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

Published online by Cambridge University Press:  20 January 2009

Päivi Mattila ,
Eero Saksman  and
Jari Taskinen
Päivi Mattila
Affiliation:
Department of Mathematics, P.O. Box 4 (Hallituskatu 15), Fin-00014, University of Helsinki, Finland
Eero Saksman
Affiliation:
Department of Mathematics, P.O. Box 4 (Hallituskatu 15), Fin-00014, University of Helsinki, Finland
Jari Taskinen
Affiliation:
Department of Mathematics, P.O. Box 4 (Hallituskatu 15), Fin-00014, University of Helsinki, Finland
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Abstract

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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.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 40 , Issue 1 , February 1997 , pp. 41 - 62
Copyright
Copyright © Edinburgh Mathematical Society 1997

References

REFERENCES

1. Ahlfors, L., Complex analysis, 3rd ed. (McGraw-Hill, New York, 1979).Google Scholar
2. Bastin, F., On bornological spaces C (X), Archiv. Math. 53 (1989), 393–398.CrossRefGoogle Scholar
3. Berenstein, C. A. and Dostal, M. A., Analytically Uniform Spaces and their Applications to Convolution Equations (Springer Lecture Notes in Math. 256, 1972).CrossRefGoogle Scholar
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.Google Scholar
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), 149–180.Google Scholar
6. Bierstedt, K. D. and Bonet, J., Dual density conditions in (DF)-spaces, I. Resultate Math. 14 (1988), 242–274.CrossRefGoogle Scholar
7. Bierstedt, K. D. and Bonet, J., Dual density conditions in (DF)-spaces, II. Bull. Soc. Roy. Sci. Liége 57 (1988), 567–589.Google Scholar
8. Bierstedt, K. D. and Bonet, J., Some recent results on VC(X), in Advances in the theory of Fréchet spaces (Kluwer, 1989), 181–194.Google Scholar
9. Bierstedt, K. D. and Meise, R., Weighted inductive limits and their projective descriptions, Doga Mat. 10, 1 (1986), 54–82. (Special issue: Proceedings of the Silivri Conference 1985).Google Scholar
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), 169–226.CrossRefGoogle Scholar
11. Bierstedt, K. D., Meise, R. and Summers, W. H., A projective description of weighted inductive limits, Trans. Amer. Math. Soc. 272 (1982), 107–160.CrossRefGoogle Scholar
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), 27–91.Google Scholar
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).Google Scholar
14. Ehrenpreis, L., Fourier Analysis in Several Complex Variables (Interscience Tracts in Math. 17, Wiley, 1970).Google Scholar
15. Horváth, J., Topological Vector Spaces and Distributions (Addison-Wesley, 1966).Google Scholar
16. Koosis, , Introduction to Hp spaces (Cambridge University Press, 1980).Google Scholar
17. Köthe, G., Topological vector spaces. Vol. 1, Second printing (Springer Verlag, 1983).Google Scholar
18. Lusky, W., On weighted spaces of harmonic and holomorphic functions, J. London Math. Soc., to appear.Google Scholar
19. Pérez Carreras, P. and Bonet, J., Barrelled Locally Convex Spaces (North-Holland Math. Studies 131, 1987).Google Scholar
20. Rudin, W., Real and complex analysis, second edition (McGraw-Hill, New York, 1974).Google Scholar
21. Shields, A. and Williams, D., Bounded projections, duality and multipliers in spaces of harmonic functions, J. Reine Angew. Math. 299/300 (1978), 259–279.Google Scholar
22. Taylor, B. A., A seminorm topology for some (DF)-spaces of entire functions, Duke Math. J. 38 (1971), 379–385.CrossRefGoogle Scholar
23. Vogt, D., Distinguished Köthe spaces, Math. Z. 202 (1989), 143–146.Google Scholar