... weighted-norm inequalities for the M-harmonicconjugate operator on the unit sphere whenever the pair (u, v) of weights satisfies the Ap-condition, and uds,vds are doubling measures, where ds is the ... the classical harmonic conjugate func-tion and so we can regard Kfas the Hilbert transform on the unit circle. The Lpboundedness property of harmonic conjugate functions on the unit circle for ... bounded on Lp( ω) if and only if the nonnegativeweight ω satisfies the Ap(S)-condition on S; i.e., the nonnegative weight ω satisfiesRim and Lee Journal of Inequalities and Applications 2011,...