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Properties of Removable Singularities for Hardy Spaces of Analytic Functions
Author(s) -
Björn Anders
Publication year - 2002
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s002461070200354x
Subject(s) - gravitational singularity , singularity , domain (mathematical analysis) , mathematics , space (punctuation) , integer (computer science) , pure mathematics , harmonic , combinatorics , mathematical analysis , physics , computer science , quantum mechanics , programming language , operating system
Removable singularities for Hardy spaces H p (Ω) = { f ∈ Hol(Ω): | f | p ⩽ u in Ω for some harmonic u }, 0 < p < ∞ are studied. A set E = Ω is a weakly removable singularity for H p (Ω\ E ) if H p (Ω\ E ) ⊂ Hol(Ω), and a strongly removable singularity for H p (Ω\ E ) if H p (Ω\ E ) = H p (Ω). The two types of singularities coincide for compact E , and weak removability is independent of the domain Ω. The paper looks at differences between weak and strong removability, the domain dependence of strong removability, and when removability is preserved under unions. In particular, a domain Ω and a set E ⊂ Ω that is weakly removable for all H p , but not strongly removable for any H p (Ω\ E ), 0 < p < ∞, are found. It is easy to show that if E is weakly removable for H p (Ω\ E ) and q > p , then E is also weakly removable for H q (Ω\ E ). It is shown that the corresponding implication for strong removability holds if and only if q / p is an integer. Finally, the theory of Hardy space capacities is extended, and a comparison is made with the similar situation for weighted Bergman spaces.