Littlewood--Paley operators and semigroup maximal operators on CMO spaces associated to Sch\"odinger operators

Abstract

Let $L=-\Delta+V$ be a Schr\"odinger operator on $\mathbb{R}^n$, where $\Delta$ is the Laplacian and $V$ satisfies the reverse H\"older inequality ${\rm RH}_q$ for some $q>n/2$. In this paper, we study the behavior of the Littlewood--Paley operators $s_L$ and $S_L$, as well as the semigroup maximal operator $T^*_L$, on the space ${\rm CMO}_L(\mathbb{R}^n)$ associated with the Schr\"odinger operator $L$. It is known from previous work that these operators are bounded on ${\rm BMO}_L(\mathbb{R}^n)$. Our main result shows that they are, in fact, mappings from ${\rm CMO}_L(\mathbb{R}^n)$ into itself. To prove this, we develop several equivalent characterizations of ${\rm CMO}_L(\mathbb{R}^n)$ and employ a refined decomposition that partitions the parameter interval at $r_B\rho(x_B)$, instead of the customary $r_B^2$ or $\rho(x_B)^2$. The new strategy allows us to overcome a key technical obstacle that arises when applying existing methods to the ${\rm CMO}_L$ setting.