欧几里得半空间中调和函数的小伍德型定理

中文摘要 AI 辅助

1927年，J.E. 小伍德证明：对于单位圆盘上的有界调和函数，通过非切向逼近区域存在几乎处处边界值的法图型结果，对于具有以下两个额外性质的任意切向逼近区域系统均不成立：该系统由终止于各边界点的曲线构成，且具有旋转不变性。然而，1984年A. 内格尔（A. Nagel）与E.M. 斯泰因（E.M. Stein）在阐述鲁丁（Rudin，1979）以及内格尔、鲁丁与夏皮罗（Nagel, Rudin and Shapiro，1982）的结果基础上，证明了存在旋转不变的切向序列系统，使得法图型结果成立，即沿该系统，每个有界调和函数几乎处处收敛到其非切向边界值。此外，他们将该结果推广到更高维欧几里得半空间。内格尔-斯泰因结果引出了一个问题：构建并证明一个也可应用于切向逼近区域（即序列式切向逼近区域）的小伍德型定理。本文针对更高维欧几里得半空间中的有界调和函数，证明了一个小伍德型定理，该定理适用于包含序列式切向逼近区域的系统。除了我们近期针对单位圆盘的结果外，这是首个此类结果。实际上，在所有其他小伍德型结果中，切向逼近区域被要求为曲线型或至少具有某种排除其为序列式可能性的拓扑性质。

英文摘要

In 1927 J.E. Littlewood proved that, for bounded harmonic functions on the unit disc, the Fatou-type result, on the existence of almost everywhere boundary values through approach regions that are nontangential, will fail for any system of tangential approach regions that has the following two additional properties: It consists of curves ending at the various boundary points, and it is rotationally invariant. However, in 1984 A. Nagel and E.M. Stein, elaborating results of Rudin (1979) and Nagel, Rudin and Shapiro (1982), proved the existence of rotationally invariant systems of tangential sequences along which a Fatou-type result holds, i.e., along which every bounded harmonic function converges a.e. to its nontangential boundary values. Moreover, they extended their result to higher-dimensional Euclidean half-spaces. The Nagel-Stein result has prompted the question of formulating and proving a Littlewood type theorem that can also be applied to tangential approach regions which are sequential. In this paper we prove a Littlewood-type theorem for bounded harmonic functions in higher-dimensional Euclidean half-spaces, for systems of tangential approach regions which includes the sequential ones. This is the first result of this kind, apart from a recent result of ours whose setting is the unit disc. Indeed, in all the other results of Littlewood type, the tangential approach regions were required to be curvilinear or at least to possess a certain topological property that excluded the possibility that they could be sequential.

发表机构

• University of Rwanda（卢旺达大学）
• Linköping University（林雪平大学）

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