Integrability of superharmonic functions in a John domain

Aikawa, Hiroaki

doi:10.1090/S0002-9939-99-04991-6

Integrability of superharmonic functions in a John domain
HTML articles powered by AMS MathViewer

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