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

Let $L=-\mathrm{\Delta }+V$ 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 $q>n\mathrm{/}2$. For $0<\alpha <2$, the Lipschitz space ${\mathrm{\Lambda }}_L^{\alpha }({\mathbb{R}}^n)$ adapted to $L$ is defined as the space of all measurable functions $f$ on ${\mathbb{R}}^n$ such that $$\mathrm{\parallel }f{\mathrm{\parallel }}_{{\mathrm{\Lambda }}_L^{\alpha }}:=\mathrm{\parallel }\rho (\cdot {)}^{-\alpha }f(\cdot ){\mathrm{\parallel }}_{L^{\mathrm{\infty }}}+\underset{z\in {\mathbb{R}}^n\mathrm{\backslash }\{0\}}{\sup }\frac{\mathrm{\parallel }f(\cdot +z)+f(\cdot -z)-2f(\cdot ){\mathrm{\parallel }}_{L^{\mathrm{\infty }}}}{\mathrm{\mid }z{\mathrm{\mid }}^{\alpha }}<\mathrm{\infty },$$ where $\rho$ is the critical radius function related to $L$. In this paper, we provide characterizations of ${\mathrm{\Lambda }}_L^{\alpha }({\mathbb{R}}^n)$ in terms of Littlewood-Paley-type decompositions and Carleson measures, for $0<\alpha <2-(n\mathrm{/}q)$.