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Weighted spaces of harmonic and holomorphic functions: sequence space representations and protective descriptions

  • Päivi Mattila (a1), Eero Saksman (a1) and Jari Taskinen (a1)

Abstract

We study the structure of inductive limits of weighted spaces of harmonic and holomorphic functions defined on the open unit disk of ℂ, and of the associated weighted locally convex spaces. Using a result of Lusky we prove, for certain radial weights on the open unit disk D of ℂ, that the spaces of harmonic and holomorphic functions are isomorphic to complemented subspaces of the corresponding Köthe sequence spaces. We also study the spaces of harmonic functions for certain non-radial weights on D. We show, under a natural sufficient condition for the weights, that the spaces of harmonic functions on D are isomorphic to corresponding spaces of continuous or bounded functions on ∂D.

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References

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1. Ahlfors, L., Complex analysis, 3rd ed. (McGraw-Hill, New York, 1979).
2. Bastin, F., On bornological spaces C (X), Archiv. Math. 53 (1989), 393398.
3. Berenstein, C. A. and Dostal, M. A., Analytically Uniform Spaces and their Applications to Convolution Equations (Springer Lecture Notes in Math. 256, 1972).
4. Bierstedt, K. D., Weighted inductive limits of spaces of holomorphic functions, in Proceedings of the 23rd Annual Iranian Congress of Mathematics, April 1992, to appear.
5. Bierstedt, K. D. and Bonet, J., Stefan Heinrich's density condition for Fréchet spaces and the characterization of the distinguished Köthe echelon spaces. Math. Nachr. 135 (1988), 149180.
6. Bierstedt, K. D. and Bonet, J., Dual density conditions in (DF)-spaces, I. Resultate Math. 14 (1988), 242274.
7. Bierstedt, K. D. and Bonet, J., Dual density conditions in (DF)-spaces, II. Bull. Soc. Roy. Sci. Liége 57 (1988), 567589.
8. Bierstedt, K. D. and Bonet, J., Some recent results on VC(X), in Advances in the theory of Fréchet spaces (Kluwer, 1989), 181194.
9. Bierstedt, K. D. and Meise, R., Weighted inductive limits and their projective descriptions, Doga Mat. 10, 1 (1986), 5482. (Special issue: Proceedings of the Silivri Conference 1985).
10. Bierstedt, K. D. and Meise, R., Distinguished echelon spaces and the projective description of weighted inductive limits of type VdC(X), in Aspects of Mathematics and its Applications (Elsevier, 1986), 169226.
11. Bierstedt, K. D., Meise, R. and Summers, W. H., A projective description of weighted inductive limits, Trans. Amer. Math. Soc. 272 (1982), 107160.
12. Bierstedt, K. D., Meise, R. and Summers, W. H., Köthe sets and Köthe sequence spaces, Functional Analysis, Holomorphy and Approximation Theory (North-Holland Math. Studies 71, 1982), 2791.
13. Bonet, J. and Taskinen, J., The subspace problem for weighted inductive limits of spaces of holomorphic functions, in Reports of the Department of Mathematics, University of Helsinki, 51 (1994).
14. Ehrenpreis, L., Fourier Analysis in Several Complex Variables (Interscience Tracts in Math. 17, Wiley, 1970).
15. Horváth, J., Topological Vector Spaces and Distributions (Addison-Wesley, 1966).
16. Koosis, , Introduction to Hp spaces (Cambridge University Press, 1980).
17. Köthe, G., Topological vector spaces. Vol. 1, Second printing (Springer Verlag, 1983).
18. Lusky, W., On weighted spaces of harmonic and holomorphic functions, J. London Math. Soc., to appear.
19. Pérez Carreras, P. and Bonet, J., Barrelled Locally Convex Spaces (North-Holland Math. Studies 131, 1987).
20. Rudin, W., Real and complex analysis, second edition (McGraw-Hill, New York, 1974).
21. Shields, A. and Williams, D., Bounded projections, duality and multipliers in spaces of harmonic functions, J. Reine Angew. Math. 299/300 (1978), 259279.
22. Taylor, B. A., A seminorm topology for some (DF)-spaces of entire functions, Duke Math. J. 38 (1971), 379385.
23. Vogt, D., Distinguished Köthe spaces, Math. Z. 202 (1989), 143146.
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Weighted spaces of harmonic and holomorphic functions: sequence space representations and protective descriptions

  • Päivi Mattila (a1), Eero Saksman (a1) and Jari Taskinen (a1)

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