Dirichlet problem for Schrödinger operators on Heisenberg groups

Abstract

We investigate the Dirichlet problem associated to the Schrödinger operator $\mathcal{L}=-{\Delta }_{{\mathbb{H}}^n}+V$ on Heisenberg group ${\mathbb{H}}^n$: \begin{align*} \begin{cases} \partial_{ss}u(g,s)-\mathcal L u(g,s)=0\,,\quad &{\rm in \,\ } \mathbb{H}^n\times\mathbb{R}^+,\\ u(g,0)=f \,,\quad &{\rm on \,\ } \mathbb{H}^n \end{cases} \end{align*} with $f$ in $L^p({\mathbb{H}}^n)$ ($1<p<\infty$) and in $H_{\mathcal{L}}^1({\mathbb{H}}^n)$, i.e., the Hardy space associated with $\mathcal{L}$. Here ${\Delta }_{{\mathbb{H}}^n}$ is the sub-Laplacian on ${\mathbb{H}}^n$ and the nonnegative potential $V$ belongs to the reverse Hölder class $B_{Q/2}$ with $Q$ the homogeneous dimension of ${\mathbb{H}}^n$. The new approach is to establish a suitable weak maximum principle, which is the key to solve this problem under the condition $V\in B_{Q/2}$. This result is new even back to ${\mathbb{R}}^n$ (the condition will become $V\in B_{n/2}$) since the previous known result requires $V\in B_{(n+1)/2}$ which went through a Liouville type theorem.

17 pages