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Transactions of the American Mathematical Society

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Weak $L_{1}$ characterizations of Poisson integrals, Green potentials and $H^{p}$ 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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Keywords: Poisson integral, Green potential, <IMG WIDTH="33" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${H^p}$"> spaces, weak <IMG WIDTH="29" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img17.gif" ALT="${L^p}$"> spaces
Article copyright: © Copyright 1977 American Mathematical Society