13 research outputs found
Multilinear Hölder-type inequalities on Lorentz sequence spaces
Get PDF[EN] We establish Holder-type inequalities for Lorentz sequence spaces and their duals. In order to achieve these and some related inequalities, we study diagonal multilinear forms in general sequence spaces, and obtain estimates for their norms. We also consider norms of multilinear forms in different Banach multilinear ideals.We would like to thank Silvia Lassalle and Andreas Defant for helpful conversations and suggestions that improved the final shape of the paper. We would also like to thank the referee for her/his valuable suggestions.
Most of the work in this axticle was performed while the third-named author was visiting the Departments of Mathematics at Universidad de Buenos Aires and Universidad de San Andres during the summer/winter of 2006. He wishes to thank all the people in and outside both Departments that made that such a delightful time.
The first-named author was partially supported by ANPCyT PICT 05 17-33042, UBACyT Grant X038 and ANPCyT PICT 06 00587. The second-named author was partially supported by ANPCyT PICT 05 17-33042. The third-named author was supported by the MEC Project MTM2008-03211 and paxtially by grants GV-AEST06/092 and UPV-PAID-00-06.Carando, D.; Dimant, V.; Sevilla Peris, P. (2009). Multilinear Hölder-type inequalities on Lorentz sequence spaces. Studia Mathematica. 195(2):127-146. https://doi.org/10.4064/sm195-2-3S127146195
Spectra of weighted algebras of holomorphic functions
Get PDF[EN] We consider weighted algebras of holomorphic functions on a Banach space. We determine conditions on a family of weights that assure that the corresponding weighted space is an algebra or has polynomial Schauder decompositions. We study the spectra of weighted algebras and endow them with an analytic structure. We also deal with composition operators and algebra homomorphisms, in particular to investigate how their induced mappings act on the analytic structure of the spectrum. Moreover, a Banach¿Stone type question is addressed.The rst author was supported by PIP 5272, UBACyT X108, PICT 05-33042 and PICT 06-00587. yThe second author was supported by the MECD Project MTM2005-08210Carando, D.; Sevilla Peris, P. (2009). Spectra of weighted algebras of holomorphic functions. Mathematische Zeitschrift. 263:887-902. https://doi.org/10.1007/s00209-008-0444-0S887902263Aron R.M., Berner P.D.: A Hahn–Banach extension theorem for analytic mappings. Bull. Soc. Math. Fr. 106, 3–24 (1978)Aron R.M., Cole B., Gamelin T.W.: Spectra of algebras of analytic functions on a Banach space. J. Reine Angew. Math. 415, 51–93 (1991)Aron R.M., Galindo P., García D., Maestre M.: Regularity and algebras of analytic functions in infinite dimensions. Trans. Am. Math. Soc. 348(2), 543–559 (1996)Bierstedt K.D., Bonet J., Galbis A.: Weighted spaces of holomorphic functions on balanced domains. Mich. Math. 40, 271–297 (1993)Bierstedt K.D., Bonet J., Taskinen J.: Associated weights and spaces of holomorphic functions. Studia Math. 127(2), 137–168 (1998)Boas R.P. Jr: Representations for entire functions of exponential type. Ann. Math. (2) 39(2), 269–286 (1938)Boas R.P. Jr: Entire Functions. Academic Press, New York (1954)Bonet, J.: Weighted spaces of holomorphic functions and operators between them. In: U. d. S. Secretariado de Publicaciones (ed.) Proceedings of the seminar of Mathematical Analysis (Univ. Málaga, Univ. Sevilla), pp. 117–138, Sevilla (2003)Bonet J., Domański P., Lindström M.: Essential norm and weak compactness of composition operators on weighted spaces of analytic functions. Canad. Math. Bull. 42(2), 139–148 (1999)Bonet J., Domański P., Lindström M., Taskinen J.: Composition operators between wighted Banach spaces of analytic functions. J. Austral. Math. Soc. (Ser. A) 64, 101–118 (1998)Bonet J., Friz M.: Weakly compact composition operators on locally convex spaces. Math. Nachr. 245, 26–44 (2002)Bonet J., Friz M., Jordá E.: Composition operators between weighted inductive limits of sapces of holomorphic functions. Publ. Math. Debrecen 67(3–4), 333–348 (2005)Cabello Sánchez F., Castillo J.M.F., García R.: Polynomials on dual-isomorphic spaces. Ark. Mat. 38(1), 37–44 (2000)Carando D., García D., Maestre M.: Homomorphisms and composition operators on algebras of analytic functions of bounded type. Adv. Math. 197(2), 607–629 (2005)Carando D., Lassalle S.: E′ and its relation with vector-valued functions on E. Ark. Mat. 42(2), 283–300 (2004)Davie A., Gamelin T.: A theorem on polynomial-star approximation. Proc. Am. Math. Soc. 106(2), 351–356 (1989)Díaz J.C., Dineen S.: Polynomials on stable spaces. Ark. Mat. 36(1), 87–96 (1998)Dineen S.: Complex Analysis on Infinite Dimensional Spaces. Springer, London (1999)Galindo P., Maestre M., Rueda P.: Biduality in spaces of holomorphic functions. Math. Scand. 86(1), 5–16 (2000)García D., Maestre M., Rueda P.: Weighted spaces of holomorphic functions on Banach spaces. Studia 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)García D., Maestre M., Sevilla-Peris P.: Weakly compact composition operators between weighted spaces. Note Mat. 25(1), 205–220 (2005/06)Garrido, M.I., Jaramillo, J.A.: Variations on the Banach–Stone theorem. Extracta Math. 17(3), 351–383. IV Course on Banach Spaces and Operators (Spanish) (Laredo, 2001) (2002)Kalton N.J.: Schauder decompositions in locally convex spaces. Proc. Cambridge Philos. Soc. 68, 377–392 (1970)Lassalle S., Zalduendo I.: To what extent does the dual Banach space E′ determine the polynomials over E?. Ark. Mat. 38(2), 343–354 (2000)Oubbi L.: Weighted algebras of continuous functions. Results Math. 24(3–4), 298–307 (1993)Vieira D.M.: Theorems of Banach–Stone type for algebras of holomorphic functions on infinite dimensional spaces. Math. Proc. R. Ir. Acad. 106A((1), 97–113 (2006
Hausdorff-Young-type inequalities for vector-valued Dirichlet series
Get PDF[EN] We study Hausdorff-Young-type inequalities for vector-valued Dirichlet series which allow us to compare the norm of a Dirichlet series in the Hardy space H-p(X) with the q-norm of its coefficients. In order to obtain inequalities completely analogous to the scalar case, a Banach space must satisfy the restrictive notion of Fourier type/cotype. We show that variants of these inequalities hold for the much broader range of spaces enjoying type/cotype. We also consider Hausdorff-Young-type inequalities for functions defined on the infinite torus T-infinity or the boolean cube {- 1, 1}(infinity). As a fundamental tool we show that type and cotype are equivalent to a hypercontractive homogeneous polynomial type and cotype, a result of independent interest.The third author was supported by MICINN and FEDER Project MTM2017-83262-C2-1-P and MECD grant PRX17/00040.Carando, D.; Marceca, F.; Sevilla Peris, P. (2020). Hausdorff-Young-type inequalities for vector-valued Dirichlet series. Transactions of the American Mathematical Society. 373(8):5627-5652. https://doi.org/10.1090/tran/8147S562756523738BLASCO, O., & PAVLOVIC, M. (2003). COMPLEX CONVEXITY AND VECTOR-VALUED LITTLEWOOD–PALEY INEQUALITIES. Bulletin of the London Mathematical Society, 35(06), 749-758. doi:10.1112/s0024609303002479Carando, D., Defant, A., & Sevilla-Peris, P. (2014). Bohr’s absolute convergence problem for ℋp-Dirichlet series in Banach spaces. Analysis & PDE, 7(2), 513-527. doi:10.2140/apde.2014.7.513Carando, D., Defant, A., & Sevilla-Peris, P. (2016). Some polynomial versions of cotype and applications. Journal of Functional Analysis, 270(1), 68-87. doi:10.1016/j.jfa.2015.09.017Davis, W. J., Garling, D. J. ., & Tomczak-Jaegermann, N. (1984). The complex convexity of quasi-normed linear spaces. Journal of Functional Analysis, 55(1), 110-150. doi:10.1016/0022-1236(84)90021-1De la Peña, V. H., & Giné, E. (1999). Decoupling. Probability and its Applications. doi:10.1007/978-1-4612-0537-1Defant, A., García, D., Maestre, M., & Pérez-García, D. (2008). Bohr’s strip for vector valued Dirichlet series. Mathematische Annalen, 342(3), 533-555. doi:10.1007/s00208-008-0246-zDefant, A., García, D., Maestre, M., & Sevilla-Peris, P. (2019). Dirichlet Series and Holomorphic Functions in High Dimensions. doi:10.1017/9781108691611Defant, A., Maestre, M., & Schwarting, U. (2012). Bohr radii of vector valued holomorphic functions. Advances in Mathematics, 231(5), 2837-2857. doi:10.1016/j.aim.2012.07.016Defant, A., Mastyło, M., & Pérez, A. (2018). On the Fourier spectrum of functions on Boolean cubes. Mathematische Annalen, 374(1-2), 653-680. doi:10.1007/s00208-018-1756-yDefant, A., & Pérez, A. (2018). Hardy spaces of vector-valued Dirichlet series. Studia Mathematica, 243(1), 53-78. doi:10.4064/sm170303-26-7Diestel, J., Jarchow, H., & Tonge, A. (1995). Absolutely Summing Operators. doi:10.1017/cbo9780511526138Garcia-Cuerva, J., Kazarian, K. S., Kolyada, V. I., & Torrea, J. L. (1998). Vector-valued Hausdorff-Young inequality and applications. Russian Mathematical Surveys, 53(3), 435-513. doi:10.1070/rm1998v053n03abeh000018Hedenmalm, H., Lindqvist, P., & Seip, K. (1997). A Hilbert space of Dirichlet series and systems of dilated
functions in L2(0,1). Duke Mathematical Journal, 86(1). doi:10.1215/s0012-7094-97-08601-4Kwapień, S., & Woyczyński, W. A. (1992). Random Series and Stochastic Integrals: Single and Multiple. doi:10.1007/978-1-4612-0425-1Pelczynski, A. (1988). Commensurate Sequences of Characters. Proceedings of the American Mathematical Society, 104(2), 525. doi:10.2307/2047005Pisier, G. (1982). Holomorphic Semi-Groups and the Geometry of Banach Spaces. The Annals of Mathematics, 115(2), 375. doi:10.2307/1971396Queffélec, H., & Queffélec, M. (2013). Diophantine Approximation and Dirichlet Series. doi:10.1007/978-93-86279-61-3Maciej Rzeszut and Michał Wojciechowski, Hoeffding Decomposition in ¹ spaces. arXiv:1906.01405.Wilf, H. S. (1970). Finite Sections of Some Classical Inequalities. doi:10.1007/978-3-642-86712-
Limit orders and multilinear forms on l(p) spaces
Get PDF[EN] Since the concept of limit order is a useful tool to study operator ideals, we
propose an analogous definition for ideals of multilinear forms. From the limit orders
of some special ideals (of nuclear, integral, r-dominated and extendible multilinear
forms) we derive some properties of them and show differences between the bilinear
and n-linear cases (n ¿ 3).The third author was supported by the MCYT and FEDER Project BFM2002-01423 and grant GV-GRUPOS04/45Carando, D.; Dimant, V.; Sevilla Peris, P. (2006). Limit orders and multilinear forms on l(p) spaces. Publications of the Research Institute for Mathematical Sciences. 42(2):507-522. https://riunet.upv.es/handle/10251/166258S50752242
Random unconditional convergence of vector-valued Dirichlet series
No full textWe study random unconditionality of Dirichlet series in vector-valued Hardy spaces Hp(X). It is shown that a Banach space X has type 2 (respectively, cotype 2) if and only if for every choice (xn)n⊂X it follows that (xnn−s)n is random unconditionally convergent (respectively, divergent) in H2(X). The analogous question on Hp(X) spaces for p≠2 is also explored. We also provide explicit examples exhibiting the differences between the unconditionality of (xnn−s)n in Hp(X) and that of (xnzn)n in Hp(X). © 2019 Elsevier Inc.The first three authors were partially supported by CONICET -PIP 11220130100329CO and ANPCyT PICT 2015-2299 . The second and third authors are also supported by a CONICET doctoral fellowship. Fourth author gratefully acknowledges support of Spanish Ministerio de Economía, Industria y Competitividad through grants MTM2016-76808-P , MTM2016-75196-P , and the “ Severo Ochoa Programme for Centres of Excellence in R&D ” ( SEV-2015-0554 ).Peer reviewe
On norm-attainment in (symmetric) tensor products
Get PDFIn this paper, we introduce a concept of norm-attainment in the projective symmetric tensor product of a Banach space X, which turns out to be naturally related to the classical norm-attainment of N -homogeneous polynomials on X. Due to this relation, we can prove that there exist symmetric tensors that do not attain their norms, which allows us to study the problem of when the set of norm-attaining elements in is dense. We show that the set of all normattaining symmetric tensors is dense in for a large set of Banach spaces such as Lp-spaces, isometric L1-predual spaces or Banach spaces with monotone Schauder basis, among others. Next, we prove that if X* satisfies the Radon-Nikodým and approximation properties, then the set of all norm-attaining symmetric tensors in is dense. From these techniques, we can present new examples of Banach spaces X and Y such that the set of all norm-attaining tensors in the projective tensor product is dense, answering positively an open question from the paperThe authors would like to thank Daniel Carando and Jorge Tomás Rodríguez for fruitful conversations on the topic of the paper. The first author was supported by the project OPVVV CAAS CZ.02.1.01/0.0/0.0/16 019/0000778 and by the Estonian Research Council grant PRG877. The second author is supported in part by the grants MTM2017-83262-C2-2-P and Fundación Séneca Región de Murcia 20906/PI/18.
The third author was supported by NRF (NRF-2019R1A2C1003857). The fourth author
was supported by Juan de la Cierva-Formación fellowship FJC2019-039973, by MTM2017-
86182-P (Government of Spain, AEI/FEDER, EU), by MICINN (Spain) Grant PGC2018-
093794-B-I00 (MCIU, AEI, FEDER, UE), by Fundación Séneca, ACyT Región de Murcia grant 20797/PI/18, by Junta de Andalucía Grant A-FQM-484-UGR18 and by Junta de Andalucía Grant FQM-0185
Spectra of weighted (LB)-algebras of entire functions on Banach spaces
No full textGiven a decreasing family of continuous weights on a Banach space X, we consider the weighted inductive limits of spaces of entire functions VH(X) and VH0(X). Motivated by recent research by D. Carando and P. Sevilla-Peris on weighted Fréchet algebras of entire functions on Banach spaces, we determine conditions on the family of weights to ensure that the corresponding weighted space is an algebra or has polynomial Schauder decompositions. We study Hörmander algebras of entire functions defined on a Banach space and we give a description of them in terms of sequence spaces. We also focus on algebra homomorphisms between these spaces and obtain a Banach-Stone type theorem for a particular decreasing family of weights. Finally, we study the spectra of these weighted algebras, endowing them with an analytic structure. © 2011 Elsevier Inc.The author was partially supported by MEC and FEDER Project MTM2010-15200 and grant F.P.U. AP2008-00604, and Conselleria d'Educacio de la GVA, Project GV/2010/040.Beltrán Meneu, MJ. (2012). Spectra of weighted (LB)-algebras of entire functions on Banach spaces. Journal of Mathematical Analysis and Applications. 387:604-617. https://doi.org/10.1016/j.jmaa.2011.09.022S60461738
Limited range multilinear extrapolation with applications to the bilinear Hilbert transform
No full textWe prove a limited range, off-diagonal extrapolation theorem that generalizes a number of results in the theory of Rubio de Francia extrapolation, and use this to prove a limited range, multilinear extrapolation theorem. We give two applications of this result to the bilinear Hilbert transform. First, we give sufficient conditions on a pair of weights w1,w2 for the bilinear Hilbert transform to satisfy weighted norm inequalities of the form (Formula presented.),where w= ww and 1p=1p1+1p2<32. This improves the recent results of Culiuc et al. by increasing the families of weights for which this inequality holds and by pushing the lower bound on p from 1 down to 23, the critical index from the unweighted theory of the bilinear Hilbert transform. Second, as an easy consequence of our method we obtain that the bilinear Hilbert transform satisfies some vector-valued inequalities with Muckenhoupt weights. This reproves and generalizes some of the vector-valued estimates obtained by Benea and Muscalu in the unweighted case. We also generalize recent results of Carando, et al. on Marcinkiewicz-Zygmund estimates for multilinear Calderón-Zygmund operators.The rst author is supported by NSF Grant DMS-1362425 and research funds from the Dean of
the College of Arts & Sciences, University of Alabama. The second author acknowledges nancial
support from the Spanish Ministry of Economy and Competitiveness, through the \Severo Ochoa"
Programme for Centres of Excellence in R&D (SEV-2015-0554). He also acknowledges that the
research leading to these results has received funding from the European Research Council under the
European Union's Seventh Framework Programme (FP7/2007-2013)/ ERC agreement no. 615112Peer Reviewe
Mean ergodic composition operators in spaces of homogeneous polynomials
Get PDF[EN] We study some dynamical properties of composition operators defined on the space P(^m X) of m-homogeneous polynomials on a Banach space X when P(^m X) is endowed with two different topologies: the one of uniform convergence on compact sets and the one defined by the usual norm. The situation is quite different for both topologies: while in the case of uniform convergence on compact sets every power bounded composition operator is uniformly mean ergodic, for the topology of the norm there is no relation between the latter properties. Several examples are given.We are indebted to Prof. Jose Bonet and Prof. Daniel Carando for helpful suggestions about this work. The research of the first author was partially supported by the project MTM2016-76647-P. The research of the second author was partially supported by the project GV Prometeo 2017/102. The research of the third author was partially supported by the project MTM2017-83262-C2-1-PJornet Casanova, D.; Santacreu, D.; Sevilla Peris, P. (2020). Mean ergodic composition operators in spaces of homogeneous polynomials. Journal of Mathematical Analysis and Applications. 483(1):1-12. https://doi.org/10.1016/j.jmaa.2019.123582S1124831Beltrán-Meneu, M. J., Gómez-Collado, M. C., Jordá, E., & Jornet, D. (2016). Mean ergodic composition operators on Banach spaces of holomorphic functions. Journal of Functional Analysis, 270(12), 4369-4385. doi:10.1016/j.jfa.2016.03.003Beltrán-Meneu, M. J., Gómez-Collado, M. C., Jordá, E., & Jornet, D. (2016). Mean ergodicity of weighted composition operators on spaces of holomorphic functions. Journal of Mathematical Analysis and Applications, 444(2), 1640-1651. doi:10.1016/j.jmaa.2016.07.039T. Bermúdez, A. Bonilla, V. Müller, A. Peris, Cesàro bounded operators in Banach spaces, J. Anal. Math. (to appear).Bonet, J., de Pagter, B., & Ricker, W. J. (2011). Mean ergodic operators and reflexive Fréchet lattices. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 141(5), 897-920. doi:10.1017/s0308210510000314Bonet, J., & Domański, P. (2011). A note on mean ergodic composition operators on spaces of holomorphic functions. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 105(2), 389-396. doi:10.1007/s13398-011-0009-7Dineen, S. (1999). Complex Analysis on Infinite Dimensional Spaces. Springer Monographs in Mathematics. doi:10.1007/978-1-4471-0869-6Kalmes, T. (2019). Power bounded weighted composition operators on function spaces defined by local properties. Journal of Mathematical Analysis and Applications, 471(1-2), 211-238. doi:10.1016/j.jmaa.2018.10.07
Operators on wighted spaces of holomorphic functions
Get PDFThe Ph.D. Thesis ¿Operators on weighted spaces of holomorphic functions¿ presented
here treats different areas of functional analysis such as spaces of holomorphic
functions, infinite dimensional holomorphy and dynamics of operators.
After a first chapter that introduces the notation, definitions and the basic results
we will use throughout the thesis, the text is divided into two parts. A first one,
consisting of Chapters 1 and 2, focused on a study of weighted (LB)-spaces of entire
functions on Banach spaces, and a second one, corresponding to Chapters 3 and
4, where we consider differentiation and integration operators acting on different
classes of weighted spaces of entire functions to study its dynamical behaviour. In
what follows, we give a brief description of the different chapters:
In Chapter 1, given a decreasing sequence of continuous radial weights on a Banach
space X, we consider the weighted inductive limits of spaces of entire functions
VH(X) and VH0(X). Weighted spaces of holomorphic functions appear naturally
in the study of growth conditions of holomorphic functions and have been investigated
by many authors since the work of Williams in 1967, Rubel and Shields
in 1970 and Shields and Williams in 1971. We determine conditions on the family
of weights to ensure that the corresponding weighted space is an algebra or
has polynomial Schauder decompositions. We study Hörmander algebras of entire
functions defined on a Banach space and we give a description of them in terms of
sequence spaces. We also focus on algebra homomorphisms between these spaces
and obtain a Banach-Stone type theorem for a particular decreasing family of
weights. Finally, we study the spectra of these weighted algebras, endowing them
with an analytic structure, and we prove that each function f ¿ VH(X) extends
naturally to an analytic function defined on the spectrum. Given an algebra homomorphism,
we also investigate how the mapping induced between the spectra
acts on the corresponding analytic structures and we show how in this setting
composition operators have a different behavior from that for holomorphic functions
of bounded type. This research is related to recent work by Carando, García,
Maestre and Sevilla-Peris. The results included in this chapter are published by
Beltrán in [14]. Chapter 2 is devoted to study the predual of VH(X) in order to linearize this space
of entire functions. We apply Mujica¿s completeness theorem for (LB)-spaces to
find a predual and to prove that VH(X) is regular and complete. We also study
conditions to ensure that the equality VH0(X) = VH(X) holds. At this point,
we will see some differences between the finite and the infinite dimensional cases.
Finally, we give conditions which ensure that a function f defined in a subset
A of X, with values in another Banach space E, and admitting certain weak
extensions in a space of holomorphic functions can be holomorphically extended
in the corresponding space of vector-valued functions. Most of the results obtained
have been published by the author in [13].
The rest of the thesis is devoted to study the dynamical behaviour of the following
three operators on weighted spaces of entire functions: the differentiation operator
Df(z) = f (z), the integration operator Jf(z) = z
0 f(¿)d¿ and the Hardy
operator Hf(z) = 1
z z
0 f(¿)d¿, z ¿ C.
In Chapter 3 we focus on the dynamics of these operators on a wide class of
weighted Banach spaces of entire functions defined by means of integrals and
supremum norms: the weighted spaces of entire functions Bp,q(v), 1 ¿ p ¿ ¿,
and 1 ¿ q ¿ ¿. For q = ¿ they are known as generalized weighted Bergman
spaces of entire functions, denoted by Hv(C) and H0
v (C) if, in addition, p = ¿.
We analyze when they are hypercyclic, chaotic, power bounded, mean ergodic
or uniformly mean ergodic; thus complementing also work by Bonet and Ricker
about mean ergodic multiplication operators. Moreover, for weights satisfying
some conditions, we estimate the norm of the operators and study their spectrum.
Special emphasis is made on exponential weights. The content of this chapter is
published in [17] and [15].
For differential operators ¿(D) : Bp,q(v) ¿ Bp,q(v), whenever D : Bp,q(v) ¿
Bp,q(v) is continuous and ¿ is an entire function, we study hypercyclicity and
chaos. The chapter ends with an example provided by A. Peris of a hypercyclic
and uniformly mean ergodic operator. To our knowledge, this is the first example
of an operator with these two properties. We thank him for giving us permission
to include it in our thesis.
The last chapter is devoted to the study of the dynamics of the differentiation and
the integration operators on weighted inductive and projective limits of spaces of
entire functions. We give sufficient conditions so that D and J are continuous on
these spaces and we characterize when the differentiation operator is hypercyclic,
topologically mixing or chaotic on projective limits. Finally, the dynamics of these
operators is investigated in the Hörmander algebras Ap(C) and A0
p(C). The results
concerning this topic are included by Bonet, Fernández and the author in [16].Beltrán Meneu, MJ. (2014). Operators on wighted spaces of holomorphic functions [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/36578TESISPremios Extraordinarios de tesis doctorale
