Hölder regularity and Liouville Theorem for the Schrödinger equation with certain critical potentials, and applications to Dirichlet problems

Abstract

Let $(X,d,\mu )$ be a metric measure space satisfying a doubling property with the upper/lower dimension $Q\ge n>1$, and admitting an $L^2$-Poincaré inequality. In this article, we establish the Hölder continuity and a Liouville-type theorem for the (elliptic-type) Schrödinger equation $$\mathbb{L}u(x,t)=-{\partial }_t^{2}u(x,t)+\mathcal{L}u(x,t)+V(x)u(x,t)=0,x\in X,t\in \mathbb{R},$$ where $\mathcal{L}$ is a non-negative operator generated by a Dirichlet form on $X$, and the non-negative potential $V$ is a Muckenhoupt weight belonging to the reverse Hölder class ${RH}_q(X)$ for some $q>\max \{Q/2,1\}$. Note that $Q/2$ is critical for the regularity theory of $-\Delta +V$ on ${\mathbb{R}}^Q$ ($Q\ge 3$) by Shen's work in 1995, which hints the critical index of $V$ for the regularity results above on $X\times \mathbb{R}$ may be $(Q+1)/2$. Our results show that this critical index is in fact $\max \{Q/2,1\}$. Our approach primarily relies on the controllable growth of $V$ and the elliptic theory for the operator $\mathbb{L}$/$-{\partial }_t^2+\mathcal{L}$ on $X\times \mathbb{R}$, rather than the analogs for $\mathcal{L}+V$/$\mathcal{L}$ on $X$, under the critical index setting. As applications, we further obtain some characterizations for solutions to the Schrödinger equation $-{\partial }_t^{2}u+\mathcal{L}u+Vu=0$ in $X\times {\mathbb{R}}_{+}$ with boundary values in BMO/CMO/Morrey spaces related to $V$, improving previous results to the critical index $q>\max \{Q/2,1\}$.

39 pages, no figures