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Báo cáo hóa học: " Research Article A Kind of Estimate of Difference Norms in Anisotropic Weighted Sobolev-Lorentz Spaces"

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 A Kind of Estimate of Difference Norms in Anisotropic Weighted Sobolev-Lorentz Spaces | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 161405 22 pages doi 10.1155 2009 161405 Research Article A Kind of Estimate of Difference Norms in Anisotropic Weighted Sobolev-Lorentz Spaces Jiecheng Chen1 and Hongliang Li1 2 1 Department of Mathematics Zhejiang University Hangzhou 310027 China 2 Department of Mathematics Zhejiang Education Institute Hangzhou 310012 China Correspondence should be addressed to Hongliang Li honglli@126.com Received 27 April 2009 Accepted 2 July 2009 Recommended by Shusen Ding We investigate the functions spaces on R for which the generalized partial derivatives Dff exist and belong to different Lorentz spaces APk Sk w where Pk 1 and w is nonincreasing and satisfies some special conditions. For the functions in these weighted Sobolev-Lorentz spaces the estimates of the Besov type norms are found. The methods used in the paper are based on some estimates of nonincreasing rearrangements and the application of Bp Bp weights. Copyright 2009 J. Chen and H. Li. 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 this paper we study functions f on R which possess the generalized partial derivatives Dĩf - ĨtẮ rk e N . 1.1 k oxf Our main goal is to obtain some norm estimates for the differences à h f x - Ệ -1 rk-ij f x jhek h e R 1.2 ek being the unit coordinate vector . 2 Journal of Inequalities and Applications The classic Sobolev embedding theorem space Wp Rn 1 p n asserts that for any function f in Sobolev Ilf II cỷ f cf 1.3 llf v C 1 dxk p q n - p. u Sobolev proved this inequality in 1938 for p 1. His method based on integral representations did not work in the case p 1. Only at the end of fifties Gagliardo and Nirenberg gave simple proofs of inequality 1.3 for all 1 p n. Inequality 1.3 has been generalized in various