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Báo cáo hóa học: " Research Article On a Hilbert-Type Operator with a Symmetric Homogeneous Kernel of −1-Order and Applications"

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 Symmetric Homogeneous Kernel of −1-Order and Applications | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 47812 9 pages doi 10.1155 2007 47812 Research Article On a Hilbert-Type Operator with a Symmetric Homogeneous Kernel of -1-Order and Applications Bicheng Yang Received 21 March 2007 Accepted 12 July 2007 Recommended by Shusen Ding Some character of the symmetric homogenous kernel of -1-order in Hilbert-type operator T lr - lr r 1 is obtained. Two equivalent inequalities with the symmetric homogenous kernel of - A-order are given. As applications some new Hilbert-type inequalities with the best constant factors and the equivalent forms as the particular cases are established. Copyright 2007 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 If the real function k x y is measurable in 0 to X 0 to satisfying k y x k x y for x y e 0 to then one calls k x y the symmetric function. Suppose that p 1 1 p 1 q 1 lr r p q are two real normal spaces and k x y is a nonnegative symmetric function in 0 to X 0 to . Define the operator T as follows for a am TO 1 e lp to Ta n k m n am n e N 1.1 m 1 or for b bn TO 1 e lq Tb m to k m n bn m e N. 1.2 n 1 The function k x y is said to be the symmetric kernel of T. 2 Journal of Inequalities and Applications If k x y is a symmetric function for e 0 small enough and x 0 set kr e x as f to x 1 e r kr e x k x t X dt r p q . 1.3 0 t In 2007 Yang 1 gave three theorems as follows. Theorem 1.1. i If for fixed x 0 and r p q the functions k x t x t 1 r are decreasing in t e 0 to and f to í x 1 r kr 0 x k x t i x dt kp 0 t p r p q 1.4 where kp is a positive constant independent of x then T e B lr lr T is called the Hilbert-type operator and IITHr kp r p q ii if for fixed x 0 e 0 and r p q the functions k x t x t 1 e r are decreasing in t e 0 to kr e x kp e r p q e 0 is