Copyright © 2010 Jaesung Lee and Kyung Soo Rim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We define the M-harmonic conjugate operator K and prove that for 1<p<∞, there is a constant Cp such that ∫S|Kf|pωdσ≤Cp∫S|f|pωdσ for all f∈Lp(ω) if and only if the nonnegative weight ω satisfies the Ap-condition. Also, we prove that if there is a constant Cp such that ∫S|Kf|pvdσ≤Cp∫S|f|pwdσ for all f∈Lp(w), then the pair of weights (v,w) satisfies the Ap-condition.