Equivalence of Haar bases associated with different dyadic systems
The authors consider a general notion of equivalence for dyadic systems on a metric space of homogeneous type that originate equivalent Haar systems for weighted Lebesgue spaces. The main reference is [\textit{M. Christ}, Colloq. Math. 60/61 (2), 601--628 (1990; Zbl 0758.42009)] and, as an application of their results, it is shown that Haar bases built on dyadic families of M. Christ with the same basic nets of points but different orders are equivalent. Basic tools are the vector-valued maximal Fefferman-Stein inequality on Lebesgue spaces with Muckenhoupt weights, and the characterization of weighted Lebesgue spaces via Haar coefficients.
scientific article; zbMATH DE number 5887815