Boundary Behaviour of Functions in Weighted Dirichlet Spaces

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• 作者：Finbarr Holland (1)
  J. Brian Twomey (1)
  
• 关键词：Weighted Dirichlet spaces ; radial variation ; radial limits ; exceptional sets ; zero capacity ; 30H05 ; 31A20
• 刊名：Computational Methods and Function Theory
• 出版年：2007
• 出版时间：December 2007
• 年：2007
• 卷：7
• 期：2
• 页码：361-370
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• 作者单位：Finbarr Holland (1)
  J. Brian Twomey (1)
  
  1. Department of Mathematics, University College Cork, Cork, Ireland
  
• ISSN：2195-3724

文摘

We consider the class $\mathcal{D}_W$ of holomorphic functions f(z) = Σ a n z n in the unit disc for which Σ W(n)|a n |2 < ? where the weight function W satisfies standard regularity conditions. We show that if Σ 1/(nW(n)) < ?and $f \in \mathcal{D}_W$ , then the radial variation L f (θ) = ?span class="a-plus-plus stack"> 0 1 |f-re iθ )| dr is finite outside an exceptional set of capacity zero, where the kernel associated with the capacity depends on W. It is known that if Σ 1/(nW(n)) = ? then there exist functions in $\mathcal{D}_W$ with L f(θ) = ?for every θ. We also show that it is a consequence of known results that if $f \in \mathcal{D}_W$ and Σ 1/W(n) = ? then f has finite radial, and non-tangential, limits outside certain exceptional sets.