Multiresolution approximations and unconditional bases on weighted Lebesgue spaces on spaces of homogeneous type

Starting from a slight modification of the dyadic sets introduced by M. Christ in [A T(b) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990) 601–628] on a space of homogeneous type (X,d,μ), an MRA type structure and a Haar system H controlled by the quasi distance d, can be constructed in this general setting in such a way that H is an orthonormal basis for L2(dμ). This paper is devoted to explore under which conditions on the measure ν, the system H is also an unconditional basis for the Lebesgue spaces Lp(dν). As a consequence, we obtain a characterization of these spaces in terms of the H–coefficients.