Bulletin of the Belgian Mathematical Society - Simon Stevin

Integrability of homogeneous polynomials on the unit ball

Piotr Kot

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Abstract

We construct some measure $\Theta^{\alpha}$ such that if $0<\alpha\leq2n-2$, $\beta=n-\frac{2+\alpha}{2}$ and $E$ is a circular set of type $G_{\delta}$ such that $E\subset\partial\Bbb B^{n}$ and $\Theta^{\alpha}(E)=0$ then there exists $f\in\Bbb O(\Bbb B^{n})\cap L^{2}(\Bbb B^{n})$ such that \[ E=E^{\beta}(f):=\left\{ z\in \partial B^{n}:\:\int_{\Bbb Dz}\left|f\right|^{2}\chi_{\beta}d\mathfrak{L}^{2}=\infty\right\} \] where $\chi_{s}:\Bbb B^{n}\ni z\longrightarrow\chi_{s}(z)=(1-\left\Vert z\right\Vert ^{2})^{s}$ and $\Bbb D$ denotes the unit disc in $\Bbb C$.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 13, Number 4 (2006), 743-762.

Dates
First available in Project Euclid: 16 January 2007

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1168957350

Digital Object Identifier
doi:10.36045/bbms/1168957350

Mathematical Reviews number (MathSciNet)
MR2300630

Zentralblatt MATH identifier
1124.32002

Subjects
Primary: 32A05: Power series, series of functions 32A35: Hp-spaces, Nevanlinna spaces [See also 32M15, 42B30, 43A85, 46J15]

Keywords
homogeneous polynomials exceptional sets highly nonintegrable holomorphic functions

Citation

Kot, Piotr. Integrability of homogeneous polynomials on the unit ball. Bull. Belg. Math. Soc. Simon Stevin 13 (2006), no. 4, 743--762. doi:10.36045/bbms/1168957350. https://projecteuclid.org/euclid.bbms/1168957350


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