Martin boundary over an isolated singularity of rotation free density

Nakai, Mitsuru

doi:10.2969/jmsj/02630483

J. Math. Soc. Japan

1974

DOI: 10.2969/jmsj/02630483

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Martin boundary over an isolated singularity of rotation free density

Mitsuru Nakai

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Cited by 27 publications

(29 citation statements)

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“…Nakai [3]). A density P = P(z)dxdy (z -x + iy) is a 2-form on Ω = Ω U dΩ with nonnegative locally Holder continuous coefficients P(z).…”

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confidence: 99%

Picard principle for finite densities on some end

Kawamura

1977

Nagoya Mathematical Journal

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Consider a parabolic end Ω of a Riemann surface in the sence of Heins [2] (cf. Nakai [3]). A density P = P(z)dxdy (z -x + iy) is a 2-form on Ω = Ω U dΩ with nonnegative locally Holder continuous coefficients P(z). A density P is said to be finite if the integralThe elliptic dimension of a density P at the ideal boundary point δ, dim P in notation, is defined (Nakai [5], [6]) to be the 'dimension' of the half module of nonnegative solutions of the equationon an end Ω with the vanishing boundary values on dΩ. The elliptic dimension of the particular density P = 0 at δ is called the harmonic dimension of δ. After Bouligand we say that the Picavd principle is valid for a density P at δ if dimP = l. For the punctured disk V: 0 < \z\ < 1, Nakai [6] showed that the Picard principle is valid for any finite density P on 0 < \z\ <; 1 at the ideal boundary 2 = 0, and he conjectured that the above theorem is valid for every general end of harmonic dimension one. The purpose of this paper is to give a partial answer in the affirmative.Heins [2] showed that the harmonic dimension of the ideal boundary δ of an end is one if Ω satisfies the condition [H]: There exists a sequence {A n } of disjoint annuli with analytic Jordan boundaries on Ω satisfying the condition that for each n,A n+1 separates A n from the ideal boundary, and A 1 separates the relative boundary dΩ from the ideal boundary? and

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“…Nakai [3]). A density P = P(z)dxdy (z -x + iy) is a 2-form on Ω = Ω U dΩ with nonnegative locally Holder continuous coefficients P(z).…”

mentioning

confidence: 99%

Picard principle for finite densities on some end

Kawamura

1977

Nagoya Mathematical Journal

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“…Then the above theorem shows that the range of the set of densities Q on fi with (1) for a rotationally invariant density P on Q also consists of 1 and N. We will recall in §1 the proof of (3) according to Nakai [3] and prove Theorem 1 in §3 by using the fundamental properties of "Martin kernels" for rotationally invariant densities P on fl with dimP = N given in § §1 and 2.…”

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confidence: 99%

“…In particular, we denote by a(P) the first singularity index Qi(P) and call it simply the singularity index of P at ÓT2. Then we have the following fundamental inequality [3]:…”

mentioning

confidence: 99%

“…For example the Picard principle for the density P\(z) = \z\~2+x is valid if and only if A G [0, oo] [3,4], the Picard principle for a density P is valid if P(z) = 0(|2|~2) as z -» 0 [2], and the Picard principle for a density Q with P(z) < Q(z) < P(z) + C\z\~2 (z e O) for a positive constant C and a rotationally invariant density P, i.e. a density P satisfying P(z) = P(|z|) (z G H), is valid if the Picard principle for P is valid [8].…”

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confidence: 99%

“…In particular the range dim Pr of the set Dr of rotationally invariant densities on Q consists of 1 and K [3] :…”

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confidence: 99%

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Picard Dimensions of Close to Rotationally Invariant Densities

Tada¹

1987

Proceedings of the American Mathematical Society

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ABSTRACT. The purpose of this paper is to show that the Picard dimensions of a rotation-free density P and a general density Q on the punctured unit disk 0 < |s| < 1 are equal to each other if |P(z) -Q(z)\ = 0(|«|~2) as z -* 0.Before stating our result we fix terminologies. We denote by fi the punctured unit disk 0 < \z\ < 1 which is viewed as an end of the punctured sphere 0 < \z] < oo so that the unit circle ]z] = 1 is the relative boundary <9fi of Í2 and the origin 2 = 0 is the ideal boundary 6Q of fi. By a density P on 0 we mean a nonnegative locally Holder continuous function P on the closure fl = f!U dQ of Q so that P may or may not have a singularity at z -0. With a density P on Ü we associate the class PP(fi; díTj of nonnegative continuous functions uonSl such that u satisfies the

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Das Picard-Prinzip und verwandte Fragen bei St�rung von harmonischen R�umen

Boukricha

1979

Math. Ann.

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Martin boundary over an isolated singularity of rotation free density

Cited by 27 publications

References 10 publications

Picard principle for finite densities on some end

Picard principle for finite densities on some end

Picard Dimensions of Close to Rotationally Invariant Densities

Das Picard-Prinzip und verwandte Fragen bei St�rung von harmonischen R�umen

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