Acta Mathematica Vietnamica

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 $\mathbb {R}^n$.

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