非自相似Sierpiński海绵上的等周不等式和poincar不等式:边界情况

IF 0.9 3区 数学 Q2 MATHEMATICS
S. Eriksson-Bique, Jasun Gong
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引用次数: 3

摘要

摘要本文构造了欧几里德空间中支持1- poincarcars不等式但内部为空的子集的一大组例子。这些例子是由一个迭代过程形成的,这个过程包括移除行为良好的域,或者更准确地说,移除那些补体在Martio和Sarvas的意义上是一致的域。虽然现有的论证依赖于Semmes曲线族的显式构造,但在Korte和Lahti之后,我们包括了一种通过使用相对等周不等式获得庞加莱不等式的新方法。为此,我们进一步引入了在给定密度水平上的等周不等式的概念和迭代这种不等式的方法。介绍了这些工具,并将其应用于一般的度量度量。我们的例子包含了Mackay, Tyson和Wildrick之前关于非自相似Sierpiński地毯的结果,并将它们扩展到许多更一般的形状以及更高的维度。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Isoperimetric and Poincaré Inequalities on Non-Self-Similar Sierpiński Sponges: the Borderline Case
Abstract In this paper we construct a large family of examples of subsets of Euclidean space that support a 1-Poincaré inequality yet have empty interior. These examples are formed from an iterative process that involves removing well-behaved domains, or more precisely, domains whose complements are uniform in the sense of Martio and Sarvas. While existing arguments rely on explicit constructions of Semmes families of curves, we include a new way of obtaining Poincaré inequalities through the use of relative isoperimetric inequalities, after Korte and Lahti. To do so, we further introduce the notion of of isoperimetric inequalities at given density levels and a way to iterate such inequalities. These tools are presented and apply to general metric measure measures. Our examples subsume the previous results of Mackay, Tyson, and Wildrick regarding non-self similar Sierpiński carpets, and extend them to many more general shapes as well as higher dimensions.
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来源期刊
Analysis and Geometry in Metric Spaces
Analysis and Geometry in Metric Spaces Mathematics-Geometry and Topology
CiteScore
1.80
自引率
0.00%
发文量
8
审稿时长
16 weeks
期刊介绍: Analysis and Geometry in Metric Spaces is an open access electronic journal that publishes cutting-edge research on analytical and geometrical problems in metric spaces and applications. We strive to present a forum where all aspects of these problems can be discussed. AGMS is devoted to the publication of results on these and related topics: Geometric inequalities in metric spaces, Geometric measure theory and variational problems in metric spaces, Analytic and geometric problems in metric measure spaces, probability spaces, and manifolds with density, Analytic and geometric problems in sub-riemannian manifolds, Carnot groups, and pseudo-hermitian manifolds. Geometric control theory, Curvature in metric and length spaces, Geometric group theory, Harmonic Analysis. Potential theory, Mass transportation problems, Quasiconformal and quasiregular mappings. Quasiconformal geometry, PDEs associated to analytic and geometric problems in metric spaces.
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