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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Weak $ L\sb{1}$ characterizations of Poisson integrals, Green potentials and $ H\sp{p}$ spaces


Author: Peter Sjögren
Journal: Trans. Amer. Math. Soc. 233 (1977), 179-196
MSC: Primary 31B10
MathSciNet review: 0463462
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Abstract: Our main result can be described as follows. A subharmonic function u in a suitable domain $ \Omega $ in $ {{\mathbf{R}}^n}$ is the difference of a Poisson integral and a Green potential if and only if u divided by the distance to $ \partial \Omega $ is in weak $ {L_1}$ in $ \Omega $.

Similar conditions are given for a harmonic function to be the Poisson integral of an $ {L_p}$ function on $ \partial \Omega $. Iterated Poisson integrals in a polydisc are also considered. As corollaries, we get weak $ {L_1}$ characterizations of $ {H^p}$ spaces of different kinds.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1977-0463462-1
PII: S 0002-9947(1977)0463462-1
Keywords: Poisson integral, Green potential, $ {H^p}$ spaces, weak $ {L^p}$ spaces
Article copyright: © Copyright 1977 American Mathematical Society