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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Note on the integrability of superharmonic functions


Author: Noriaki Suzuki
Journal: Proc. Amer. Math. Soc. 118 (1993), 415-417
MSC: Primary 31B05
MathSciNet review: 1126201
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Abstract: Let $ D$ be a domain in $ {{\mathbf{R}}^n}$ and let $ {S^ + }(D)$ be the set of all nonnegative superharmonic functions on $ D$. It is shown that if $ {S^ + }(D) \subset {L^p}(D)$ with some $ p > 0$, then for each $ {x_0} \in D$ there is a constant $ C = C(D,p,{x_0}) > 0$ such that the inequality

$\displaystyle \int_D {u{{(x)}^p}dx \leqslant Cu{{({x_0})}^p}} $

holds for all $ u \in {S^ + }(D)$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1126201-1
PII: S 0002-9939(1993)1126201-1
Article copyright: © Copyright 1993 American Mathematical Society