用平面 Lipschitz 域上的全纯函数表达边界值的新方法

Steven R. Bell, Loredana Lanzani, Nathan A. Wagner
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摘要

我们将简单相连的利普斯奇茨域 $Omega \subset \mathbb C$ 边界上的 $p$ - 可积分函数分解为两个唯一确定的全纯函数的边界值之和,其中一个在 $\Omega$ 中是全纯的,而另一个在 $\mathbb C\setminus\overline{\Omega}$ 中是全纯的,并且在无穷远处消失。在光滑的情况下,分解的唯一性是基本的,但将其扩展到 $L^p$ 设置依赖于全形哈代空间 $h^p(b\Omega)$ 的正则性结果,即使对于光滑的 $\Omega$ 来说,这个结果似乎也是新的。我们的结果的直接后果将是作用于 $L^p(b\Omega)$ 的考奇变换的核的新表征。这些结果为谐函数的经典狄利克特问题和泊松公式提供了新的视角,即使在圆盘的情况下也是如此。本文还介绍了进一步的应用以及未来的工作方向。
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A new way to express boundary values in terms of holomorphic functions on planar Lipschitz domains
We decompose $p$ - integrable functions on the boundary of a simply connected Lipschitz domain $\Omega \subset \mathbb C$ into the sum of the boundary values of two, uniquely determined holomorphic functions, where one is holomorphic in $\Omega$ while the other is holomorphic in $\mathbb C\setminus \overline{\Omega}$ and vanishes at infinity. This decomposition has been described previously for smooth functions on the boundary of a smooth domain. Uniqueness of the decomposition is elementary in the smooth case, but extending it to the $L^p$ setting relies upon a regularity result for the holomorphic Hardy space $h^p(b\Omega)$ which appears to be new even for smooth $\Omega$. An immediate consequence of our result will be a new characterization of the kernel of the Cauchy transform acting on $L^p(b\Omega)$. These results give a new perspective on the classical Dirichlet problem for harmonic functions and the Poisson formula even in the case of the disc. Further applications are presented along with directions for future work.
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