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Báo cáo hóa học: " Research Article On a Hilbert-Type Operator with a Class of Homogeneous Kernels"

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 On a Hilbert-Type Operator with a Class of Homogeneous Kernels | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 572176 9 pages doi 10.1155 2009 572176 Research Article On a Hilbert-Type Operator with a Class of Homogeneous Kernels Bicheng Yang Department of Mathematics Guangdong Education Institute Guangzhou Guangdong 510303 China Correspondence should be addressed to Bicheng Yang bcyang@pub.guangzhou.gd.cn Received 15 September 2008 Accepted 20 February 2009 Recommended by Patricia J. Y. Wong By using the way of weight coefficient and the theory of operators we define a Hilbert-type operator with a class of homogeneous kernels and obtain its norm. As applications an extended basic theorem on Hilbert-type inequalities with the decreasing homogeneous kernels of -l-degree is established and some particular cases are considered. Copyright 2009 Bicheng Yang. 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. 1. Introduction In 1908 Weyl published the well-known Hilbert s inequality as the following. If an TO 1 bn TO 1 are real sequences 0 YfTO 1 an TO and 0 DTO 1 bn TO then 1 TO TO n 1 m 1 ambn m n TO TO 1 2 A an bA 1.1 where the constant factor n is the best possible. In 1925 Hardy gave an extension of 1.1 by introducing one pair of conjugate exponents p q 1 p 1 q 1 as 2 . If p 1 an bn 0 0 STO 1 an TO and 0 2TO 1 bn TO then TOTO n 1 m 1 ambn m n TO 1 p TO 1 q n y ap y bq sin n p 1an Ề 1.2 2 Journal of Inequalities and Applications where the constant factor n sin n p is the best possible. We named 1.2 Hardy-Hilbert s inequality. In 1934 Hardy et al. 3 gave some applications of 1.1 - 1.2 and a basic theorem with the general kernel see 3 Theorem 318 . Theorem A. Suppose that p 1 1 p 1 q 1 k x y is a homogeneous function of -1-degree and k fTOk u 1 u-1 p du is a positive number. If both k u 1 u-1 p and k 1 u u 1 q are strictly .