Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
MathSciNet review: 1054163
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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$.

References [Enhancements On Off] (What's this?)

  • [1] C. Fefferman and E. M. Stein, $ {H^p}$ spaces of several variables, Acta Math. 129 (1972), 137-193. MR 0447953 (56:6263)
  • [2] L. L. Helms, Introduction to potential theory, Wiley-Interscience, New York, 1969. MR 0261018 (41:5638)
  • [3] F.-Y. Maeda and N. Suzuki, The integrability of superharmonic functions on Lipschitz domains, Bull. London Math. Soc. 21 (1989), 270-278. MR 986371 (90b:31004)
  • [4] N. Suzuki, Nonintegrability of harmonic functions in a domain, Japan. J. Math. (N.S.) 16 (1990). MR 1091161 (91m:31003)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 31B05

Retrieve articles in all journals with MSC: 31B05

Additional Information

Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society