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A Characterization of Hardy Spaces on the Unit Ball of C n
Author(s) -
Stoll Manfred
Publication year - 1993
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/jlms/s2-48.1.126
Subject(s) - hardy space , unit sphere , holomorphic function , invariant (physics) , mathematics , combinatorics , ball (mathematics) , mathematical analysis , physics , mathematical physics
A function f holomorphic in the unit ball B of C n lies in the Hardy space H p , 0 < p < ∞, if and only if∫ B( 1 −| z | 2 ) n| f ( z ) |p − 2| ∇ ˜ f ( z ) | 2 d λ ( z ) < ∞ where ∇ ˜ and λ denote the invariant gradient and invariant measure on B , respectively. Furthermore, if f ε H p , thenlim ( 1 − r 2 ) n∫ B r| f ( z ) |v − 2| ∇ ˜ f ( z ) | 2 d λ ( z ) = 0 . An analogous characterization is also given for invariant harmonic functions for the case 1 < p ⩽ ∞.