Abstract: 

We provide characterisations of boundedness and compactness of the commutator [b,T] of a non-degenerate Calderón-Zygmund operator T and a pointwise multiplier b, between two weighted Lebesgue spaces, when the domain exponent is smaller than the codomain exponent. This is done in the Bloom setting. That is, assuming the weights are in the A_p and A_q classes respectively, the characterisations are made through b belonging to a weighted fractional BMO (for boundedness) or VMO class (for compactness). The weight that defines the class is a fractional Bloom weight built of the two original weights.

Our results form a quite natural extension of earlier research by others, where boundedness and compactness have been studied when the exponents are the same. Our approach allows complex-valued functions b, while the arguments based on the median of b on a set inherently require real values. The talk is based on joint work with Tuomas Hytönen and Tuomas Oikari.