Endpoint boundedness of singular integrals: CMO space associated to Schrödinger operators
Xueting Han, Ji Li, Liangchuan Wu
TL;DR
This work studies endpoint boundedness for classical singular integrals on the Schrödinger-operator- adapted vanishing mean oscillation space ${\rm CMO}_{\mathcal L}(\mathbb{R}^n)$ with ${\mathcal L}=-\Delta+V$ and $V\in RH_q$, $q\ge \frac{n}{2}$. The authors establish three main results: (i) the Hardy-Littlewood maximal operator $M$ is bounded on ${\rm CMO}_{\mathcal L}$, (ii) the adjoint Riesz transforms $R_j^*$ map $C_0(\mathbb{R}^n)$ into ${\rm CMO}_{\mathcal L}$, and (iii) the Poisson and heat semigroup–driven approximations to the identity characterize ${\rm CMO}_{\mathcal L}$. The proofs rely on sharp heat-kernel estimates and the tent-space framework for ${\rm CMO}_{\mathcal L}$, rather than classical Calderón–Zygmund theory, and yield a Riesz-type representation for functionals on ${\rm CMO}_{\mathcal L}$. When $V\equiv0$, these results recover the classical ${\rm CMO}$ theory, illustrating how the potential complicates the analysis yet preserves a parallel structure. The findings have implications for harmonic analysis in Schrödinger settings and for associated function-space dualities.
Abstract
Let $ \mathcal{L} = -Δ+ V $ be a Schrödinger operator acting on $ L^2(\mathbb{R}^n) $, where the nonnegative potential $ V $ belongs to the reverse Hölder class $ RH_q $ for some $ q \geq n/2 $. This article is primarily concerned with the study of endpoint boundedness for classical singular integral operators in the context of the space $ \mathrm{CMO}_{\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{CMO}_{\mathcal{L}}(\mathbb{R}^n) $; (ii) for each $ j = 1, \ldots, n$, the adjoint of the Riesz transform $ \partial_j \mathcal{L}^{-1/2} $ is bounded from $ C_0(\mathbb{R}^n) $ into $ \mathrm{CMO}_{\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{CMO}_{\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{CMO}_{\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].