Boundedness and compactness for commutators of singular integrals related to a critical radius function

Abstract

We work in the general framework of a family of singular integrals with kernels controlled in terms of a critical radius function ρ. This family models the harmonic analysis derived from the Schr¨odinger operator L = −∆ + V , where the non-negative potential V satisfies an appropriate reverse H¨older condition. For their commutators, we find sufficient conditions on the symbols for boundedness and/or compactness when acting on weighted Lp spaces. In all cases, the classes of symbols and weights are larger than their classical counterparts, BMO, CMO and Ap. When these general results are applied to the Schr¨odinger context, we obtain boundedness and compactness for commutators of operators like ∇L−1/2 , ∇2L−1 , V 1/2L−1/2 , V 1/2∇L−1 , V L−1 and Liα. As in Uchiyama’s classical paper, we give a full description of the class for compactness, CMO∞ρ , assuming ρ to be bounded. Finally, we provide examples showing that CMO is strictly contained in CMO∞ρ for any ρ, bounded or not.