Poisson integrals for standard weighted Laplacians in the unit disc
Poisson integrals for standard weighted Laplacians in the unit disc
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DOI：
10.2969/jmsj/06520447
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发表时间：
2013-04
影响因子：
0.7
通讯作者：
A. Olofsson;Jens Wittsten
中科院分区：
文献类型：
--
作者：
A. Olofsson;Jens Wittsten
本文对单位圆盘上的一类标准加权拉普拉斯微分算子，给出了经典Poisson积分公式的一个对应公式。在此过程中，相应的Dirichlet边值问题是解决任意分布的边界数据。边界极限和相关的解决方案的表示研究齐次Banach空间的框架内。特别强调的是把所谓的相对完备的齐次Banach空间。0.设Ω是复平面C中的一个区域，其权函数w：Ω →（0，∞）为我们提供了一种计算加权面积的方法，即利用加权面积元（0.1）dAw（z）= w（z）dA（z），z = x+ iy ∈ Ω，其中dA（z）= dxdy是通常的平面勒贝格面积测度。通过几何考虑，面积元（0.1）是由（黎曼）度量（0.2）dsw（z）2 = w（z）诱导的面积元。|DZ|，z ∈ Ω，其中|DZ|是复平面的通常弧长元素。我们在这里提到小林和Nomizu [24，25]作为一个广泛的背景微分几何。复分析中的一个基本研究对象是柯西-黎曼微分算子z = z = 1 2（x − 1 i y），z = x+ iy ∈ C，我们可以把它看作复平面上的向量场C z。度量（0.2）建议我们考虑加权Cauchy-Riemann微分算子w，z = w（z）z，z ∈ Ω。日期：2011年8月16日。2010年数学学科分类。小学：31 A05;中学：35 J25。
In this paper a counterpart of the classical Poisson integral formula is found for a class of standard weighted Laplace differential operators in the unit disc. In the process the corresponding Dirichlet boundary value problem is solved for arbitrary distributional boundary data. Boundary limits and representations of the associated solutions are studied within a framework of homogeneous Banach spaces. Special emphasis is put on the so-called relative completion of a homogeneous Banach space. 0. Introduction Let Ω be a domain in the complex plane C equipped with a weight function w : Ω → (0,∞) which provides us with a way to calculate weighted area using the weighted area element (0.1) dAw(z) = w(z)dA(z), z = x+ iy ∈ Ω, where dA(z) = dxdy is the usual planar Lebesgue area measure. By geometric considerations the area element (0.1) is the area element induced by the (Riemannian) metric (0.2) dsw(z) 2 = w(z)|dz|, z ∈ Ω, where |dz| is the usual arclength element of the complex plane. We mention here Kobayashi and Nomizu [24, 25] as an extensive background on differential geometry. A fundamental object of study in complex analysis is the Cauchy-Riemann differential operator ∂̄z = ∂ ∂z̄ = 1 2 ( ∂ ∂x − 1 i ∂ ∂y ) , z = x+ iy ∈ C, which we can think of as a vector field C ∋ z 7→ ∂̄z over the complex plane. The metric (0.2) suggests us to consider the weighted Cauchy-Riemann differential operator ∂̄w,z = w(z) ∂̄z, z ∈ Ω. Date: August 16, 2011. 2010 Mathematics Subject Classification. Primary: 31A05; Secondary: 35J25.