Haar type systems and Banach functions spaces on spaces of homogeneous type

Abstract

In this note we prove that the Haar type systems defined on spaces of homogeneous type are unconditional bases for a wide family of Banach function spaces. Also, we give a characterization of these spaces via Haar coefficients. The main tool used in the proof is a generalization of the technique of extrapolation of Rubio de Francia in Banach function spaces defined on spaces of homogeneous type.