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: BEST CONSTANTS FOR CERTAIN MULTILINEAR INTEGRAL OPERATORS | BEST CONSTANTS FOR CERTAIN MULTILINEAR INTEGRAL OPERATORS ARPAD BENYI AND CHOONGHONG TADAHIRO OH Received 7 December 2004 Revised 8 March 2005 Accepted 27 March 2005 We provide explicit formulas in terms of the special function gamma for the best constants in nontensorial multilinear extensions of some classical integral inequalities due to Hilbert Hardy and Hardy-Littlewood-Polya. Copyright 2006 A. Benyi and C. Tadahiro Oh. 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 and statement of results In his lectures on integral equations Hilbert initiated the study of maxima of bilinear and multilinear forms. The case p 2 in the following inequality is nowadays known as Hilbert s double series theorem. If p 1 p p p - 1 Sm i am A and Xn 1 bn B then 00 00 m 1 n 1 ambn m n ncsc f n A1 pB1 p . p Hilbert s proof apart from the determination of the best possible constant ncsc n p was published by Weyl 7 . The calculation of the constant and the integral analogue of Hilbert s double series theorem for p 2 are due to Schur 6 . The generalizations to other p s of both the discrete and integral versions of this result were discovered later on by Hardy and Riesz and published by Hardy in 4 . The statement of the integral analogue of the theorem above is the following. If p 1 p p p - 1 r fp x dx F V gp y dy G then JJ f y dx dy n csc n F1 pG1 p . Other proofs and generalizations in different directions were given by several authors see the book by Hardy et al. 5 page 227 . Applications of Hilbert s theorem range from the Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2006 Article ID 28582 Pages 1-12 DOI JIA 2006 28582 2 Best constants for certain multilinear integral operators theory of analytic functions to results about the moments of real variable functions
