Properties of Removable Singularities for Hardy Spaces of Analytic Functions |
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Authors: | Bjorn Anders |
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Institution: | Department of Mathematics, Linköping University SE-581 83 Linköping, Sweden, anbjo{at}mai.liu.se |
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Abstract: | Removable singularities for Hardy spaces Hp( ) = {f Hol( ): |f|p u in for some harmonic u}, 0 < p < are studied. A setE = is a weakly removable singularity for Hp( \E) if Hp( \E) Hol( ), and a strongly removable singularity for Hp( \E) if Hp( \E)= Hp( ). The two types of singularities coincide for compactE, and weak removability is independent of the domain . The paper looks at differences between weak and strong removability,the domain dependence of strong removability, and when removabilityis preserved under unions. In particular, a domain and a setE that is weakly removable for all Hp, but not strongly removablefor any Hp( \E), 0 < p < , are found. It is easy to show that if E is weakly removable for Hp( \E)and q > p, then E is also weakly removable for Hq( \E). Itis shown that the corresponding implication for strong removabilityholds if and only if q/p is an integer. Finally, the theory of Hardy space capacities is extended, anda comparison is made with the similar situation for weightedBergman spaces. |
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