Littlewood--Paley operators and semigroup maximal operators on CMO spaces associated to Schödinger operators

Let $L=-\mathrm{\Delta }+V$ be a Schrödinger operator on ${\mathbb{R}}^n$, where $\mathrm{\Delta }$ is the Laplacian and $V$ satisfies the reverse Hölder inequality ${\mathrm{R}\mathrm{H}}_q$ for some $q>n\mathrm{/}2$. In this paper, we study the behavior of the Littlewood–Paley operators $s_L$ and $S_L$, as well as the semigroup maximal operator $T_L^{\ast }$, on the space ${\mathrm{C}\mathrm{M}\mathrm{O}}_L({\mathbb{R}}^n)$ associated with the Schrödinger operator $L$. It is known from previous work that these operators are bounded on ${\mathrm{B}\mathrm{M}\mathrm{O}}_L({\mathbb{R}}^n)$. Our main result shows that they are, in fact, mappings from ${\mathrm{C}\mathrm{M}\mathrm{O}}_L({\mathbb{R}}^n)$ into itself. To prove this, we develop several equivalent characterizations of ${\mathrm{C}\mathrm{M}\mathrm{O}}_L({\mathbb{R}}^n)$ 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 ${\mathrm{C}\mathrm{M}\mathrm{O}}_L$ setting.