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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; holomorphic function; Muckenhoupt weight; removable singularity; singular set; Sobolev space; weight
Summary:
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, $\mathop {\mathrm BMO}$ 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 $w^{\prime }=w^{1/(1-p)}$.
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