Removable singularities for weighted Bergman spaces

Anders Björn

Czechoslovak Mathematical Journal (2006)

  • Volume: 56, Issue: 1, page 179-227
  • ISSN: 0011-4642

Abstract

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We develop a theory of removable singularities for the weighted Bergman space 𝒜 μ p ( Ω ) = { f analytic in Ω ∫ Ω | f | p d μ < ∞ } , where μ is a Radon measure on ℂ . The set A is weakly removable for 𝒜 μ p ( Ω ∖ A ) if 𝒜 μ p ( Ω ∖ A ) ⊂ Hol ( Ω ) , and strongly removable for 𝒜 μ p ( Ω ∖ A ) if 𝒜 μ p ( Ω ∖ A ) = 𝒜 μ p ( Ω ) . The general theory developed is in many ways similar to the theory of removable singularities for Hardy H p spaces, B M O and locally Lipschitz spaces of analytic functions, including the existence of counterexamples to many plausible properties, e.g. the union of two compact removable singularities needs not be removable. In the case when weak and strong removability are the same for all sets, in particular if μ is absolutely continuous with respect to the Lebesgue measure m , we are able to say more than in the general case. In this case we obtain a Dolzhenko type result saying that a countable union of compact removable singularities is removable. When d μ = w d m and w is a Muckenhoupt A p weight, 1 < p < ∞ , the removable singularities are characterized as the null sets of the weighted Sobolev space capacity with respect to the dual exponent p ' = p / ( p - 1 ) and the dual weight w ' = w 1 / ( 1 - p ) .

How to cite

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Björn, Anders. "Removable singularities for weighted Bergman spaces." Czechoslovak Mathematical Journal 56.1 (2006): 179-227. <http://eudml.org/doc/31023>.

@article{Björn2006,
abstract = {We develop a theory of removable singularities for the weighted Bergman space $\{\mathcal \{A\}\}^p_\mu (\Omega )=\lbrace f \text\{analytic\} \text\{in\} \Omega \: \int _\Omega |f|^p \mathrm \{d\}\mu < \infty \rbrace $, where $\mu $ is a Radon measure on $\mathbb \{C\}$. The set $A$ is weakly removable for $\{\mathcal \{A\}\}^p_\mu (\Omega \setminus A)$ if $\{\mathcal \{A\}\}^p_\mu (\Omega \setminus A) \subset \text\{Hol\}(\Omega )$, and strongly removable for $\{\mathcal \{A\}\}^p_\mu (\Omega \setminus A)$ if $\{\mathcal \{A\}\}^p_\mu (\Omega \setminus A) = \{\mathcal \{A\}\}^p_\mu (\Omega )$. The general theory developed is in many ways similar to the theory of removable singularities for Hardy $H^p$ spaces, $\mathop \{\mathrm \{B\}MO\}$ and locally Lipschitz spaces of analytic functions, including the existence of counterexamples to many plausible properties, e.g. the union of two compact removable singularities needs not be removable. In the case when weak and strong removability are the same for all sets, in particular if $\mu $ is absolutely continuous with respect to the Lebesgue measure $m$, we are able to say more than in the general case. In this case we obtain a Dolzhenko type result saying that a countable union of compact removable singularities is removable. When $\mathrm \{d\}\mu = w\mathrm \{d\}m$ and $w$ is a Muckenhoupt $A_p$ weight, $1<p<\infty $, the removable singularities are characterized as the null sets of the weighted Sobolev space capacity with respect to the dual exponent $p^\{\prime \}=p/(p-1)$ and the dual weight $w^\{\prime \}=w^\{1/(1-p)\}$.},
author = {Björn, Anders},
journal = {Czechoslovak Mathematical Journal},
keywords = {analytic continuation; analytic function; Bergman space; capacity; exceptional set; holomorphic function; Muckenhoupt weight; removable singularity; singular set; Sobolev space; weight; analytic continuation; analytic function; Bergman space; capacity; exceptional set},
language = {eng},
number = {1},
pages = {179-227},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Removable singularities for weighted Bergman spaces},
url = {http://eudml.org/doc/31023},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Björn, Anders
TI - Removable singularities for weighted Bergman spaces
JO - Czechoslovak Mathematical Journal
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 56
IS - 1
SP - 179
EP - 227
AB - We develop a theory of removable singularities for the weighted Bergman space ${\mathcal {A}}^p_\mu (\Omega )=\lbrace f \text{analytic} \text{in} \Omega \: \int _\Omega |f|^p \mathrm {d}\mu < \infty \rbrace $, where $\mu $ is a Radon measure on $\mathbb {C}$. The set $A$ is weakly removable for ${\mathcal {A}}^p_\mu (\Omega \setminus A)$ if ${\mathcal {A}}^p_\mu (\Omega \setminus A) \subset \text{Hol}(\Omega )$, and strongly removable for ${\mathcal {A}}^p_\mu (\Omega \setminus A)$ if ${\mathcal {A}}^p_\mu (\Omega \setminus A) = {\mathcal {A}}^p_\mu (\Omega )$. The general theory developed is in many ways similar to the theory of removable singularities for Hardy $H^p$ spaces, $\mathop {\mathrm {B}MO}$ and locally Lipschitz spaces of analytic functions, including the existence of counterexamples to many plausible properties, e.g. the union of two compact removable singularities needs not be removable. In the case when weak and strong removability are the same for all sets, in particular if $\mu $ is absolutely continuous with respect to the Lebesgue measure $m$, we are able to say more than in the general case. In this case we obtain a Dolzhenko type result saying that a countable union of compact removable singularities is removable. When $\mathrm {d}\mu = w\mathrm {d}m$ and $w$ is a Muckenhoupt $A_p$ weight, $1<p<\infty $, the removable singularities are characterized as the null sets of the weighted Sobolev space capacity with respect to the dual exponent $p^{\prime }=p/(p-1)$ and the dual weight $w^{\prime }=w^{1/(1-p)}$.
LA - eng
KW - analytic continuation; analytic function; Bergman space; capacity; exceptional set; holomorphic function; Muckenhoupt weight; removable singularity; singular set; Sobolev space; weight; analytic continuation; analytic function; Bergman space; capacity; exceptional set
UR - http://eudml.org/doc/31023
ER -

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