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Best filters for the general Fatou boundary limit theorem. (English) Zbl 0817.31001
In contrast with a negative result by J. L. Doob it is shown that after a suitable normalization there is a “best” family of filters for which the Fatou boundary limit theorem holds. The normalization assigns to each positive harmonic function a set of boundary points at which that function must vanish. The result which is new even for harmonic functions on the unit disk is formulated in terms of a general potential theoretical setting.

MSC:
31A20 Boundary behavior (theorems of Fatou type, etc.) of harmonic functions in two dimensions
31B25 Boundary behavior of harmonic functions in higher dimensions
31C35 Martin boundary theory
30E25 Boundary value problems in the complex plane
30F25 Ideal boundary theory for Riemann surfaces
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[1] J. Bliedtner and W. Hansen, Potential theory, Universitext, Springer-Verlag, Berlin, 1986. An analytic and probabilistic approach to balayage. · Zbl 0706.31001
[2] Jürgen Bliedtner and Peter A. Loeb, A measure-theoretic boundary limit theorem, Arch. Math. (Basel) 43 (1984), no. 4, 373 – 376. · Zbl 0603.31009 · doi:10.1007/BF01196663 · doi.org
[3] J. Bliedtner and P. Loeb, A reduction technique for limit theorems in analysis and probability theory, Ark. Mat. 30 (1992), no. 1, 25 – 43. · Zbl 0757.28006 · doi:10.1007/BF02384860 · doi.org
[4] J. L. Doob, Boundary approach filters for analytic functions, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 3, 187 – 213 (English, with French summary). · Zbl 0251.30034
[5] J. L. Doob, Classical potential theory and its probabilistic counterpart, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 262, Springer-Verlag, New York, 1984. · Zbl 0549.31001
[6] L. L. Helms, Introduction to potential theory, Pure and Applied Mathematics, Vol. XXII, Wiley-Interscience A Division of John Wiley & Sons, New York-London-Sydney, 1969. · Zbl 0188.17203
[7] Peter A. Loeb, A regular metrizable boundary for solutions of elliptic and parabolic differential equations, Math. Ann. 251 (1980), no. 1, 43 – 50. · Zbl 0435.31008 · doi:10.1007/BF01420279 · doi.org
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