Asymptotically Mean Value Harmonic Functions in Doubling Metric Measure Spaces

Pub Date : 2020-05-28 DOI:10.1515/agms-2022-0143
Tomasz Adamowicz, Antoni Kijowski, Elefterios Soultanis
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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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