Invariant subspaces in Bergman spaces and Hedenmalm's boundary value problem

D. Khavinson Department of Mathematical Sciences, University of Arkansas H. S. Shapiro Department of Mathematics, Royal Institute of Technology

TBD mathscidoc:1701.332818

Arkiv for Matematik, 32, (2), 309-321, 1993.3
A function$G$in a Bergman space$A$^{$p$}, 0<$p$<∞, in the unit disk$D$is called$A$^{$p$}-inner if |$G$|^{$p$}−1 annihilates all bounded harmonic functions in$D$. Extending a recent result by Hedenmalm for$p$=2, we show (Thm. 2) that the unique compactly-supported solution Φ of the problem $$\Delta \Phi = \chi _D (|G|^p - 1),$$ where χ_{$D$}denotes the characteristic function of$D$and$G$is an arbitrary$A$^{$p$}-inner function, is continuous in$C$, and, moreover, has a vanishing normal derivative in a weak sense on the unit circle. This allows us to extend all of Hedenmalm's results concerning the invariant subspaces in the Bergman space$A$^{2}to a general$A$^{$p$}-setting.
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@inproceedings{d.1993invariant,
  title={Invariant subspaces in Bergman spaces and Hedenmalm's boundary value problem},
  author={D. Khavinson, and H. S. Shapiro},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203543668264627},
  booktitle={Arkiv for Matematik},
  volume={32},
  number={2},
  pages={309-321},
  year={1993},
}
D. Khavinson, and H. S. Shapiro. Invariant subspaces in Bergman spaces and Hedenmalm's boundary value problem. 1993. Vol. 32. In Arkiv for Matematik. pp.309-321. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203543668264627.
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