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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 Some Quasimetrics and Their Applications | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 167403 12 pages doi 10.1155 2009 167403 Research Article On Some Quasimetrics and Their Applications Imed Bachar1 and Habib Maagli2 1 Mathematics Department College of Sciences King Saud University P.O. Box 2455 Riyadh 11451 Saudi Arabia 2 Departement de Mathématiques Faculte des Sciences de Tunis Campus Universitaire 2092 Tunis Tunisia Correspondence should be addressed to Imed Bachar abachar@ksu.edu.sa Received 27 September 2009 Accepted 8 December 2009 Recommended by Shusen Ding We aim at giving a rich class of quasi-metrics from which we obtain as an application an interesting inequality for the Greens function of the fractional Laplacian in a smooth domain in R . Copyright 2009 I. Bachar and H. Maagli. 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 Let D be a bounded smooth domain in R n 1 or D R x xn e R xn 0 the half space. We denote by GD the Green s function of the operator u -A au with Dirichlet or Navier boundary conditions where a is a positive integer or 0 a 1. The following inequality called 3G inequality has been proved by several authors see 1 2 for a 1 or 3 4 for a 1 or 5 for 0 a 1 . There exists a constant C 0 such that for each x y z e D gd x z GD z y X z X z GtCy - C iixiDMiy GDlyZ 1-1 where x X x is a positive function which depends on the Euclidean distance between x and dD and the exponent a. More precisely to prove this inequality the authors showed that the function x y p x y X x X y GD x y is a quasi-metric on D see Definition 2.1 . We emphasis that the generalized 3G inequality is crucial for various applications see e.g. 1 Theorem 1.2 6 Lemma 7.1 . It is also very interesting tools for analysts working on pde s. In 7 Theorem 5.1 the authors used the standard 3G .