Schatten classes and commutators of Riesz transforms in the two weight setting

We characterize the Schatten class $S^p$ of the commutator of Riesz transforms $[b,R_j]$ in $R^n$ ($j=1,\dots ,n$) in the two weight setting for $n<p<\infty$, by introducing the condition that the symbol b is 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 $S^{n,\infty }$ 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' quantized derivative in the two weight setting.