On the boundedness and Schatten class property of noncommutative martingale paraproducts and operator-valued commutators

Abstract: We study the Schatten class membership of semicommutative martingale paraproducts and use the transference method to describe Schatten class membership of purely noncommutative martingale paraproducts, especially for CAR algebras and $\mathop{\otimes}\limits_{k=1}^{\infty}\mathbb{M}_{d}$ in terms of martingale Besov spaces. Using Hyt\"{o}nen's dyadic martingale technique, we also obtain sufficient conditions on the Schatten class membership and the boundedness of operator-valued commutators involving general singular integral operators. We establish the complex median method, which is applicable to complex-valued functions. We apply it to get the optimal necessary conditions on the Schatten class membership of operator-valued commutators associated with non-degenerate kernels in Hyt\"{o}nen's sense. This resolves the problem on the characterization of Schatten class membership of operator-valued commutators. Our results are new even in the scalar case. 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 goes much further beyond the scalar case and can be applied to the semicommutative setting. By a weak-factorization type decomposition, we get some necessary but not optimal conditions on the boundedness of operator-valued commutators. In addition, we give a new proof of the boundedness of commutators still involving general singular integral operators concerning $BMO$ spaces in the commutative setting.