Some characterizations of BMO and Lipschitz spaces in the Schrödinger setting

Abstract

We consider the Schrödinger operator $\mathcal{L}=-\hat{I}”+V$ on ${\mathbb{R}}^d$, $d\ge 3$, where the nonnegative potential $V$ belongs to the reverse Hölder class $R{H}_s$ for some $s\ge d/2$. A real-valued function $f\in L_{\mathrm{loc}}^1({\mathbb{R}}^d)$ belongs to the (BMO) space ${\mathrm{BMO}}_{\ddot{I},\hat{I}¸}({\mathbb{R}}^d)$ with $0<\hat{I}¸<\infty$ if \begin{equation*} \|f\|_{\mathrm{BMO}_{ρ,θ}} :=\sup_{B(x_0,r)}\bigg(1+\frac{r}{ρ(x_0)}\bigg)^{-θ}\bigg(\frac{1}{|B(x_0,r)|}\int_{B(x_0,r)}\big|f(x)-f_{B}\big|\,dx\bigg), \end{equation*} where the supremum is taken over all balls $B(x_0,r)\subset {\mathbb{R}}^d$, $\ddot{I}(\cdot )$ is the critical radius function in the Schrödinger context and \begin{equation*} f_{B}:=\frac{1}{|B(x_0,r)|}\int_{B(x_0,r)}f(y)\,dy. \end{equation*} A real-valued function $f\in L_{\mathrm{loc}}^1({\mathbb{R}}^d)$ belongs to the (Lipschitz) space $\mathrm{Lip}_β^{ρ,θ}(\mathbb R^d)$ with $0<{\hat{I}}^2<1$ and $0<\hat{I}¸<\infty$ if \begin{equation*} \|f\|_{\mathrm{Lip}_β^{ρ,θ}} :=\sup_{B(x_0,r)}\bigg(1+\frac{r}{ρ(x_0)}\bigg)^{-θ} \bigg(\frac{1}{|B(x_0,r)|^{1+β/d}}\int_{B(x_0,r)}\big|f(x)-f_{B}\big|\,dx\bigg). \end{equation*} It can be easily seen that ${\mathrm{BMO}}_{\ddot{I},\hat{I}¸}({\mathbb{R}}^d)$ (or $\mathrm{Lip}_β^{ρ,θ}(\mathbb R^d)$) is a function space which is larger than the classical BMO (or Lipschitz) space. In this paper, we give some new characterizations of BMO and Lipschitz spaces associated with the Schrödinger operator $\mathcal{L}$. We extend some previous works of Bongioanni--Harboure--Salinas and Liu--Sheng to the weighted case. The classes of weights considered here are larger than the classical Muckenhoupt classes.

18 pages