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Superharmonic extensions, mean values and Riesz measures. (English) Zbl 0785.31002
Let \(Z\) be a relatively closed polar subset of an open set \(D\) in \(\mathbb{R}^ n\). The classical extension theorem asserts that, if \(u\) is superharmonic on \(D \backslash Z\) and locally bounded below on \(D\), then \(u\) has a superharmonic extension to \(D\).
The starting point for this paper is a generalization of this result in which: (i) the set \(Z\) is no longer required to be closed, and (ii) the lower boundedness condition is replaced by the requirement that there is a positive superharmonic function \(v\) on \(D\) for which \(u/v\) has a nonnegative lower limit at all points of \(Z\). Extension results for harmonic functions are deduced from this. There then follows an extensive collection of related results based on the limiting behaviour of spherical means of superharmonic (or \(\delta\)-subharmonic) functions as the radius tends to 0. Some of these give sufficient conditions for a polar set to be positive for the Riesz measure of a \(\delta\)-subharmonic function. Several of the results involve Hausdorff measure conditions. Complementary extension results have been given independently by the reviewer [J. Lond. Math. Soc., II. Ser. 48, 515-525 (1993)].
MSC:
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
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[1] Armitage, D. H.: ?Mean values and associated measures of superharmonic functions?,Hiroshima Math. J. 13 (1983), 53-63. · Zbl 0512.31009
[2] Arsove, M. and Leutwiler, H.: ?Quasi-bounded and singular functions?,Trans. Amer. Math. Soc. 189 (1974), 276-302. · Zbl 0287.31008
[3] Besicovitch, A. S.: ?On sufficient conditions for a function to be analytic, and on behaviour of analytic functions in the neighbourhood of non-isolated singular points?,Proc. London Math. Soc. 32(2) (1931), 1-9. · JFM 56.0272.01
[4] Besicovitch, A. S.: ?A general form of the covering principle and relative differentiation of additive functions?,Proc. Cambridge Phil. Soc. 41 (1945), 103-110. · Zbl 0063.00352
[5] Besicovitch, A. S.: ?A general form of the covering principle and relative differentiation of additive functions II?,Proc. Cambridge Phil. Soc. 42 (1946), 1-10. · Zbl 0063.00353
[6] Besicovitch, A. S.: ?On the definition of tangents to sets of infinite linear measure?,Proc. Cambridge Phil. Soc. 52 (1956), 20-29. · Zbl 0070.05402
[7] Brelot, M.: ?Refinements on the superharmonic continuation?,Hokkaido Math. J. 10 (1981), 68-88. · Zbl 0512.31007
[8] Doob, J. L.:Classical Potential Theory and Its Probabilistic Counterpart, Springer-Verlag (1984). · Zbl 0549.31001
[9] Falconer, K. J.:The Geometry of Fractal Sets, Cambridge University Press (1985). · Zbl 0587.28004
[10] Federer, H.:Geometric Measure Theory, Springer-Verlag (1969). · Zbl 0176.00801
[11] Gardiner, S. J.: ?Removable singularities for subharmonic functions?,Pacific J. Math. 147 (1991), 71-80. · Zbl 0663.31004
[12] Hayman, W. K. and Kennedy, P. B.:Subharmonic Functions, Vol. 1, Academic Press (1976). · Zbl 0419.31001
[13] Kaufman, R. and Wu, J.-M.: ?Removable singularities for analytic or subharmonic functions?,Ark. Mat. 18 (1980), 107-116. Correction,ibid. 21 (1983), 1. · Zbl 0444.30002
[14] Kuran, Ü.: ?Some extension theorems for harmonic, superharmonic and holomorphic functions?,J. London Math. Soc. 22(2) (1980), 269-284. · Zbl 0446.31004
[15] Marstrand, J. M.: ?The (?,s)-regular subsets ofn-space?,Trans. Amer. Math. Soc. 113 (1964), 369-392. · Zbl 0144.04902
[16] Wallin, H.: ?Hausdorff measures and generalized differentiation?,Math. Ann. 183 (1969), 275-286. · Zbl 0181.05701
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