We work in the general framework of a family of singular integrals with kernels controlled in terms of a critical radius function \(\rho \) . This family models the harmonic analysis derived from the Schrödinger operator \(L= -\Delta +V\) , where the non-negative potential V satisfies an appropriate reverse Hölder condition. For their commutators, we find sufficient conditions on the symbols for boundedness and/or compactness when acting on weighted \(L^p\) spaces. In all cases, the classes of symbols and weights are larger than their classical counterparts, \(BMO \) , \(CMO \) and \(A_p\) . When these general results are applied to the Schrödinger context, we obtain boundedness and compactness for commutators of operators like \(\nabla L^{-1/2}\) , \(\nabla ^2 L^{-1}\) , \(V^{1/2} L^{-1/2}\) , \(V^{1/2} \nabla L^{-1}\) , \(V L^{-1}\) and \(L^{i\alpha }\) . As in Uchiyama’s classical paper, we give a full description of the class for compactness, \(CMO ^\infty _\rho \) , under mild conditions on \(\rho \) . Finally, we provide examples showing that \(CMO \) is strictly contained in \(CMO ^\infty _\rho \) .
