On generalized Bergman spaces

• Autorzy: Wolfgang Lusky
• Języki publikacji: EN

Abstrakty

EN

Let D be the open unit disc and μ a positive bounded measure on [0,1]. Extending results of Mateljević/Pavlović and Shields/Williams we give Banach-space descriptions of the classes of all harmonic (holomorphic) functions f: D → ℂ satisfying $ʃ_{0}^{1} (ʃ_{0}^{2π} |f(re^{iφ})|^p dφ)^{q/p} dμ(r) < ∞$.

Kategorie tematyczne

• 46E15: Banach spaces of continuous, differentiable or analytic functions
• 46B45: Banach sequence spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1996
• Tom: 119
• Numer: 1
• Strony: 77-95

Daty

wydano

1996

otrzymano

1995-09-28

poprawiono

1996-02-16

Twórcy

autor

Wolfgang Lusky

• Fachbereich 17, Universität-Gesamthochschule, Warburger Straße 100, D-33098 Paderborn, Germany

Bibliografia

• [1] S. Axler, Bergman spaces and their operators, in: Survey of Some Recent Results in Operator Theory, B. Conway and B. Morrel (eds.), Pitman Res. Notes, 1988, 1-50.
• [2] K. D. Bierstedt and W. H. Summers, Biduals of weighted Banach spaces of analytic functions, J. Austral. Math. Soc. Sec. A 54 (1993), 70-79.
• [3] O. Blasco, Multipliers on weighted Besov spaces of analytic functions, in: Contemp. Math. 144, Amer. Math. Soc., 1993, 23-33.
• [4] R. R. Coifman and R. Rochberg, Representation theorems for holomorphic and harmonic functions in $L^p$, Astérisque 77 (1980), 12-66.
• [5] P. L. Duren, Theory of $H^p$-Spaces, Academic Press, New York, 1970.
• [6] T. M. Flett, The dual of an inequality of Hardy and Littlewood and some related inequalities, J. Math. Anal. Appl. 38 (1972), 746-765.
• [7] T. M. Flett, Lipschitz spaces of functions on the circle and the disc, ibid. 39 (1972), 125-158.
• [8] G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals II, Math. Z. 34 (1932), 403-439.
• [9] G. H. Hardy and J. E. Littlewood, Theorems concerning mean values of analytic or harmonic functions, Quart. J. Math. 12 (1941), 221-256.
• [10] J. Lindenstrauss and A. Pełczyński, Contributions to the theory of classical Banach spaces, J. Funct. Anal. 8 (1971), 225-249.
• [11] W. Lusky, On the structure of $Hv_0(D)$ and $hv_0(D)$, Math. Nachr. 159 (1992), 279-289.
• [12] W. Lusky, On weighted spaces of harmonic and holomorphic functions, J. London Math. Soc. (2) 51 (1995), 309-320.
• [13] M. Mateljević and M. Pavlović, $L^p$-behaviour of the integral means of analytic functions, Studia Math. 77 (1984), 219-237.
• [14] L. A. Rubel and A. L. Shields, The second duals of certain spaces of analytic functions, J. Austral. Math. Soc. 11 (1970), 276-280.
• [15] A. L. Shields and D. L. Williams, Bounded projections, duality and multipliers in spaces of analytic functions, Trans. Amer. Math. Soc. 162 (1971), 287-302.
• [16] A. L. Shields and D. L. Williams, Bounded projections, duality and multipliers in spaces of harmonic functions, J. Reine Angew. Math. 299/300 (1978), 256-279.
• [17] A. L. Shields and D. L. Williams, Bounded projections and the growth of harmonic conjugates in the unit disc, Michigan Math. J. 29 (1982), 3-25.
• [18] A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Academic Press, New York, 1986.
• [19] P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge University Press, 1991.
• [20] P. Wojtaszczyk, On unconditional polynomial bases in $L_p$ and Bergman spaces, Constr. Approx., to appear.