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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Weak $L_{1}$ characterizations of Poisson integrals, Green potentials and $H^{p}$ spaces
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by Peter Sjögren PDF
Trans. Amer. Math. Soc. 233 (1977), 179-196 Request permission

Abstract:

Our main result can be described as follows. A subharmonic function u in a suitable domain $\Omega$ in ${{\mathbf {R}}^n}$ is the difference of a Poisson integral and a Green potential if and only if u divided by the distance to $\partial \Omega$ is in weak ${L_1}$ in $\Omega$. Similar conditions are given for a harmonic function to be the Poisson integral of an ${L_p}$ function on $\partial \Omega$. Iterated Poisson integrals in a polydisc are also considered. As corollaries, we get weak ${L_1}$ characterizations of ${H^p}$ spaces of different kinds.
References
  • Arne Beurling, A minimum principle for positive harmonic functions, Ann. Acad. Sci. Fenn. Ser. A I No. 372 (1965), 7. MR 0188466
  • Björn Dahlberg, A minimum principle for positive harmonic functions, Proc. London Math. Soc. (3) 33 (1976), no. 2, 238–250. MR 409847, DOI 10.1112/plms/s3-33.2.238
  • C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215
  • Günter Hellwig, Partielle Differentialgleichungen: Eine Einführung, Mathematische Leitfäden, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1960 (German). MR 0114986, DOI 10.1007/978-3-663-11002-6
  • V. G. Maz′ja, On Beurling’s theorem on the minimum principle for positive harmonic functions, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 30 (1972), 76–90 (Russian). Investigations on linear operators and the theory of functions, III. MR 0330484
  • Walter Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0255841
  • P. Sjögren, La convolution dans ${L^1}$ faible de ${R^n}$, Séminaire Choquet: Initiation à l’analyse, 13ième année (1973-74), exposé 14, Secrétariat mathématique, Paris, 10 pp. —, Noyaux singuliers positifs et ensembles exceptionnels, Séminaire Choquet: Initiation à l’analyse, 14ième année (1974-75), exposé 8, Secrétariat mathématique, Paris, 23 pp.
  • Peter Sjögren, Une propriété des fonctions harmoniques positives, d’après Dahlberg, Séminaire de Théorie du Potentiel de Paris, No. 2 (Univ. Paris, Paris, 1975–1976) Lecture Notes in Math., Vol. 563, Springer, Berlin, 1976, pp. 275–282 (French). MR 0588344
  • —, Characterizations of Poisson integrals and ${H^p}$ spaces, Report 1975-5, Dept. of Math., Chalmers Univ. of Technology and the Univ. of Göteborg, Sweden.
  • Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
  • Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
  • 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
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 233 (1977), 179-196
  • MSC: Primary 31B10
  • DOI: https://doi.org/10.1090/S0002-9947-1977-0463462-1
  • MathSciNet review: 0463462