Integrability of superharmonic functions and subharmonic functions
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- by Hiroaki Aikawa PDF
- Proc. Amer. Math. Soc. 120 (1994), 109-117 Request permission
Abstract:
We apply the coarea formula to obtain integrability of superharmonic functions and nonintegrability of subharmonic functions. The results involve the Green function. For a certain domain, say Lipschitz domain, we estimate the Green function and restate the results in terms of the distance from the boundary.References
- Alano Ancona, On strong barriers and an inequality of Hardy for domains in $\textbf {R}^n$, J. London Math. Soc. (2) 34 (1986), no. 2, 274–290. MR 856511, DOI 10.1112/jlms/s2-34.2.274
- On the global integrability of superharmonic functions in balls, J. London Math. Soc. (2) 4 (1971), 365–373. MR 291484, DOI 10.1112/jlms/s2-4.2.365
- D. H. Armitage, Further results on the global integrability of superharmonic functions, J. London Math. Soc. (2) 6 (1972), 109–121. MR 313524, DOI 10.1112/jlms/s2-6.1.109
- D. A. Brannan and W. K. Hayman, Research problems in complex analysis, Bull. London Math. Soc. 21 (1989), no. 1, 1–35. MR 967787, DOI 10.1112/blms/21.1.1
- Peter W. Jones, A geometric localization theorem, Adv. in Math. 46 (1982), no. 1, 71–79. MR 676987, DOI 10.1016/0001-8708(82)90054-8
- Peter W. Jones and Thomas H. Wolff, Hausdorff dimension of harmonic measures in the plane, Acta Math. 161 (1988), no. 1-2, 131–144. MR 962097, DOI 10.1007/BF02392296
- 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, A distortion theorem for conformal mappings with an application to subharmonic functions, Hiroshima Math. J. 20 (1990), no. 2, 341–350. MR 1063368
- 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
- Vladimir G. Maz’ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR 817985, DOI 10.1007/978-3-662-09922-3
- 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, Nonintegrability of harmonic functions in a domain, Japan. J. Math. (N.S.) 16 (1990), no. 2, 269–278. MR 1091161, DOI 10.4099/math1924.16.269
- Noriaki Suzuki, Nonintegrability of superharmonic functions, Proc. Amer. Math. Soc. 113 (1991), no. 1, 113–115. MR 1054163, DOI 10.1090/S0002-9939-1991-1054163-2 —, Note on the integrability of superharmonic functions, preprint.
- Kjell-Ove Widman, Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations, Math. Scand. 21 (1967), 17–37 (1968). MR 239264, DOI 10.7146/math.scand.a-10841
- Jang-Mei Wu, Boundary density and the Green function, Michigan Math. J. 38 (1991), no. 2, 175–189. MR 1098855, DOI 10.1307/mmj/1029004327
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 120 (1994), 109-117
- MSC: Primary 31B05
- DOI: https://doi.org/10.1090/S0002-9939-1994-1169019-7
- MathSciNet review: 1169019