Schatten Class Estimates for Paraproducts in Multi-parameter setting and applications

Abstract

Let ${\hat{I}}_b$ be a bounded $n$ parameter paraproduct with symbol $b$. We demonstrate that this operator is in the Schatten class $S^p$, $0<p<\infty$, if the symbol is in the $n$ parameter Besov space $B_p$. Our result covers both the dyadic and continuous version of the paraproducts in the multiparameter setting.

Updated version. The characterisation of [b,H_1H_2] via Besov type space in terms of the Slobodeckij type norm is not clear