Properties of Removable Singularities for Hardy Spaces of Analytic Functions

doi:10.1112/S002461070200354X

Journal of the London Mathematical Society 2002 66(3):651-670; doi:10.1112/S002461070200354X
© 2002 by London Mathematical Society

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Properties of Removable Singularities for Hardy Spaces of Analytic Functions

Anders Björn

Department of Mathematics, Linköping University SE-581 83 Linköping, Sweden, anbjo{at}mai.liu.se

Received 2 April 2001. Revision received 7 January 2002.

Removable singularities for Hardy spaces H^p( ${Omega}$ ) = {f $isin$ Hol( ${Omega}$ ): |f|^p $≤$ u in ${Omega}$ for some harmonic u}, 0 < p < ${infty}$ are studied. A setE = ${Omega}$ is a weakly removable singularity for H^p( ${Omega}$ \E) if H^p( ${Omega}$ \E) $sub$ Hol( ${Omega}$ ), and a strongly removable singularity for H^p( ${Omega}$ \E) if H^p( ${Omega}$ \E)= H^p( ${Omega}$ ). The two types of singularities coincide for compactE, and weak removability is independent of the domain ${Omega}$ .

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 ${Omega}$ and a setE $sub$ ${Omega}$ that is weakly removable for all H^p, but not strongly removablefor any H^p( ${Omega}$ \E), 0 < p < ${infty}$ , are found.

It is easy to show that if E is weakly removable for H^p( ${Omega}$ \E)and q > p, then E is also weakly removable for H^q( ${Omega}$ \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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