# MathSciDoc: An Archive for Mathematician ∫

#### TBDmathscidoc: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.
@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.