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Integrability of superharmonic functions, uniform domains, and Hölder domains

Author(s): Yasuhiro Gotoh
Journal: Proc. Amer. Math. Soc. 127 (1999), 1443-1451.
MSC (1991): Primary 46E15
Posted: January 29, 1999
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Abstract: Let $S^+(D)$ denote the space of all positive superharmonic functions on a domain $D \subset \mathbf R^n$. Lindqvist showed that $\log S^+(D)$ is a bounded subset of $BMO(D)$. Using this, we give a characterization of finitely connected $2$-dimensional uniform domains and remarks on Hölder domains.

References:

1.
H. Aikawa, Integrability of superharmonic functions and subharmonic functions, Proc. Amer. Math. Soc., 120 (1994), 109-117. MR 94b:31003

2.
F. W. Gehring and B. G. Osgood, Uniform domains and the quasihyperbolic metric, J. Anal. Math., 36 (1979), 50-74. MR 81k:30023

3.
Y. Gotoh, On global integrability of $BMO$ functions on general domains, to appear in J. Anal. Math.

4.
P. Lindqvist, Global integrability and degenerate quasilinear elliptic equations, J. Anal. Math. 61 (1993), 283-292. MR 95b:35062

5.
F. -Y. Maeda and N. Suzuki, The integrability of superharmonic functions on Lipschitz domains, Bull. London Math. Soc., 21 (1989), 270-278. MR 90b:31004

6.
M. Masumoto, Integrability of superharmonic functions on plane domains, J. London Math. Soc., 45 (1992), 62-78. MR 93f:31002

7.
M. Masumoto, Integrability of superharmonic functions on Hölder domains of the plane, Proc. Amer. Math. Soc., 117 (1993), 1083-1088. MR 93e:31001

8.
H. M. Reimann and T. Rychener, Funktionen beschränkter mittelerer Oszillation, Lecture Notes in Math. 489, Springer, 1975. MR 58:23564

9.
W. Smith and D. Stegenga, Exponential integrability of the quasihyperbolic metric on Hölder Domains, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 16 (1991), 345-360. MR 93b:30016

10.
W. Smith and D. Stegenga, Sobolev imbedding and the integrability of harmonic functions on Hölder domains, (Proc. Interna. Conf. Potential Theory, Nagoya 1990), Walter de Gruyter, Berlin, 1992, pp. 303-313. MR 93d:46056

11.
D. A. Stegenga and D. C. Ullrich, Superharmonic functions in Hölder domains, Rocky Mountain J. Math. 29 (1995), 1539-1556. MR 97j:31003

12.
N. Suzuki, Note on the integrability of superharmonic functions, Proc. Amer. Math. Soc., 118 (1993), 415-417. MR 93g:31007

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Additional Information:

Yasuhiro Gotoh
Affiliation: Department of Mathematics, National Defense Academy, Hashirimizu 1-10-20 Yokosuka 239, Japan
Email: gotoh@cc.nda.ac.jp

DOI: 10.1090/S0002-9939-99-04670-5
PII: S 0002-9939(99)04670-5
Keywords: BMO, quasihyperbolic metric, uniform domain, H\"older domain, superharmonic function, harmonic function
Received by editor(s): May 7, 1997
Received by editor(s) in revised form: August 25, 1997
Posted: January 29, 1999
Communicated by: Albert Baernstein II
Copyright of article: Copyright 1999, American Mathematical Society

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