Norm inequalities for the conjugate operator in two-weighted Lebesgue spaces

Abstract

In this article, first, we prove that weighted-norm inequalities for the M-harmonic conjugate operator on the unit sphere whenever the pair (u, v) of weights satisfies the A p -condition, and uds, vds are doubling measures, where ds is the rotationinvariant positive Borel measure on the unit sphere with total measure 1. Then, we drive cross-weighted norm inequalities between the Hardy-Littlewood maximal function and the sharp maximal function whenever (u, v) satisfies the A p -condition, and vds does a certain regular condition. 2000 MSC: primary 32A70; secondary 47G10.