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

We characterize the Schatten class Sᵖ of the commutator of Riesz transforms [b,Rⱼ] in Rⁿ (j=1,…, n) in the two weight setting for n< p<∞, 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 being in the weak Schatten class Sn,∞ is characterized by the symbol b being 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.