Mean type of functions of bounded characteristic and Martin functions in Denjoy domains

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• 作者：Alexander Volberg^a ; ¹ ; ^{volberg@math.msu.edu" class="auth_mail" title="E-mail the corresponding author} ; Peter Yuditskii^b ; ² ; ^{peter.yuditskii@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Martin function ; Denjoy domains ; Bounded characteristic ; Mean type ; Widom domains ; Harmonic measure
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：26 February 2016
• 年：2016
• 卷：290
• 期：Complete
• 页码：860-887
• 全文大小：489 K

文摘

Functions of bounded characteristic in simply connected domains have a classical factorization to Blaschke, outer and singular inner parts. The latter has a singular measure on the boundary assigned to it. The exponential speed of change of a function when approaching a point of a boundary (mean type) corresponds to a point mass at this point. In this paper we consider the analogous relation for functions in infinitely connected (Denjoy) domains. The factorization result holds of course with one important addition: all functions involved become multiple valued even though the initial function was single valued. The mean type now can be measured by using the Martin function of the domain. But this result does not follow from the lifting to the universal covering of the domain because of the simple (but interesting) reason that the mean types of the original and the lifted functions can be completely different.