AI 中文总结
本文针对海森堡群上的薛定谔算子$L$，引入两类适配的利普希茨空间，证明其在特定参数下等价并给出应用，丰富了相关算子的正则性研究。
AI 中文摘要
设$L =-\Delta_{\mathbb{H}^n} +V$为海森堡群$\mathbb{H}^n$上的薛定谔算子，其中$\Delta_{\mathbb{H}^n}$为次拉普拉斯算子，$V$为属于逆赫尔德类$RH_q(\mathbb{H}^n)$的非负位势，$q>Q/2$，$Q:=2n+2$为$\mathbb{H}^n$的齐次维数。本文受De León-Contreras与Torrea的研究工作启发，引入适配于$L$的利普希茨空间$\Lambda_L^\alpha (\mathbb{H}^n)$（$0< \alpha <2$），该空间通过涉及与$V$相关的临界半径函数$\rho$的逐点二阶差分条件定义；还引入另一类适配于$L$的利普希茨空间$\Gamma_L^{\alpha/2}(\mathbb{H}^n)$（$0< \alpha <\infty$），该空间基于热半群$e^{-tL}$定义。本文证明，当$0< \alpha <2-(Q/q)$时，$\Lambda_{L}^\alpha (\mathbb{H}^n) =\Gamma_L^{\alpha/2} (\mathbb{H}^n)$且二者范数等价；同时给出$\Gamma_L^{\alpha/2}(\mathbb{H}^n)$在算子$L$的分数次幂正则性中的应用。
英文摘要
Let $L =-Δ_{\mathbb{H}^n} +V$ be the Schödinger operator on the Heisenberg group $\mathbb{H}^n$, where $Δ_{\mathbb{H}^n}$ is the sub-Laplacian, and $V$ is a nonnegative potential belonging to the reverse Hölder class $RH_q(\mathbb{H}^n)$ for some $q > Q/2$, where $Q:=2n+2$ is the homogeneous dimension of $\mathbb{H}^n$. In this paper, motivated by the work of De León-Contreras and Torrea \cite{DT}, we introduce the Lipschitz spaces $Λ_L^α(\mathbb{H}^n)$, $0< α<2$, adapted to $L$ via a pointwise second-order difference condition involving the critical radius function $ρ$ related to $V$, and also introduce another type of Lipschitz spaces $Γ^{α/2}_L(\mathbb{H}^n)$, $0< α<\infty$, adapted to $L$ in terms of the heat semigroup $e^{-tL}$. We show that for $0< α<2-(Q/q)$, $Λ_{L}^α(\mathbb{H}^n) =Γ_L^{α/2} (\mathbb{H}^n)$ with equivalent norms. Applications of $Γ^{α/2}_L(\mathbb{H}^n)$ to the regularity of the fractional powers of the operator $L$ are also given.
Comments19 pages
