Extrapolation of solvability of the regularity and the Poisson regularity problems in rough domains

IF 1.7 2区 数学 Q1 MATHEMATICS Journal of Functional Analysis Pub Date : 2024-09-13 DOI:10.1016/j.jfa.2024.110672
Josep M. Gallegos , Mihalis Mourgoglou , Xavier Tolsa
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Abstract

Let ΩRn+1, n2, be an open set satisfying the corkscrew condition with n-Ahlfors regular boundary ∂Ω, but without any connectivity assumption. We study the connection between solvability of the regularity problem for divergence form elliptic operators with boundary data in the Hajłasz-Sobolev space M1,1(Ω) and the weak-A property of the associated elliptic measure. In particular, we show that solvability of the regularity problem in M1,1(Ω) is equivalent to the solvability of the regularity problem in M1,p(Ω) for some p>1. We also prove analogous extrapolation results for the Poisson regularity problem defined on tent spaces. Moreover, under the hypothesis that ∂Ω supports a weak (1,1)-Poincaré inequality, we show that the solvability of the regularity problem in the Hajłasz-Sobolev space M1,1(Ω) is equivalent to a stronger solvability in a Hardy-Sobolev space of tangential derivatives.

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粗糙域中正则性和泊松正则性问题的可解性外推法
假设Ω⊂Rn+1, n≥2 是满足开方条件的开放集,具有 n-Ahlfors 正则边界∂Ω,但没有任何连通性假设。我们研究了在 Hajłasz-Sobolev 空间 M1,1(∂Ω) 中具有边界数据的发散形式椭圆算子正则性问题的可解性与相关椭圆度量的弱 A∞ 特性之间的联系。我们特别证明了 M1,1(∂Ω) 中的正则性问题的可解性等同于对于某个 p>1 的 M1,p(∂Ω) 中的正则性问题的可解性。 我们还证明了定义在帐篷空间上的泊松正则性问题的类似外推法结果。此外,在 ∂Ω 支持弱 (1,1)-Poincaré 不等式的假设下,我们证明了正则性问题在 Hajłasz-Sobolev 空间 M1,1(∂Ω) 中的可解性等同于切向导数的 Hardy-Sobolev 空间中更强的可解性。
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来源期刊
CiteScore
3.20
自引率
5.90%
发文量
271
审稿时长
7.5 months
期刊介绍: The Journal of Functional Analysis presents original research papers in all scientific disciplines in which modern functional analysis plays a basic role. Articles by scientists in a variety of interdisciplinary areas are published. Research Areas Include: • Significant applications of functional analysis, including those to other areas of mathematics • New developments in functional analysis • Contributions to important problems in and challenges to functional analysis
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