2024/10/04 by Li, Bo, Li, Ji, Wu, Liangchuan
#35J10 #42B35 #43A85 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2410.03418
Let (X,d,μ) be a metric measure space satisfying a doubling property with the upper/lower dimension Q≥ n>1, and admitting an L2-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)=-∂2tu(x,t)+\mathcal L u(x,t)+V(x)u(x,t)=0, x∈ X, t∈\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 RHq(X) for some q>max\Q/2,1\. Note that Q/2 is critical for the regularity theory of -Δ+V on ℝQ (Q≥3) by Shen's work in 1995, which hints the critical index of V for the regularity results above on X× \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/-∂2t+L on X× \mathbb R, rather than the analogs for \mathcal L+V/L on X, under the critical index setting. As applications, we further obtain some characterizations for solutions to the Schrödinger equation -∂2tu+\mathcal L u+Vu=0 in X× \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\.