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

HUGO AIMAR; ANA BERNARDIS; BIBIANA IAFFEI

JOURNAL OF APPROXIMATION THEORY

Elsevier

Año: 2007 vol. 148 p. 12 - 34

0021-9045

Starting from a slight modification of the dyadic sets introducedby M. Christ in [M. Christ, A T(b) theorem with remarks onanalytic capacity and the Cauchy integral, Colloq. Math. 60/61(1990) 601--628] on a space of homogeneous type $(X,d,mu)$, anMRA type structure and a Haar system $mathcal{H}$ controlled bythe quasi distance $d$, can be constructed in this general settingin such a way that $mathcal{H}$  is an orthonormal basis for$L^2(dmu)$. This paper is devoted to explore under whichconditions on the measure $ u$, the system $mathcal{H}$ is alsoan unconditional basis for the Lebesgue spaces $L^p(d u)$. As aconsequence, we obtain a characterization of these spaces in termsof the $mathcal{H}$--coefficients.