Schatten classes and commutators in the two weight setting, II. Riesz transforms

We characterize the Schatten class $S^p$ of the commutator of Riesz transforms $[b,R_j]$ in ${\mathbb{R}}^n$ ($j=1,\dots ,n$) in the two weight setting for $n<p<\mathrm{\infty }$, by introducing the condition that the symbol $b$ being in Besov spaces associated with the given two weights. At the critical index $p=n$, the commutator $[b,R_j]$ belongs to Schatten class $S^n$ if and only if $b$ is a constant, and to the weak Schatten class $S^{n,\mathrm{\infty }}$ if and only if $b$ is in an oscillation sequence space associated with the given two weights. As a direct application, we have the Schatten class estimate for A. Connes’ quantised derivative in the two weight setting.