论文标题
Beurling-Carleson集，内部功能和半线性方程
Beurling-Carleson sets, inner functions and a semi-linear equation
论文作者
论文摘要
Beurling-Carleson集合已经出现在许多复杂分析的领域中，例如分析函数的零边界零集，内瓦林纳纳类中具有导数的内部功能，加权伯格曼空间中的循环性，富是否Widom-type组的紫红色群和有量的Banach Algebras中的Corona问题。在调查了这些事态发展之后，我们给出了beurling-Carleson集的一般定义，并讨论了它们的一些基本属性。我们表明，罗伯特的分解表征了不收取beurling-carleson套装的度量。 对于单位圆上的积极单数度量$μ$，让$s_μ$表示单数尺寸$μ$的单数内部功能。在本文的第二部分中，我们使用电晕型分解与单位圆圈上的许多奇异措施的属性相关联，例如Nevanlinna类$ \ Mathcal $ \ Mathcal n $中的$ s'__ $的会员资格，$s_μ$和WEWPAIMALY级别的区域条件。众所周知，这些特性中的每一个都符合集中在贝尔林 - 卡利森集中的措施。我们表明，这些属性中的每一个都意味着$μ$生活在可数的 - 卡勒森套装的可数联合中。我们还描述了涉及Hardy Space $ H^p $的$S'_μ$的局部关系，在Besov Space $ b^p $和$ b^p $和$（1-p）$ - beurling-beurling-carleson套件中的会员资格，并给出了许多示例，这些示例表明我们的结果是最佳的。 最后，我们表明，在$α$ -Bebeerling-Carleson套件的可计数工会上进行的措施几乎进行了两次培养，当$ΔU= u^u^p \ cdotχ_{u> 0} $时，当$ p> 3 $和$ p> 3 $和$α= = \ frac {p-3} $时。
Beurling-Carleson sets have appeared in a number of areas of complex analysis such as boundary zero sets of analytic functions, inner functions with derivative in the Nevanlinna class, cyclicity in weighted Bergman spaces, Fuchsian groups of Widom-type and the corona problem in quotient Banach algebras. After surveying these developments, we give a general definition of Beurling-Carleson sets and discuss some of their basic properties. We show that the Roberts decomposition characterizes measures that do not charge Beurling-Carleson sets. For a positive singular measure $μ$ on the unit circle, let $S_μ$ denote the singular inner function with singular measure $μ$. In the second part of the paper, we use a corona-type decomposition to relate a number of properties of singular measures on the unit circle such as membership of $S'_μ$ in the Nevanlinna class $\mathcal N$, area conditions on level sets of $S_μ$ and wepability. It was known that each of these properties holds for measures concentrated on Beurling-Carleson sets. We show that each of these properties implies that $μ$ lives on a countable union of Beurling-Carleson sets. We also describe partial relations involving the membership of $S'_μ$ in the Hardy space $H^p$, membership of $S_μ$ in the Besov space $B^p$ and $(1-p)$-Beurling-Carleson sets and give a number of examples which show that our results are optimal. Finally, we show that measures that live on countable unions of $α$-Beurling-Carleson sets are almost in bijection with nearly-maximal solutions of $Δu = u^p \cdot χ_{u > 0}$ when $p > 3$ and $α= \frac{p-3}{p-1}$.
