Littlewood-Paley and Carleson measure characterizations of Lipschitz spaces adapted to Schr\"{o}dinger operators

Abstract

Let $L =-\Delta +V$ be a Schr\"{o}dinger operator on $\mathbb{R}^n$, $n \geq 3$, with the potential $V$ being nonnegative and belonging to the reverse H\"{o}lder class $RH_q$ 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 \[ \|f\|_{\Lambda_L^\alpha}:= \|\rho(\cdot)^{-\alpha}f(\cdot)\|_{L^\infty}+ \sup_{z \in \mathbb{R}^n \backslash \{0\}} \frac{\|f(\cdot + z) + f(\cdot -z) -2 f(\cdot)\|_{L^\infty}}{|z|^\alpha} <\infty, \] where $\rho$ is the critical radius function related to $L$. In this paper, we provide characterizations of $\Lambda^\alpha_L(\mathbb{R}^n)$ in terms of Littlewood-Paley-type decompositions and Carleson measures, for $0< \alpha < 2 -(n /q)$.