Lipschitz spaces adapted to Schrödinger operators on the Heisenberg group

Abstract

Let $L=-{\Delta }_{{\mathbb{H}}^n}+V$ be the Schödinger operator on the Heisenberg group ${\mathbb{H}}^n$, where ${\Delta }_{{\mathbb{H}}^n}$ is the sub-Laplacian, and $V$ is a nonnegative potential belonging to the reverse Hölder class $R{H}_q({\mathbb{H}}^n)$ 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 ${\Lambda }_L^{\alpha }({\mathbb{H}}^n)$, $0<\alpha <2$, adapted to $L$ via a pointwise second-order difference condition involving the critical radius function $\rho$ related to $V$, and also introduce another type of Lipschitz spaces ${\Gamma }_L^{\alpha /2}({\mathbb{H}}^n)$, $0<\alpha <\infty$, adapted to $L$ in terms of the heat semigroup $e^{-tL}$. We show that for $0<\alpha <2-(Q/q)$, ${\Lambda }_L^{\alpha }({\mathbb{H}}^n)={\Gamma }_L^{\alpha /2}({\mathbb{H}}^n)$ with equivalent norms. Applications of ${\Gamma }_L^{\alpha /2}({\mathbb{H}}^n)$ to the regularity of the fractional powers of the operator $L$ are also given.

19 pages