Weighted Calderón–Zygmund Decomposition with Some Applications to Interpolation

Abstract

Let X be an A1-regular lattice of measurable functions and let Q be a projection that is also a Calderon–Zygmund operator. In this case, it is possible to define a space XQ consisting of functions f ∈ X for which Qf = f in a certain sense. By using the Bourgain approach to interpolation, we show that the couple (L 1 , XQ) is K-closed in (L1, X). This result is sharp in the sense that, in general, A1-regularity cannot be replaced by weaker conditions such as Ap-regularity for p > 1.