论文标题
Schrödinger运营商Heisenberg Groups的Dirichlet问题
Dirichlet problem for Schrödinger operators on Heisenberg groups
论文作者
论文摘要
我们研究了与schrödingeroperator $ \ mathcal l =-Δ_ {\ Mathbb {h}^n}+v $相关的dirichlet问题u（g，s）= 0 \ ,, \ quad＆{\ rm in \，\} \ Mathbb {h} \ end {align*}，$ f $ in $ l^p（\ mathbb {h}^n）$（$ 1 <p <\ infty $）和$ h^1 _ {\ mathcal l}（\ mathcal l}（\ mathbb {h h}^n）$，I.E.这里$δ_ {\ Mathbb {h}^n} $是$ \ Mathbb h^n $上的sub-laplacian，而非负势$ v $属于反向Hölder类$ b_ {q/2} $，带有$ q $的同质尺寸$ \ mathbbbbbbbbbbbbb {新方法是建立一个合适的弱最大原理，这是在b_ {q/2} $中的条件$ v \下解决此问题的关键。该结果是新的，甚至回到$ \ Mathbb r^n $（条件将变成b_ {n/2} $中的$ v \），因为先前的已知结果需要$ v \ in B _ {（n+1）/2} $，它通过了liouville type Theorem。
We investigate the Dirichlet problem associated to the Schrödinger operator $\mathcal L=-Δ_{\mathbb{H}^n}+V$ 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 $H^1_{\mathcal L}(\mathbb{H}^n)$, i.e., the Hardy space associated with $\mathcal L$. Here $Δ_{\mathbb{H}^n}$ 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.
