On weighted spaces of holomorphic functions of several variables

Peer reviewe

Weighted Banach spaces of harmonic functions

“The final publication is available at Springer via http://dx.doi.org/10.1007/s13398-012-0109-z."We study Banach spaces of harmonic functions on open sets of or endowed with weighted supremum norms. We investigate the harmonic associated weight defined naturally as the analogue of the holomorphic associated weight introduced by Bierstedt, Bonet, and Taskinen and we compare them. We study composition operators with holomorphic symbol between weighted Banach spaces of pluriharmonic functions characterizing the continuity, the compactness and the essential norm of composition operators among these spaces in terms of associated weights.The research of the first author was partially supported by MEC and FEDER Project MTM2010-15200 and by GV project ACOMP/2012/090.Jorda Mora, E.; Zarco García, AM. (2014). Weighted Banach spaces of harmonic functions. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas. 108(2):405-418. https://doi.org/10.1007/s13398-012-0109-zS4054181082Axler, S., Bourdon, P., Ramey, W.: Harmonic Function Theory, 2nd edn. Springer, Berlin (2001)Bierstedt, K.D., Bonet, J., Galbis, A.: Weighted spaces of holomorphic functions on balanced domains. Mich. Math. J. 40(2), 271–297 (1993)Bierstedt, K.D., Bonet, J., Taskinen, J.: Associated weights and spaces of holomorphic functions. Stud. Math. 127(2), 137–168 (1998)Bierstedt, K.D., Summers, W.H.: Biduals of weighted Banach spaces of analytic functions. J. Aust. Math. Soc. Ser. A 54(1), 70–79 (1993)Bonet, J., Domański, P., Lindström, M.: Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions. Can. Math. Bull. 42(2), 139–148 (1999)Bonet, J., Domański, P., Lindström, M.: Weakly compact composition operators on weighted vector-valued Banach spaces of analytic mappings. Ann. Acad. Sci. Fenn. Math. Ser. A I 26, 233–248 (2001)Bonet, J., Domański, P., Lindström, M., Taskinen, J.: Composition operators between weighted Banach spaces of analytic functions. J. Aust. Math. Soc. Ser. A 64, 101–118 (1998)Bonet, J., Friz, M., Jordá, E.: Composition operators between weighted inductive limits of spaces of holomorphic functions. Publ. Math. Debr. Ser. A 67, 333–348 (2005)Boyd, C., Rueda, P.: The v-boundary of weighted spaces of holomorphic functions. Ann. Acad. Sci. Fenn. Math. 30, 337–352 (2005)Boyd, C., Rueda, P.: Complete weights and v-peak points of spaces of weighted holomorphic functions. Isr. J. Math. 155, 57–80 (2006)Boyd, C., Rueda, P.: Isometries of weighted spaces of harmonic functions. Potential Anal. 29(1), 37–48 (2008)Carando, D., Sevilla-Peris, P.: Spectra of weighted algebras of holomorphic functions. Math. Z. 263, 887–902 (2009)Contreras, M.D., Hernández-Díaz, G.: Weighted composition operators in weighted Banach spaces of analytic functions. J. Aust. Math. Soc. Ser. A 69(1), 41–60 (2000)García, D., Maestre, M., Rueda, P.: Weighted spaces of holomorphic functions on Banach spaces. Stud. Math. 138(1), 1–24 (2000)García, D., Maestre, M., Sevilla-Peris, P.: Composition operators between weighted spaces of holomorphic functions on Banach spaces. Ann. Acad. Sci. Fenn. Math. 29, 81–98 (2004)Gunning, R., Rossi, H.: Analytic Functions of Several Complex Variables. AMS Chelsea Publishing, Providence (2009)Hoffman, K.: Banach Spaces of Analytic Functions. Prentice-Hall, Englewood Cliffs (1962)Krantz, S.G.: Function Theory of Several Complex Variables. AMS, Providence (2001)Lusky, W.: On weighted spaces of harmonic and holomorphic functions. J. Lond. Math. Soc. 51, 309–320 (1995)Lusky, W.: On the isomorphism classes of weighted spaces of harmonic and holomorphic functions. Stud. Math. 175(1), 19–45 (2006)Meise, R., Vogt, D.: Introduction to Functional Analysis. Oxford University Press, Oxford (1997)Montes-Rodríguez, A.: Weight composition operators on weighted Banach spaces of analytic functions. J. Lond. Math. Soc. 61(2), 872–884 (2000)Ng, K.F.: On a theorem of Diximier. Math. Scand. 29, 279–280 (1972)Rudin, W.: Real and Complex Analysis. MacGraw-Hill, NY (1970)Rudin, W.: Functional analysis. In: International series in pure and applied mathematics, 2nd edn. McGraw-Hill, Inc., New York (1991)Shields, A.L., Williams, D.L.: Bounded projections, duality and multipliers in spaces of harmonic functions. J. Reine Angew. Math. 299(300), 256–279 (1978)Shields, A.L., Williams, D.L.: Bounded projections and the growth of harmonic conjugates in the unit disc. Mich. Math. J. 29, 3–25 (1982)Zheng, L.: The essential norms and spectra of composition operators on $$H^\infty$$ . Pac. J. Math. 203(2), 503–510 (2002

Classical operators on the Hörmander algebras

We study the integration operator, the differentiation operator and more general differential operators on radial Fr´echet or (LB) H¨ormander algebras of entire functions. We analyze when these operators are power bounded, hypercyclic and (uniformly) mean ergodic.This research was partially supported by MEC and FEDER Project MTM2010-15200. The research of M. J. Beltran was also supported by grant F.P.U. AP2008-00604 and Programa de Apoyo a la Investigacion y Desarrollo de la UPV PAID-06-12, and the research of J. Bonet and C. Fernandez, by GVA under Project PROMETEOII/2013/013.Beltrán Meneu, MJ.; Bonet Solves, JA.; Fernández, C. (2015). Classical operators on the Hörmander algebras. Discrete and Continuous Dynamical Systems - Series A. 35(2):637-652. https://doi.org/10.3934/dcds.2015.35.637S63765235

A note on completeness of weighted normed spaces of analytic functions

[EN] Given a non-negative weight v, not necessarily bounded or strictly positive, defined on a domain G in the complex plane, we consider the weighted space H-v(infinity) (G)of all holomorphic functions on G such that the product v vertical bar f vertical bar is bounded in G and study the question of when such a space is complete under the canonical sup-seminorm. We obtain both some necessary and some sufficient conditions in terms of the weight v, exhibit several relevant examples, and characterize completeness in the case of spaces with radial weights on balanced domains.The first author was partially supported by MTM2013-43540-P and MTM2016-76647-P by MINECO/FEDER-EU and GVA Prometeo II/2013/013. The second author was partially supported by the MINECO/FEDER-EU Grant MTM2015-65792-P. Both authors were partially supported by Thematic Research Network MTM2015-69323-REDT, MINECO, Spain.Bonet Solves, JA.; Vukotic, D. (2017). A note on completeness of weighted normed spaces of analytic functions. Results in Mathematics. 72(1-2):263-279. https://doi.org/10.1007/s00025-017-0696-2S263279721-2Arcozzi, N., Björn, A.: Dominating sets for analytic and harmonic functions and completeness of weighted Bergman spaces. Math. Proc. R. Ir. Acad. 102A, 175–192 (2002)Berenstein, C.A., Gay, R.: Complex Variables, An Introduction. Springer, New York (1991)Bierstedt, K.D., Bonet, J., Galbis, A.: Weighted spaces of holomorphic functions on bounded domains. Mich. Math. J. 40, 271–297 (1993)Bierstedt, K.D., Bonet, J., Taskinen, J.: Associated weights and spaces of holomorphic functions. Stud. Math. 127, 137–168 (1998)Björn, A.: Removable singularities for weighted Bergman spaces. Czechoslov. Math. J. 56, 179–227 (2006)Bonet, J., Domański, P., Lindström, M.: Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions. Can. Math. Bull. 42(2), 139–148 (1999)Bonet, J., Vogt, D.: Weighted spaces of holomorphic functions and sequence spaces. Note Mat. 17, 87–97 (1997)Conway, J.B.: Functions of One Complex Variable, Second Edition, Graduate Texts in Mathematics, vol. 11. Springer, New York (1978)Gaier, D.: Lectures on Complex Approximation. Birkhäuser, Boston (1987)Grosse-Erdmann, K.-G.: A weak criterion for vector-valued holomorphic functions. Math. Proc. Camb. Philos. Soc. 136, 399–411 (2004)Hörmander, L.: An Introduction to Complex Analysis in Several Variables. North-Holland, Amsterdam (1979)Horváth, J.: Topological Vector Spaces and Distributions. Addison-Wesley, Reading (1966)Lusky, W.: On weighted spaces of harmonic and holomorphic functions. J. Lond. Math. Soc. 51, 309–320 (1995)Lusky, W.: On the isomorphism classes of weighted spaces of harmonic and holomorphic functions. Stud. Math. 175, 19–45 (2006)Nakazi, T.: Weighted Bloch spaces which are Banach spaces. Rend. Circ. Mat. Palermo 62, 427–440 (2013)Shields, A.L., Williams, D.L.: Bounded projections, duality and multipliers in spaces of analytic functions. Trans. Am. Math. Soc. 162, 287–302 (1971

Superposition operators between weighted Banach spaces of analytic functions of controlled growth

The final publication is available at Springer via: http://dx.doi.org/10.1007/s00605-012-0441-6[EN] We characterize the entire functions which transform a weighted Banach space of holomorphic functions on the disc of type H∞ into another such space by superposition. We also show that all the superposition operators induced by such entire functions map bounded sets into bounded sets and are continuous. Superposition operators that map bounded sets into relatively compact sets are also considered. © 2012 Springer-Verlag Wien.The research of Bonet was partially supported by MICINN and FEDER Project MTM2010-15200, by GV project Prometeo/2008/101, and by ACOMP/2012/090. The research of Vukotic was partially supported by MICINN grant MTM2009-14694-C02-01, Spain and by the European ESF Network HCAA ("Harmonic and Complex Analysis and Its Applications").Bonet Solves, JA.; Vukotić, D. (2013). Superposition operators between weighted Banach spaces of analytic functions of controlled growth. Monatshefte für Mathematik. 170(3-4):311-323. https://doi.org/10.1007/s00605-012-0441-6S3113231703-4Álvarez, V., Márquez, M.A., Vukotić, D.: Superposition operators between the Bloch space and Bergman spaces. Ark. Mat. 42, 205–216 (2004)Appell, J., Zabrejko, P.P.: Nonlinear Superposition Operators, Cambridge Tracts in Mathematics 95. Cambridge University Press, London (1990)Appell, J., Zabrejko, P.P.: Remarks on the superposition operator problem in various function spaces. Complex Var. Elliptic Equ. 55(8–10), 727–737 (2010)Bierstedt, K.D., Bonet, J., Galbis, A.: Weighted spaces of holomorphic functions on bounded domains. Michigan Math. J. 40, 271–297 (1993)Bierstedt, K.D., Bonet, J., Taskinen, J.: Associated weights and spaces of holomorphic functions. Studia Math. 127, 137–168 (1998)Bonet, J., Domański, P., Lindström, M.: Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions. Can. Math. Bull. 42(2), 139–148 (1999)Bonet, J., Domański, P., Lindström, M., Taskinen, J.: Composition operators between weighted Banach spaces of analytic functions. J. Aust. Math. Soc. (Ser. A) 64, 101–118 (1998)Boyd, C., Rueda, P.: Holomorphic superposition operators between Banach function spaces. Preprint (2011)Boyd, C., Rueda, P.: Superposition operators between weighted spaces of analytic functions. Preprint (2011)Buckley, S.M., Fernández, J.L., Vukotić, D.: Superposition operators on Dirichlet type spaces. In: Papers on Analysis: A Volume dedicaed to Olli Martio on the occasion of his 60th birthday. Rep. Univ. Jyväskyla Dept. Math. Stat, vol. 83, pp. 41–61. Univ. Jyväskyla, Jyväskyla (2001)Buckley, S.M., Vukotić, D.: Univalent interpolation in Besov spaces and superposition into Bergman spaces. Potential Anal. 29(1), 1–16 (2008)Cámera, G.A.: Nonlinear superposition on spaces of analytic functions. In: Harmonic Analysis and Operator Theory (Carácas, 1994), Contemp. Math, vol. 189, pp. 103–116. Am. Math. Soc, Providence (1995)Cámera, G.A., Giménez, J.: The nonlinear superposition operators acting on Bergman spaces. Compositio Math. 93, 23–35 (1994)Castillo, R.E., Ramos Fernández, J.C., Salazar, M.: Bounded superposition operators between Bloch-Orlicz and $$\alpha$$ -Bloch spaces. Appl. Math. Comp. 218, 3441–3450 (2011)Dineen, S.: Complex Analysis in Locally Convex Spaces, vol. 57. North-Holland Math. Studies, Amsterdam (1981)Girela, D., Márquez, M.A.: Superposition operators between $$Q_p$$ spaces and Hardy spaces. J. Math. Anal. Appl. 364, 463–472 (2010)Grosse-Erdmann, K.-G.: A weak criterion for vector-valued holomorphic functions. Math. Proc. Camb. Publ. Soc. 136, 399–41 (2004)Harutyunyan, A., Lusky, W.: On the boundedness of the differentiation operator between weighted spaces of holomorphic functions. Studia Math. 184, 233–247 (2008)Langenbruch, M.: Continuation of Gevrey regularity for solutions of partial differential operators. In: Functional Analysis (Trier, 1994), pp. 249–280. de Gruyter, Berlin (1996)Levin, B.Ya.: Lectures on Entire Functions. Translations of Mathematical Monographs, vol. 150, Amer. Math. Soc., Providence (1996).Lusky, W.: On weighted spaces of harmonic and holomorphic functions. J. Lond. Math. Soc. 51, 309–320 (1995)Lusky, W.: On the isomorphism classes of weighted spaces of harmonic and holomorphic functions. Studia Math. 175, 19–45 (2006)Pommerenke, Ch.: Boundary Behaviour of Conformal Maps. Springer, Berlin (1992)Ramos Fernández, J.C.: Bounded superposition operators between weighted Banach spaces of analytic functions. Preprint, Available from http://arxiv.org/abs/1203.5857Shields, A.L., Williams, D.L.: Bounded projections, duality and multipliers in spaces of analytic functions. Trans. Am. Math. Soc. 162, 287–302 (1971)Vukotić, D.: Integrability, growth of conformal maps, and superposition operators, Technical Report 10. 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Duality and distance formulas in spaces defined by means of oscillation

For the classical space of functions with bounded mean oscillation, it is well known that VMO∗∗=BMOVMO∗∗=BMO and there are many characterizations of the distance from a function f in BMOBMO to VMOVMO. When considering the Bloch space, results in the same vein are available with respect to the little Bloch space. In this paper such duality results and distance formulas are obtained by pure functional analysis. Applications include general Möbius invariant spaces such as QK-spaces, weighted spaces, Lipschitz–Hölder spaces and rectangular BMOBMO of several variables

On realcompact topological vector spaces

[EN] This survey paper collects some of older and quite new concepts and results from descriptive set topology applied to study certain infinite-dimensional topological vector spaces appearing in Functional Analysis, including Frechet spaces, (L F)-spaces, and their duals, (D F)-spaces and spaces of continuous real-valued functions C(X) on a completely regular Hausdorff space X. Especially (L F)-spaces and their duals arise in many fields of Functional Analysis and its applications, for example in Distributions Theory, Differential Equations and Complex Analysis. The concept of a realcompact topological space, although originally introduced and studied in General Topology, has been also studied because of very concrete applications in Linear Functional Analysis.The research for the first named author was (partially) supported by Ministry of Science and Higher Education, Poland, Grant no. NN201 2740 33 and for the both authors by the project MTM2008-01502 of the Spanish Ministry of Science and Innovation.Kakol, JM.; López Pellicer, M. (2011). On realcompact topological vector spaces. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas. 105(1):39-70. https://doi.org/10.1007/s13398-011-0003-0S39701051Argyros S., Mercourakis S.: On weakly Lindelöf Banach spaces. Rocky Mountain J. Math. 23(2), 395–446 (1993). doi: 10.1216/rmjm/1181072569Arkhangel’skii, A. V.: Topological Function Spaces, Mathematics and its Applications, vol. 78, Kluwer, Dordrecht (1992)Batt J., Hiermeyer W.: On compactness in L p (μ, X) in the weak topology and in the topology σ(L p (μ, X), L p (μ,X′)). Math. Z. 182, 409–423 (1983)Baumgartner J.E., van Douwen E.K.: Strong realcompactness and weakly measurable cardinals. Topol. 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Complut. 2(Supplementary Issue), 179–199 (1989)Pérez Carreras P., Bonet J.: Barrelled Locally Convex Spaces, Mathematics Studies 131. North-Holland, Amsterdam (1987)Pfister H.H.: Bemerkungen zum Satz über die separabilität der Fréchet-Montel Raüme. Arch. Math. (Basel) 27, 86–92 (1976). doi: 10.1007/BF01224645Robertson N.: The metrisability of precompact sets. Bull. Aust. Math. Soc. 43(1), 131–135 (1991). doi: 10.1017/S0004972700028847Rogers C.A., Jayne J.E., Dellacherie C., Topsøe F., Hoffman-Jørgensen J., Martin D.A., Kechris A.S., Stone A.H.: Analytic Sets. Academic Press, London (1980)Saxon S.A.: Nuclear and product spaces, Baire-like spaces, and the strongest locally convex topology. Math. Ann. 197(2), 87–106 (1972). doi: 10.1007/BF01419586Schawartz L.: Radom Measures on Arbitrary Topological Spaces and Cylindrical Measures. Oxford University Press, Oxford (1973)Schlüchtermann G., Wheller R.F.: On strongly WCG Banach spaces. Math. 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Springer, Berlin (1974

A characterization of weighted (LB)(LB)-spaces of holomorphic functions having the dual density condition

summary:We characterize when weighted $(LB)$-spaces of holomorphic functions have the dual density condition, when the weights are radial and grow logarithmically