gradients of the heat kernels of \(-\Delta \) and the Schrödinger operators \(-\Delta + \mathbb {V}\) on a Lie group, where \(\mathbb {V}\) is a non-negative polynomial potential. These results have significant implications for gradient estimates of solutions to the parabolic Schrödinger equation. Consequently, we prove the boundedness of the Riesz transform on the \(\textrm{BMO}\)-type space associated with \(\mathbb {L}\) and provide an application to the regularity of solutions of the Schrödinger equation. Some of these findings are novel even in the Euclidean setting \(\mathbb {R}^n\).