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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


Integral representations for Riesz systems in the unit ball and some applications

Author: Ashot Djrbashian
Journal: Proc. Amer. Math. Soc. 117 (1993), 395-403
MSC: Primary 42B99; Secondary 31B10, 46E15
MathSciNet review: 1116256
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Abstract: We introduce $ A_\alpha ^p$ spaces of systems of harmonic functions satisfying Cauchy-Riemann equations in $ {{\mathbf{R}}^{\mathbf{n}}}$ and find integral representations. Using these representations and estimates for the integral kernel we prove boundedness of the representation operator in $ {L^p}$ and Lipschitz classes.

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  • [1] A. È. Dzhrbashyan, Integral representations and continuous projectors in some spaces of harmonic functions, Mat. Sb. (N.S.) 121(163) (1983), no. 2, 259–271 (Russian). MR 703328 (85g:31004)
  • [2] A. È. Dzhrbashyan, Classes 𝐴^{𝑝}_{𝛼} of harmonic functions in half-spaces, and an analogue of a theorem of M. Riesz, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 22 (1987), no. 4, 386–398, 414 (Russian, with English and Armenian summaries). MR 931892 (89e:31002)
  • [3] Ashot E. Djrbashian and Faizo A. Shamoian, Topics in the theory of 𝐴^{𝑝}_{𝛼} spaces, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 105, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1988. With German, French and Russian summaries. MR 1021691 (91k:46019)
  • [4] A. È. Dzhrbashyan, Integral representations for classes of harmonic vector functions in the unit ball, Akad. Nauk Armyan. SSR Dokl. 88 (1989), no. 3, 112–116 (Russian, with Armenian summary). MR 1040129 (91b:31008)
  • [5] -, Integral representations for Riesz systems in half-spaces and balls, preprint 89-01, Inst. Math. Acad. Sci. Armenian SSR, Yerevan 1989.
  • [6] Peter L. Duren, Theory of 𝐻^{𝑝} spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655 (42 #3552)
  • [7] Adam Korányi and Stephen Vági, Cauchy-Szegö integrals for systems of harmonic functions, Ann. Scuola Norm. Sup. Pisa (3) 26 (1972), 181–196. MR 0390259 (52 #11085)
  • [8] Ewa Ligocka, The Hölder continuity of the Bergman projection and proper holomorphic mappings, Studia Math. 80 (1984), no. 2, 89–107. MR 781328 (86e:32030)
  • [9] Fulvio Ricci and Guido Weiss, A characterization of 𝐻¹(Σ_{𝑛-1}), Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 289–294. MR 545268 (80m:30043)
  • [10] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095 (44 #7280)
  • [11] Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR 0304972 (46 #4102)
  • [12] K. Yosida, Functional analysis, Springer-Verlag, Berlin, 1965.

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Additional Information

PII: S 0002-9939(1993)1116256-2
Keywords: Riesz systems, $ A_\alpha ^p$ spaces, bounded projections, Lipschitz classes
Article copyright: © Copyright 1993 American Mathematical Society