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

Abstract

Let $L =-Δ+V$ be a Schrödinger operator on $\mathbb{R}^n$, $n \geq 3$, with the potential $V$ being nonnegative and belonging to the reverse Hölder class $RH_q$ for some $q >n/2$. For $0< α<2$, the Lipschitz space $Λ_L^α(\mathbb{R}^n)$ 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 $ρ$ is the critical radius function related to $L$. In this paper, we provide characterizations of $Λ^α_L(\mathbb{R}^n)$ in terms of Littlewood-Paley-type decompositions and Carleson measures, for $0< α< 2 -(n /q)$.