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Transactions of the American Mathematical Society

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About boundary values in $ A(\Omega)$


Author: Piotr Kot
Journal: Trans. Amer. Math. Soc. 363 (2011), 4063-4079
MSC (2010): Primary 32A05, 32A40
DOI: https://doi.org/10.1090/S0002-9947-2011-05083-X
Published electronically: March 21, 2011
MathSciNet review: 2792980
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Abstract: Assume that $ \Omega\subset\mathbb{C}^{n}$ is a balanced bounded domain with a holomorphic support function (e.g. strictly pseudoconvex domain with $ C^{2}$ boundary). We denote $ \left\Vert f\right\Vert _{z}^{2}:=\int_{0}^{1}\vert f(e^{2\pi it}z)\vert^{2}dt$. Let $ \varepsilon>0$ and $ \sigma$ be a circular invariant Borel probability measure on $ \partial\Omega$. If $ g\in A(\Omega)$ and $ h$ is a continuous function on $ \partial\Omega$ with $ \vert g\vert<h$ on $ \partial\Omega$, then we construct nonconstant functions $ f_{1},f_{2}\in A(\Omega)$ with $ \left\Vert g+f_{1}\right\Vert _{z}\leq\left\Vert h\right\Vert _{z}$, $ \vert(g+f_{2})(z)\vert\leq\max_{\vert\lambda\vert=1}h(\lambda z)$ for $ z\in\partial\Omega$ and

$\displaystyle \sigma\left(\left\{ z\in\partial\Omega:\left\Vert g+f_{1}\right\V... ... z)\vert \ne\max_{\vert\lambda\vert=1}h(\lambda z)\right\} \right)<\varepsilon.$

Additionally if $ \Omega$ is a circular, bounded, strictly convex domain with $ C^{2}$ boundary, then we give the construction of $ f_{3}\in\mathbb{O}(\Omega)$, the holomorphic function with: $ \left\Vert h-\vert g+f_{3}^{*}\vert\right\Vert _{z}=0$ for all $ z\in\partial\Omega$, where $ f^{*}$ denotes the radial limit of $ f$. We also construct $ f_{4}\in A(\Omega)$ with $ \left\Vert g+f_{4}\right\Vert _{z}=\left\Vert h\right\Vert _{z}$ for $ z\in\partial\Omega$.

In all cases we can make $ f_{i}$ arbitrarily small on a given compact subset $ F\subset\Omega$ and make it vanish to a given order at the point 0.


References [Enhancements On Off] (What's this?)

  • 1. A.B. Aleksandrov: The existence of inner functions in the unit ball. Math. Sb. 117, 147-163 (1982). MR 658785 (83i:32002)
  • 2. P. Kot: A Holomorphic Function with Given Almost All Boundary Values on a Domain with Holomorphic Support Function, Journal of Convex Analysis 14, no. 4, 693-704 (2007). MR 2350811 (2008m:32009)
  • 3. P. Kot: Homogeneous polynomials on strictly convex domains, Proc. Amer. Math. Soc. 135, 3895-3903 (2007). MR 2341939 (2008m:32010)
  • 4. P. Kot: Bounded holomorphic functions with given maximum modulus on all circles, Proc. Amer. Math. Soc. 137, 179-187 (2009). MR 2439439 (2009f:32008)
  • 5. E. Löw: Inner Functions and Boundary Values in $ H^{\infty}(\Omega)$ and $ A(\Omega)$ in Smoothly Bounded Pseudoconvex Domains, Math. Z. 185, 191-210 (1984). MR 731340 (85d:32032)
  • 6. B. Stensönes Henriksen: A peak set of Hausdorff dimension $ 2n-1$ for the algebra $ A(D)$ in the boundary of a domain $ D$ with $ C^{\infty}$-boundary in $ \mathbb{C}^{n}$, Math. Ann. 259, 271-277 (1982). MR 656666 (83k:32028)
  • 7. E.L. Stout, Th. Duchamp: Maximum modulus sets, Ann. Inst. Fourier (Grenoble) 31.3, 37-69 (1981). MR 638616 (83d:32019)
  • 8. E.L. Stout: The dimension of peak interpolation sets, Proc. Amer. Math. Soc. 86, no. 3, 413-416 (1982). MR 671206 (84e:32015)

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Additional Information

Piotr Kot
Affiliation: Instytut Matematyki, Politechnika Krakowska, ul. Warszawska 24, 31-155 Kraków, Poland
Email: pkot@pk.edu.pl

DOI: https://doi.org/10.1090/S0002-9947-2011-05083-X
Keywords: Maximum modulus set, inner function.
Received by editor(s): December 16, 2007
Received by editor(s) in revised form: April 13, 2009
Published electronically: March 21, 2011
Article copyright: © Copyright 2011 American Mathematical Society

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