Integrability of superharmonic functions in a John domain
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- by Hiroaki Aikawa PDF
- Proc. Amer. Math. Soc. 128 (2000), 195-201 Request permission
Abstract:
The integrability of positive superharmonic functions on a bounded fat John domain is established. No exterior conditions are assumed. For a general bounded John domain the $L^{p}$-integrability is proved with the estimate of $p$ in terms of the John constant.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
- H. Aikawa, Norm estimate of Green operator, perturbation of Green function and integrability of superharmonic functions, Math. Ann. 312 (1998), 289–318.
- H. Aikawa, Norm estimate for the Green operator with applications, Proceedings of Complex Analysis and Differential Equations, Marcus Wallenberg Symposium in honor of Matts Essén, Uppsala University (1997) (to appear).
- Hiroaki Aikawa and Minoru Murata, Generalized Cranston-McConnell inequalities and Martin boundaries of unbounded domains, J. Anal. Math. 69 (1996), 137–152. MR 1428098, DOI 10.1007/BF02787105
- D. H. Armitage, On the global integrability of superharmonic functions in balls, J. London Math. Soc. (2) 4 (1971), 365–373.
- N. A. Lukaševič, On the theory of Painlevé’s equations, Differencial′nye Uravnenija 6 (1970), 425–430 (Russian). MR 0267176
- F. W. Gehring and O. Martio, Lipschitz classes and quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 203–219. MR 802481, DOI 10.5186/aasfm.1985.1022
- Y. Gotoh, Integrability of superharmonic functions, uniform domains, and Hölder domains, Proc. Amer. Math. Soc. (to appear).
- B. Gustafsson, M. Sakai and H. S. Shapiro, On domains in which harmonic functions satisfy generalized mean value properties, Potential Analysis 7 (1997), 467–484.
- 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
- Shinji Yamashita, Univalent analytic functions and the Poincaré metric, Kodai Math. J. 13 (1990), no. 2, 164–175. MR 1061918, DOI 10.2996/kmj/1138039216
- 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
- Wayne Smith and David A. Stegenga, Hölder domains and Poincaré domains, Trans. Amer. Math. Soc. 319 (1990), no. 1, 67–100. MR 978378, DOI 10.1090/S0002-9947-1990-0978378-8
- 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
- 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
- William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR 1014685, DOI 10.1007/978-1-4612-1015-3
Additional Information
- Hiroaki Aikawa
- Affiliation: Department of Mathematics, Shimane University, Matsue 690-8504, Japan
- Email: haikawa@math.shimane-u.ac.jp
- Received by editor(s): March 17, 1998
- Published electronically: May 27, 1999
- Additional Notes: This work was supported in part by Grant-in-Aid for Scientific Research (B) (No. 09440062), Japanese Ministry of Education, Science and Culture.
- Communicated by: Albert Baernstein II
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 195-201
- MSC (1991): Primary 31B05
- DOI: https://doi.org/10.1090/S0002-9939-99-04991-6
- MathSciNet review: 1622765