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1998 | 127 | 2 | 137-168
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Associated weights and spaces of holomorphic functions

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EN
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EN
When treating spaces of holomorphic functions with growth conditions, one is led to introduce associated weights. In our main theorem we characterize, in terms of the sequence of associated weights, several properties of weighted (LB)-spaces of holomorphic functions on an open subset $G ⊂ ℂ^N$ which play an important role in the projective description problem. A number of relevant examples are provided, and a "new projective description problem" is posed. The proof of our main result can also serve to characterize when the embedding of two weighted Banach spaces of holomorphic functions is compact. Our investigations on conditions when an associated weight coincides with the original one and our estimates of the associated weights in several cases (mainly for G = ℂ or D) should be of independent interest.
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autor
  • Department of Mathematics, P.O. Box 4 (Yliopistonkatu 5), FIN-00014 University of Helsinki, Finland , taskinen@cc.helsinki.fi
  • Departamento de Matemática Aplicada, Universidad Politécnica de Valencia, E-46071 Valencia, Spain , jbonet@pleiades.upv.es
Bibliografia
  • [1] A. V. Abanin, Criteria for weak sufficiency, Math. Notes 40 (1986), 757-764.
  • [2] J. M. Anderson and J. Duncan, Duals of Banach spaces of entire functions, Glasgow Math. J. 32 (1990), 215-220.
  • [3] F. Bastin and J. Bonet, Locally bounded noncontinuous linear forms on strong duals of nondistinguished Köthe echelon spaces, Proc. Amer. Math. Soc. 108 (1990), 769-774.
  • [4] C. A. Berenstein and F. Gay, Complex Variables, An Introduction, Grad. Texts in Math. 125, Springer, 1991.
  • [5] K. D. Bierstedt, An introduction to locally convex inductive limits, in: Functional Analysis and its Applications (Proc. CIMPA Autumn School, Nice, 1986), World Scientific, 1988, 35-133.
  • [6] K. D. Bierstedt and J. Bonet, Some recent results on VC(X), in: Advances in the Theory of Fréchet Spaces (Istanbul, 1988), NATO Adv. Sci. Inst. Ser. C 287, Kluwer Acad. Publ., 1989, 181-194.
  • [7] K. D. Bierstedt and J. Bonet, Projective descriptions of weighted inductive limits: The vector-valued cases, ibid., 195-221.
  • [8] K. D. Bierstedt, J. Bonet and A. Galbis, Weighted spaces of holomorphic functions on balanced domains, Michigan Math. J. 40 (1993), 271-297.
  • [9] K. D. Bierstedt and R. Meise, Weighted inductive limits and their projective descriptions, Proc. Silivri Conference in Functional Analysis 1985, Doğa Tr. J. Math. 10 (1986), 54-82.
  • [10] K. D. Bierstedt and R. Meise, Distinguished echelon spaces and the projective description of weighted inductive limits of type $V_d C(X)$, in: Aspects of Mathematics and its Applications, North-Holland Math. Library 34, North-Holland, 1986, 169-226.
  • [11] K. D. Bierstedt, R. Meise and W. H. Summers, A projective description of weighted inductive limits, Trans. Amer. Math. Soc. 272 (1982), 107-160.
  • [12] K. D. Bierstedt and W. H. Summers, Biduals of weighted Banach spaces of analytic functions, J. Austral. Math. Soc. Ser. A 54 (1993), 70-79.
  • [13] J. Bonet, P. Domański, M. Lindström and J. Taskinen, Composition operators between weighted Banach spaces of analytic functions, J. Austral. Math. Soc. Ser. A, to appear.
  • [14] J. Bonet and S. N. Melikhov, Interpolation of entire functions and projective descriptions, J. Math. Anal. Appl. 205 (1997), 454-460.
  • [15] J. Bonet and J. Taskinen, The subspace problem for weighted inductive limits of spaces of holomorphic functions, Michigan Math. J. 42 (1995), 255-268.
  • [16] R. W. Braun, R. Meise and B. A. Taylor, Ultradifferentiable functions and Fourier analysis, Results Math. 17 (1990), 206-237.
  • [17] J. Clunie and T. Kővári, On integral functions having prescribed asymptotic growth II, Canad. J. Math. 20 (1968), 7-20.
  • [18] P. Erdős and T. Kővári, On the maximum modulus of entire functions, Acta Math. Acad. Sci. Hungar. 7 (1957), 305-317.
  • [19] L. Hörmander, An Introduction to Complex Analysis in Several Variables, North-Holland Math. Library 7, North-Holland, 1973.
  • [20] L. Hörmander, The Analysis of Linear Partial Differential Operators II, Grundlehren Math. Wiss. 257, Springer, 1983.
  • [21] W. Lusky, On weighted spaces of harmonic and holomorphic functions, J. London Math. Soc. (2) 51 (1995), 309-320.
  • [22] P. Mattila, E. Saksman and J. Taskinen, Weighted spaces of harmonic and holomorphic functions: Sequence space representations and projective descriptions, Proc. Edinburgh Math. Soc. 40 (1997), 41-62.
  • [23] P. Pérez Carreras and J. Bonet, Barrelled Locally Convex Spaces, North-Holland Math. Stud. 131, North-Holland, 1987.
  • [24] W. Rudin, Real and Complex Analysis, McGraw-Hill, 1970.
  • [25] 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.
  • [26] A. L. Shields and D. L. Williams, Bounded projections and the growth of harmonic conjugates in the unit disk, Michigan Math. J. 29 (1982), 3-25.
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Bibliografia
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bwmeta1.element.bwnjournal-article-smv127i2p137bwm
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