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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The duals of harmonic Bergman spaces

Authors: Charles V. Coffman and Jonathan Cohen
Journal: Proc. Amer. Math. Soc. 110 (1990), 697-704
MSC: Primary 46E15; Secondary 31B05
MathSciNet review: 1028042
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Abstract: In this paper we show that for $ \Omega $, a starlike Lipschitz domain, the dual of the space of harmonic functions in $ {L^p}(\Omega )$ need not be the harmonic functions in $ {L^q}(\Omega )$, where $ 1/p + 1/q = 1$. We show that, as a consequence, the harmonic Bergman projection for $ \Omega $ need not extend to a bounded operator on $ {L^p}(\Omega )$ for all $ 1 < p < \infty $. The duality result is a partial answer to a question of Nakai and Sario [9] posed initially in the Proceedings of the London Mathematical Society in 1978. We treat the duality question as a biharmonic problem, and our result follows from the failure of uniqueness for the biharmonic Dirichlet problem in domains with sharp intruding corners.

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Article copyright: © Copyright 1990 American Mathematical Society