Integrability of superharmonic functions, uniform domains, and Hölder domains

Gotoh, Yasuhiro

doi:10.1090/S0002-9939-99-04670-5

Integrability of superharmonic functions, uniform domains, and Hölder domains
HTML articles powered by AMS MathViewer

by Yasuhiro Gotoh PDF

Proc. Amer. Math. Soc. 127 (1999), 1443-1451 Request permission

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

Hiroaki Aikawa, Integrability of superharmonic functions and subharmonic functions, Proc. Amer. Math. Soc. 120 (1994), no. 1, 109–117. MR 1169019, DOI 10.1090/S0002-9939-1994-1169019-7
F. W. Gehring and B. G. Osgood, Uniform domains and the quasihyperbolic metric, J. Analyse Math. 36 (1979), 50–74 (1980). MR 581801, DOI 10.1007/BF02798768
Y. Gotoh, On global integrability of $BMO$ functions on general domains, to appear in J. Anal. Math.
Peter Lindqvist, Global integrability and degenerate quasilinear elliptic equations, J. Anal. Math. 61 (1993), 283–292. MR 1253445, DOI 10.1007/BF02788845
Fumi-Yuki Maeda and Noriaki Suzuki, The integrability of superharmonic functions on Lipschitz domains, Bull. London Math. Soc. 21 (1989), no. 3, 270–278. MR 986371, DOI 10.1112/blms/21.3.270
Makoto Masumoto, Integrability of superharmonic functions on plane domains, J. London Math. Soc. (2) 45 (1992), no. 1, 62–78. MR 1157552, DOI 10.1112/jlms/s2-45.1.62
Makoto Masumoto, Integrability of superharmonic functions on Hölder domains of the plane, Proc. Amer. Math. Soc. 117 (1993), no. 4, 1083–1088. MR 1152284, DOI 10.1090/S0002-9939-1993-1152284-9
Hans Martin Reimann and Thomas Rychener, Funktionen beschränkter mittlerer Oszillation, Lecture Notes in Mathematics, Vol. 487, Springer-Verlag, Berlin-New York, 1975 (German). MR 0511997, DOI 10.1007/BFb0081825
Wayne Smith and David A. Stegenga, Exponential integrability of the quasi-hyperbolic metric on Hölder domains, Ann. Acad. Sci. Fenn. Ser. A I Math. 16 (1991), no. 2, 345–360. MR 1139802, DOI 10.5186/aasfm.1991.1625
Wayne Smith and David A. Stegenga, Sobolev imbeddings and integrability of harmonic functions on Hölder domains, Potential theory (Nagoya, 1990) de Gruyter, Berlin, 1992, pp. 303–313. MR 1167248
David A. Stegenga and David C. Ullrich, Superharmonic functions in Hölder domains, Rocky Mountain J. Math. 25 (1995), no. 4, 1539–1556. MR 1371353, DOI 10.1216/rmjm/1181072160
Noriaki Suzuki, Note on the integrability of superharmonic functions, Proc. Amer. Math. Soc. 118 (1993), no. 2, 415–417. MR 1126201, DOI 10.1090/S0002-9939-1993-1126201-1