Littlewood-Paley and Carleson measure characterizations of Lipschitz spaces adapted to Schrödinger operators

Abstract

Let $L=-\Delta +V$ be a Schrödinger operator on ${\mathbb{R}}^n$, $n\ge 3$, with the potential $V$ being nonnegative and belonging to the reverse Hölder class $R{H}_q$ for some $q>n/2$. For $0<\alpha <2$, the Lipschitz space ${\Lambda }_L^{\alpha }({\mathbb{R}}^n)$ adapted to $L$ is defined as the space of all measurable functions $f$ on ${\mathbb{R}}^n$ such that \[ \|f\|_{Λ_L^α}:= \|ρ(\cdot)^{-α}f(\cdot)\|_{L^\infty}+ \sup_{z \in \mathbb{R}^n \backslash \{0\}} \frac{\|f(\cdot + z) + f(\cdot -z) -2 f(\cdot)\|_{L^\infty}}{|z|^α} <\infty, \] where $\rho$ is the critical radius function related to $L$. In this paper, we provide characterizations of ${\Lambda }_L^{\alpha }({\mathbb{R}}^n)$ in terms of Littlewood-Paley-type decompositions and Carleson measures, for $0<\alpha <2-(n/q)$.

27 pages