Endpoint boundedness of singular integrals: CMO space associated to Schrödinger operators

Let $\mathcal{L}=-\mathrm{\Delta }+V$ be a Schrödinger operator acting on $L^2({\mathbb{R}}^n)$, where the nonnegative potential $V$ belongs to the reverse Hölder class $R{H}_q$ for some $q\ge n\mathrm{/}2$. This article is primarily concerned with the study of endpoint boundedness for classical singular integral operators in the context of the space ${\mathrm{C}\mathrm{M}\mathrm{O}}_{\mathcal{L}}({\mathbb{R}}^n)$, consisting of functions of vanishing mean oscillation associated with $\mathcal{L}$. We establish the following main results: (i) the standard Hardy–Littlewood maximal operator is bounded on ${\mathrm{C}\mathrm{M}\mathrm{O}}_{\mathcal{L}}({\mathbb{R}}^n)$; (ii) for each $j=1,\dots ,n$, the adjoint of the Riesz transform ${\mathrm{\partial }}_j{\mathcal{L}}^{-1\mathrm{/}2}$ is bounded from $C_0({\mathbb{R}}^n)$ into ${\mathrm{C}\mathrm{M}\mathrm{O}}_{\mathcal{L}}({\mathbb{R}}^n)$; and (iii) the approximation to the identity generated by the Poisson and heat semigroups associated with $\mathcal{L}$ characterizes ${\mathrm{C}\mathrm{M}\mathrm{O}}_{\mathcal{L}}({\mathbb{R}}^n)$ appropriately. These results recover the classical analogues corresponding to the Laplacian as a special case. However, the presence of the potential $V$ introduces substantial analytical challenges, necessitating tools beyond the scope of classical Calderón–Zygmund theory. Our approach leverages precise heat kernel estimates and the structural properties of ${\mathrm{C}\mathrm{M}\mathrm{O}}_{\mathcal{L}}({\mathbb{R}}^n)$ established by Song and the third author in [J. Geom. Anal. 32 (2022), no. 4, Paper No. 130, 37 pp].