Complex median method and Schatten class membership of commutators

This article is devoted to the study of the Schatten class membership of commutators involving singular integral operators. We utilize martingale paraproducts and Hyt\"{o}nen's dyadic martingale technique to obtain sufficient conditions on the weak-type and strong-type Schatten class membership of commutators in terms of Sobolev spaces and Besov spaces respectively. We also establish the complex median method, which is applicable to complex-valued functions. We apply it to get the optimal necessary conditions on the weak-type and strong-type Schatten class membership of commutators associated with non-degenerate kernels. This resolves the problem on the characterization of the weak-type and strong-type Schatten class membership of commutators. Our new approach is built on Hyt\"{o}nen's dyadic martingale technique and the complex median method. Compared with all the previous ones, this new one is more powerful in several aspects: $(a)$ it permits us to deal with more general singular integral operators with little smoothness; $(b)$ it allows us to deal with commutators with complex-valued kernels; $(c)$ it turns out to be powerful enough to deal with the weak-type and strong-type Schatten class of commutators in a universal way.