TY - JOUR

T1 - On Carleman formulas and on the class of holomorphic functions representable by them

AU - Aizenberg, Lev

AU - Vidras, Alekos

PY - 2002

Y1 - 2002

N2 - Carleman formulas, unlike the Cauchy formula, restore a function holomorphic in a domain D by its values on a part M of the boundary ∂D, provided that M is of positive Lebesgue measure. An extensive survey of Carleman formulas is found in [AIZ]. In the present paper new Carleman formulas are obtained for domains in ℂn and the question about a description of the class of holomorphic functions that are represented by Carleman formula is investigated. In [AIZTV] we considered the simplest Carleman formulas in one and several complex variables on particular simply connected domains. It was shown there that a necessary and sufficient condition for a holomorphic function f to be represented by Carleman formula over the set M is that f must belong to "the Hardy class ℋ1 near the set M". In the present paper we look at the case of Fok-Kuni integral representation formula. This is a particular form of abstract Carleman formula, but it involves simply connected domains, not covered by previous results. Furthermore we initiate the study of the same questions for non-simply connected domains. We obtain the description of the class of functions representable by Carleman formula on annulii in ℂ and their generalizations, the Reinhardt domains in ℂn.

AB - Carleman formulas, unlike the Cauchy formula, restore a function holomorphic in a domain D by its values on a part M of the boundary ∂D, provided that M is of positive Lebesgue measure. An extensive survey of Carleman formulas is found in [AIZ]. In the present paper new Carleman formulas are obtained for domains in ℂn and the question about a description of the class of holomorphic functions that are represented by Carleman formula is investigated. In [AIZTV] we considered the simplest Carleman formulas in one and several complex variables on particular simply connected domains. It was shown there that a necessary and sufficient condition for a holomorphic function f to be represented by Carleman formula over the set M is that f must belong to "the Hardy class ℋ1 near the set M". In the present paper we look at the case of Fok-Kuni integral representation formula. This is a particular form of abstract Carleman formula, but it involves simply connected domains, not covered by previous results. Furthermore we initiate the study of the same questions for non-simply connected domains. We obtain the description of the class of functions representable by Carleman formula on annulii in ℂ and their generalizations, the Reinhardt domains in ℂn.

KW - Analytic continuation

KW - Carleman formulas

UR - http://www.scopus.com/inward/record.url?scp=0036066654&partnerID=8YFLogxK

U2 - 10.1002/1522-2616(200204)237:1<5::AID-MANA5>3.0.CO;2-F

DO - 10.1002/1522-2616(200204)237:1<5::AID-MANA5>3.0.CO;2-F

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AN - SCOPUS:0036066654

SN - 0025-584X

VL - 237

SP - 5

EP - 25

JO - Mathematische Nachrichten

JF - Mathematische Nachrichten

ER -