Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Estimates of M-Harmonic Conjugate Operator | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 435450 13 pages doi 2010 435450 Research Article Estimates of M-Harmonic Conjugate Operator Jaesung Lee and Kyung Soo Rim Department of Mathematics Sogang University 1 Sinsu-dong Mapo-gu Seoul 121-742 South Korea Correspondence should be addressed to Jaesung Lee jalee@ Received 30 November 2009 Revised 23 February 2010 Accepted 17 March 2010 Academic Editor Shusen Ding Copyright 2010 J. Lee and K. S. 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 OT there is a constant Cp such that Js Kf pwdơ Cp Js If pwdơ for all f e lp co if and only if the nonnegative weight w satisfies the Ap-condition. Also we prove that if there is a constant Cp such that Js Kf pvdơ Cp Js f pwdơ for all f e ư w then the pair of weights v w satisfies the Ap-condition. 1. Introduction Let B be the unit ball of Cn with norm z z zỸ where is the Hermitian inner product let s be the unit sphere and Ơ be the rotation-invariant probability measure on s. In 1 for z e B ị e s we defined the kernel K z ị by iK z ị 2C z ị - P z ị - 1 where C z ị 1 - z ị n is the Cauchy kernel and P z ị 1 - z 2 n 1 - z ị 2n is the invariant Poisson kernel. Thus for each ị e s the kernel K ị is M-harmonic. And for all f e A B the ball algebra such that f 0 is real the reproducing property of 2C z ị - 1 of 2 gives K z ị Re f ị ãơ ị -i f z - Re f z Im f z . s For that reason K z ị is called the M-harmonic conjugate kernel. 2 Journal of Inequalities and Applications For f e LfS Kf the M-harmonic conjugate function of f on S is defined by Kf Z lim í. K rU f dơ ị r 1 S since the limit exists almost everywhere. For n 1 the definition of Kf is the same as the .
