Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 435450, 13 pages
Research Article

Estimates of M-Harmonic Conjugate Operator

Department of Mathematics, Sogang University, 1 Sinsu-dong, Mapo-gu, Seoul 121-742, South Korea

Received 30 November 2009; Revised 23 February 2010; Accepted 17 March 2010

Academic Editor: Shusen Ding

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.


We define the M-harmonic conjugate operator K and prove that for 1<p<, there is a constant Cp such that S|Kf|pωdσCpS|f|pωdσ for all fLp(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|pvdσCpS|f|pwdσ for all fLp(w), then the pair of weights (v,w) satisfies the Ap-condition.