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 Rⁿ (j=1,…,n) in the two weight setting for n<p<∞, 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 Sⁿ if and only if b is a constant, and to the weak Schatten class S^n,∞ 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.