Sets of determination for harmonic functions in an NTA domain

Hiroaki AIKAWA

doi:10.2969/jmsj/04820299

Hiroaki AIKAWA

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JOURNAL FREE ACCESS

1996 Volume 48 Issue 2 Pages 299-315

DOI https://doi.org/10.2969/jmsj/04820299

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Published: 1996 Received: June 20, 1994 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) H. Aikawa, Quasiadditivity of Riesz capacity, Math. Scand., 69 (1991), 15-30.
2) H. Aikawa, Quasiadditivity of capacity and minimal thinness, Ann. Acad. Sci. Fenn. Ser. AI Math., 18 (1993), 65-75.
3) A. Ancona, Positive harmonic functions and hyperbolicity, Potential Theory- Surveys and Problems, Lecture Notes in Math., 1344, Springer-Verlag, 1987, pp. 1-23.
4) A. Beurling, A minimum principle for positive harmonic functions, Ann. Acad. Sci. Fenn. Ser. AI, 372 (1965).
5) M. Brelot, On topologies and boundaries in potential theory, Lecture Notes in Math., 174, Springer-Verlag, 1971.
6) F. F. Bonsall and D. Walsh, Vanishing l¹-sums of the Poisson kernel and sums with positive coefficients, Proc. Edinburgh Math. Soc., 32 (1989), 431-447.
7) B. E. J, Dahlberg, A minimum principle for positive harmonic functions, Proc. London Math. Soc. (3), 33 (1976), 238-250.
8) J. L. Doob, Classical potential theory and its probabilistic counterpart, Springer- Verlag, 1984.
9) N. F. Dudley Ward, Atomic decompositions of integrable or continuous functions, PhD thesis (1991), University of York.
10) M. Essén, On minimal thinness, reduced functions and Green potentials, Proc. Edinburgh Math. Soc., 36 (1992), 87-106,
11) S. J. Gardiner, Sets of determination for harmonic functions, Trans. Amer. Math. Soc., 338 (1993), 233-243.
12) W. K. Hayman, Subharmonic functions, Vol. 2, Academic Press, 1989.
13) W. K. Hayman and T. J. Lyons, Bases for positive continuous functions, J. London Math. Soc., 42 (1990), 292-308.
14) R. R. Hunt and R. L. Wheeden, Positive harmonic functions on Lipschitz domains, Trans. Amer. Math. Soc., 147 (1970), 507-527.
15) D. S. Jerison and C. E. Kenig, Boundary behavior of harmonic functions in non- tangentially accessible domains, Adv. in Math., 46 (1982), 80-147.
16) N. S. Landkof, Foundations of modern potential theory, Springer-Verlag, 1972,
17) J. Lelong-Ferrand, Étude au voisinage de la frontière des fonctions surharmoniques positive dans un demi-espace, Ann. Sci. École Norm. Sup., 66 (1949), 125-159,
18) V. G. Maz'ya, Beurling's theorem on a minimum principle for positive harmonic functions, (Russian), Zapiski Nauchnykh Seminarov LOMI, 30 (1972), 76-90; (English translation), J. Soviet Math., 4 (1975), 367-379.
19) L. Naïm, Sur le rôle de la frontière de R. S. Martin dans la théorie du potentiel, Ann. Inst. Fourier (Grenoble), 7 (1957), 183-281.
20) P, Sjögren, Weak L₁ characterization of Poisson integrals, Green potentials and H^p spaces, Trans. Amer. Math. Soc., 233 (1977), 179-196.
21) Y. Zhang, Ensembles équivalents a un point frontière dans un domaine lipshitzien, Séminaire de théorie du potential, Paris No. 9, Lecture Notes in Math., 1393, Springer-Verlag, 1989, pp. 256-265.
22) W.P. Ziemer, Weakly differentiable functions, Springer-Verlag, 1989.

Right : [1] H. Aikawa, Quasiadditivity of Riesz capacity, Math. Scand., 69 (1991), 15-30.
[2] H. Aikawa, Quasiadditivity of capacity and minimal thinness, Ann. Acad. Sci. Fenn. Ser. AI Math., 18 (1993), 65-75.
[3] A. Ancona, Positive harmonic functions and hyperbolicity, Potential Theory-Surveys and Problems, Lecture Notes in Math., 1344, Springer-Verlag, 1987, pp. 1-23.
[4] A. Beurling, A minimum principle for positive harmonic functions, Ann. Acad. Sci. Fenn. Ser. AI, 372 (1965).
[5] M. Brelot, On topologies and boundaries in potential theory, Lecture Notes in Math., 174, Springer-Verlag, 1971.
[6] F. F. Bonsall and D. Walsh, Vanishing l¹-sums of the Poisson kernel and sums with positive coefficients, Proc. Edinburgh Math. Soc., 32 (1989), 431-447.
[7] B. E. J, Dahlberg, A minimum principle for positive harmonic functions, Proc. London Math. Soc. (3), 33 (1976), 238-250.
[8] J. L. Doob, Classical potential theory and its probabilistic counterpart, Springer-Verlag, 1984.
[9] N. F. Dudley Ward, Atomic decompositions of integrable or continuous functions, PhD thesis (1991), University of York.
[10] M. Essén, On minimal thinness, reduced functions and Green potentials, Proc. Edinburgh Math. Soc., 36 (1992), 87-106,
[11] S. J. Gardiner, Sets of determination for harmonic functions, Trans. Amer. Math. Soc., 338 (1993), 233-243.
[12] W. K. Hayman, Subharmonic functions, Vol. 2, Academic Press, 1989.
[13] W. K. Hayman and T. J. Lyons, Bases for positive continuous functions, J. London Math. Soc., 42 (1990), 292-308.
[14] R. R. Hunt and R. L. Wheeden, Positive harmonic functions on Lipschitz domains, Trans. Amer. Math. Soc., 147 (1970), 507-527.
[15] D. S. Jerison and C. E. Kenig, Boundary behavior of harmonic functions in non-tangentially accessible domains, Adv. in Math., 46 (1982), 80-147.
[16] N. S. Landkof, Foundations of modern potential theory, Springer-Verlag, 1972,
[17] J. Lelong-Ferrand, Étude au voisinage de la frontière des fonctions surharmoniques positive dans un demi-espace, Ann. Sci. École Norm. Sup., 66 (1949), 125-159,
[18] V. G. Maz'ya, Beurling's theorem on a minimum principle for positive harmonic functions, (Russian), Zapiski Nauchnykh Seminarov LOMI, 30 (1972), 76-90; (English translation), J. Soviet Math., 4 (1975), 367-379.
[19] L. Naïm, Sur le rôle de la frontière de R. S. Martin dans la théorie du potentiel, Ann. Inst. Fourier (Grenoble), 7 (1957), 183-281.
[20] P, Sjögren, Weak L₁ characterization of Poisson integrals, Green potentials and H^p spaces, Trans. Amer. Math. Soc., 233 (1977), 179-196.
[21] Y. Zhang, Ensembles équivalents a un point frontière dans un domaine lipshitzien, Séminaire de théorie du potential, Paris No. 9, Lecture Notes in Math., 1393, Springer-Verlag, 1989, pp. 256-265.
[22] W. P. Ziemer, Weakly differentiable functions, Springer-Verlag, 1989.

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