TAILIEUCHUNG - Đề tài " The Poincar´e inequality is an open ended condition "

Let p 1 and let (X, d, μ) be a complete metric measure space with μ Borel and doubling that admits a (1, p)-Poincar´ inequality. Then there exists e ε 0 such that (X, d, μ) admits a (1, q)-Poincar´ inequality for every q p−ε, e quantitatively. 1. Introduction Metric spaces of homogeneous type, introduced by Coifman and Weiss [7], [8], have become a standard setting for harmonic analysis related to singular integrals and Hardy spaces. Such metric spaces are often referred to as a metric measure space with a doubling measure. An advantage of working. | Annals of Mathematics The Poincar e inequality is an open ended condition By Stephen Keith and Xiao Zhong Annals of Mathematics 167 2008 575 599 The Poincare inequality is an open ended condition By Stephen Keith and Xiao Zhong Abstract Let p 1 and let X d p be a complete metric measure space with p Borel and doubling that admits a 1 p -Poincare inequality. Then there exists e 0 such that X d p admits a 1 q -Poincare inequality for every q p e quantitatively. 1. Introduction Metric spaces of homogeneous type introduced by Coifman and Weiss 7 8 have become a standard setting for harmonic analysis related to singular integrals and Hardy spaces. Such metric spaces are often referred to as a metric measure space with a doubling measure. An advantage of working with these spaces is the wide collection of examples see 6 47 . A second advantage is that many classical theorems from Euclidean space still remain true including the Vitali covering theorem the Lebesgue differentiation theorem the Hardy-Littlewood maximal theorem and the John-Nirenberg lemma see 20 47 . However theorems that rely on methods beyond zero-order calculus are generally unavailable. To move into the realm of first-order calculus requires limiting attention to fewer metric measure spaces and is often achieved by requiring that a Poincare inequality is admitted. Typically a metric measure space is said to admit a Poincare inequality or inequalities if a significant collection of real-valued functions defined over the space observes Poincare inequalities as in in some uniform sense. There are many important examples of such spaces see 28 26 and many classical first-order theorems from Euclidean space remain true in this setting. These include results from second-order partial differential equations quasiconformal mappings geometric measure theory and Sobolev . was partially supported by the Academy of Finland project 53292 and the Australian Research Council. . was partially supported by the

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