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Proceedings of the American Mathematical Society

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

 
 

 

Nonintegrability of superharmonic functions


Author: Noriaki Suzuki
Journal: Proc. Amer. Math. Soc. 113 (1991), 113-115
MSC: Primary 31B05
DOI: https://doi.org/10.1090/S0002-9939-1991-1054163-2
MathSciNet review: 1054163
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Abstract: In this article we prove the following: If $ u$ is a nonzero superharmonic function on a proper subdomain $ D$ in $ {R^n}$, then

$\displaystyle {\int_D {\left\vert {u(x)} \right\vert} ^p}{\delta _D}{(x)^{np - n - 2p}}dx = \infty ,$

where $ 0 < p \leq 1$ and $ {\delta _D}(x)$ denotes the distance between $ x$ and the boundary of $ D$.

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DOI: https://doi.org/10.1090/S0002-9939-1991-1054163-2
Article copyright: © Copyright 1991 American Mathematical Society

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