Isomorphisms on weighed Banach spaces of harmonic and holomorphic functions

Abstract

1 6 2013 ; S ; For an arbitrary open subset U subset of R-d or U subset of C-d and a continuous function v : U ->]0,infinity[ we show that the space h(v0) (U) of weighed harmonic functions is almost isometric to a (closed) subspace of c(0), thus extending a theorem due to Bonet and Wolf for spaces of holomorphic functions H-v0 (U) on open sets U subset of C-d. Inspired by recent work of Boyd and Rueda, we characterize in terms of the extremal points of the dual of h(v0) (U) when h(v0) (U) is isometric to a subspace of c(0). Some geometric conditions on an open set U subset of C-d and convexity conditions on a weight v on U are given to ensure that neither H-v0 (U) nor h(v0) (U) are rotund. The authors are grateful to J. Bonet for a lot of ideas and discussions during all this work. They are also indebted to P. Rueda, who besides giving them some references provided them some unpublished work which has inspired a big part of this paper. They also thank M. Maestre for his careful reading of the paper and helpful discussions. Finally, they thank the referees for the careful analysis of the paper and the important suggestions which they gave us. The research of the first author was supported by MICINN and FEDER, Project MTM2010-15200. The research of both authors is partially supported by Programa de Apoyo a la Investigacion y Desarrollo de la UPV PAID-06-12. Jorda Mora, E.; Zarco García, AM. (2013). Isomorphisms on weighed Banach spaces of harmonic and holomorphic functions. Journal of Function Spaces and Applications. 2013:1-6. https://doi.org/10.1155/2013/178460 Bierstedt, K. D., Bonet, J., & Galbis, A. (1993). Weighted spaces of holomorphic functions on balanced domains. The Michigan Mathematical Journal, 40(2), 271-297. doi:10.1307/mmj/1029004753 Bonet, J., Dománski, P., & Lindström, M. (1999). Essential Norm and Weak Compactness of Composition Operators on Weighted Banach Spaces of Analytic Functions. Canadian Mathematical Bulletin, 42(2), 139-148. doi:10.4153/cmb-1999-016-x Bonet, J., Domański, P., ...