Asymptotically Mean Value Harmonic Functions in Doubling Metric Measure Spaces

IF 0.9 3区数学 Q2 MATHEMATICS

Analysis and Geometry in Metric Spaces Pub Date : 2020-05-28 DOI:10.1515/agms-2022-0143

Tomasz Adamowicz, Antoni Kijowski, Elefterios Soultanis

引用次数: 3

Abstract

Abstract We consider functions with an asymptotic mean value property, known to characterize harmonicity in Riemannian manifolds and in doubling metric measure spaces. We show that the strongly amv-harmonic functions are Hölder continuous for any exponent below one. More generally, we define the class of functions with finite amv-norm and show that functions in this class belong to a fractional Hajłasz–Sobolev space and their blow-ups satisfy the mean-value property. Furthermore, in the weighted Euclidean setting we find an elliptic PDE satisfied by amv-harmonic functions.

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双度量测度空间中的渐近均值调和函数

摘要考虑具有渐近均值性质的函数，这些函数在黎曼流形和双度量度量空间中具有调和性。我们证明了强谐波函数对于任何低于1的指数都是Hölder连续的。更一般地，我们定义了一类具有有限amv-范数的函数，并证明了该类函数属于分数阶Hajłasz-Sobolev空间，并且它们的膨胀满足中值性质。此外，在加权欧几里得环境下，我们得到了一个由谐波函数满足的椭圆偏微分方程。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Analysis and Geometry in Metric Spaces Mathematics-Geometry and Topology

CiteScore

1.80

自引率

0.00%

发文量

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.