On weighted L_p-spaces of vector-valued entire analytic functions

[EN] The weighted L p -spaces of entire analytic functions are generalized to the vector-valued setting. In particular, it is shown that the dual of the space LKp,¿(E) is isomorphic to L¿Kp¿,¿¿1(E¿) when the function ¿ K is an L p,¿ (E)-Fourier multiplier. This result allows us to give some new characterizations of the so-called UMD-property and to represent several ultradistribution spaces by means of spaces of vector sequences.Motos Izquierdo, J.; Planells Gilabert, MJ.; Talavera Usano, CF. (2008). On weighted L_p-spaces of vector-valued entire analytic functions. Mathematische Zeitschrift. 260(2):451-472. doi:10.1007/s00209-007-0283-4S4514722602Aldous D.J. (1979). Unconditional bases and martingales in L p (F). Math. Proc. Camb. Phil. Soc. 85: 117–123Amann H. (1997). Operator-valued Fourier multipliers, vector-valued Besov spaces and applications. Math. Nachr. 186: 5–56Björck G. (1966). Linear partial differential operators and generalized distributions. Ark. Mat. 6: 351–407Bourgain J. (1984). Some remarks on Banach spaces in which martingale difference sequences are unconditional. Ark. Mat. 22: 163–168Burkholder, D.L.: A geometrical condition that implies the existence of certain singular integrals of Banach-space-valued functions. In: Proceedings of conference on Harmonic analysis in Honour of Antoni Zygmund, Chicago 1981 (Belmont Cal., Wadsworth), pp. 270–286 (1983)Capon M. (1983). Primarité de L p (X). Trans. Am. Math. Soc. 276: 431–487Coifman R.R. and Fefferman C. (1974). Weighted norm inequalities for maximal functions and singular integrals. Stud. Math. 51: 241–250Defant M. (1989). On the vector-valued Hilbert transform. Math. Nachr. 141: 251–265Diestel, J., Uhl, J.J.:Vector measures, Mathematical Surveys, no. 15, Am. Math. Soc., Providence (1977)Dinculeanu N. (1967). Vector Measures, International Series of Monographs in Pure and Applied Mathematics, no. 95. Pergamon, New YorkFefferman C. (1971). The multiplier problem for the ball. Ann. Math. 94: 330–336Frazier M., Jawerth B., Weiss G. (1991) Littlewood–Paley theory and the study of function spaces. In: CBMS Regional Conference on Series Mathematics, no. 79. AMS, ProvidenceGarcía-Cuerva, J., Rubio de Francia, J.L.: Weighted norm inequalities and related topics, North Holland Mathematical Studies, vol. 116, Amsterdam (1985)Grothendieck, A.: Produits tensoriels topologiques et espaces nucléaires. Mem. Am. Math. Soc., no. 16, AMS, Providence (1955)Grudzinski O. (1979). Temperierte Beurling-distributionen. Math. Nachr. 91: 297–320Holub J.R. (1971). Hilbertian operators and reflexive tensor products. Pacific J. Math. 36: 185–194Hörmander L. (1960). Estimates for translation invariant operators on L p spaces. Acta Math. 104: 93–139Hörmander L. (1983). The Analysis of Linear Partial Differential Operators II, Grundlehren, no. 257. Springer, BerlinKomatsu H. (1973). Ultradistributions I. Structure theorems and a characterization. J. Fac. Sci. Univ. Tokyo. Sect. IA Math. 20: 25–105Komatsu H. (1982). Ultradistributions III. Vector-valued ultradistributions and the theory of kernels. J. Fac. Sci. Univ. Tokyo. Sect. IA Math. 29: 653–718Lindenstrauss J. and Pełczyński A. (1968) Absolutely summing operators in $${\mathcal{L}}_p$$ spaces and their applications. Stud. Math. 29: 275–326Lindenstrauss J. and Tzafriri L. (1977). Classical Banach spaces I. Springer, HeidelbergLöfstrom J. (1982) Interpolation of weighted spaces of differentiable functions on $${\mathbb{R}}^d$$ . Ann. Math. Pura Appl. 132: 189–214Mitiagin B.S. (1972). On idempotent multipliers in symmetric functional spaces. Funkcional Anal. i Priložen 6: 81–82Motos, J., Planells, M.J., Talavera, C.F.: On some iterated weighted spaces. J. Math. Anal. Appl. 338, 162–174 (2008) doi: 10.1016/j.jmaa.2007.05.009Muckenhoupt B. (1972). Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165: 207–226Nikol’skij S.M. (1975). Approximation of Functions of Several Variables and Imbedding Theorems. Springer, BerlinOdell E. (1976). On complemented subspaces of (∑ l 2) l p . Israel J. Math. 23: 353–367Rosenthal, H.P.: Projections onto translation-invariant subspaces of L p (G). Memoirs Am. Math. Soc., no. 63, AMS, Providence (1966)Rubio de Francia J.L. (1986). Martingale and integral transforms of Banach space valued functions. In: Bastero, J. and San Miguel, M. (eds) Probability and Banach Spaces, Lecture Notes in Mathematics, no. 1221, pp 195–222. Springer, BerlinRubio de Francia J.L., Ruiz F.J. and Torrea J.L. (1986). Calderón–Zygmund theory for operator-valued kernels. Adv. Math. 62: 7–48Schmeisser, H.J.: Vector-valued Sobolev and Besov spaces, Teubner-Text. Math., no. 96, Teubner, Leipzig, pp. 4–44 (1987)Schmeisser H.J. and Triebel H. (1987). Topics in Fourier analysis and Function Spaces. Chichester, WileySchwartz L. (1954). Espaces de fonctions différentiables à valeurs vectorielles. J. Anal. Math. 4: 88–148Schwartz L. (1957). Théorie des distributions à valeurs vectorielles. Ann. Inst. Fourier 7: 1–141Trèves F. (1967). Topological Vector Spaces, Distributions and Kernels. Academic Press, New YorkTriebel H. (1973). Über die Existenz von Schauderbasen in Sobolev–Besov–Raümen. Isomorphiebeziehungen Stud. Math. 46: 83–100Triebel, H.: Fourier analysis and function spaces, Teubner Texte Math., no. 7, Teubner, Leipzig (1977)Triebel H. (1978). Interpolation Theory, Function Spaces, Differential Operators. North-Holland, AmsterdamTriebel H. (1983). Theory of Function Spaces. Birkhäuser, BaselTriebel H. (1997). Fractals and Spectra. Birkhäuser, BaselVogt, D.: Sequence space representations of spaces of test functions and distributions. In: Zapata, G.I. (ed.) Functional analysis, holomorphy and approximation theory, Lecture Notes in Pure and Applied Mathematics, no. 83, pp. 405–443 (1983)Wojtaczczyk P. (1991). Banach Spaces for Analysts. Cambridge University Press, Cambridg

The approximation property of some vector valued Sobolev-Slobodeckij spaces

In this paper we consider the Sobolev-Slobodeckij spaces Wm,p(ℜn,E) where E is a strict (LF)-space, m∈(0,∞)\ℕ and p∈[1,∞). We prove that Wm,p(ℜn,E) has the approximation property provided E has it, furthermore if E is a Banach space with the strict approximation property then Wm,p(ℜn,E) has this property

On sequence space representations of Hörmander-Beurling spaces

[EN] It is shown that B-p'.1/(k) over bar(loc) (Omega) is isomorphic to (B-p,k(c) (Omega))(b)' (Omega open set in R-n, 1 <= p <= infinity, k Beurling-Bjorck weight) extending a Hormander's result (the proof we give is valid in the vector-valued case, too). As a consequence, and using Vogt's representation theorems and weighted L-p-spaces of entire analytic functions, a number of results on sequence space representations of Hormander-Beurling are given. (C) 2008 Elsevier Inc. All rights reserved.The author is partially supported by DGES, Spain, Project MTM2005-08350-C03-03.Motos Izquierdo, J.; Planells Gilabert, MJ. (2008). On sequence space representations of Hörmander-Beurling spaces. Journal of Mathematical Analysis and Applications. 348(1):395-403. https://doi.org/10.1016/j.jmaa.2008.07.031S395403348

On some iterated weighted spaces

[EN] It is proved that Hormander Bp,kloc (Omega 1 x Omega 2) and B-p,k1(loc) (Omega 1, B-p,k2(loc) (Omega(2))) spaces (Omega(1) subset of R-n, Omega(2) subset of R-m open sets, 1 <= p < infinity, k(i) Beurling-Bjorck weights, k = k(1) circle times k(2)) are isomorphic whereas the iterated spaces B-p,k1(loc) (Omega 1, B-p,k2(loc) (Omega(2))) and B-p,k2(loc) (Omega 1, B-p,k1(loc) (Omega(1))) are not if 1 < p not equal q < infinity. A similar result for weighted L-p-spaces of entire analytic functions is also obtained. Finally a result on iterated Besov spaces is given: B-2,q(s) (R-n, B-2,q(s) (R-m)) and B-2,q(s)(Rn+m) are not isomorphic when 1 < q not equal 2 < infinity. (c) 2007 Elsevier Inc. All rights reserved.The author is partially supported by DGES, Spain, Project MTM2005-08350.Motos Izquierdo, J.; Planells Gilabert, MJ.; Talavera Usano, CF. (2008). On some iterated weighted spaces. Journal of Mathematical Analysis and Applications. 338(1):162-174. https://doi.org/10.1016/j.jmaa.2007.05.009S162174338

Reducing the environmental impact of surgery on a global scale: systematic review and co-prioritization with healthcare workers in 132 countries

Abstract Background Healthcare cannot achieve net-zero carbon without addressing operating theatres. The aim of this study was to prioritize feasible interventions to reduce the environmental impact of operating theatres. Methods This study adopted a four-phase Delphi consensus co-prioritization methodology. In phase 1, a systematic review of published interventions and global consultation of perioperative healthcare professionals were used to longlist interventions. In phase 2, iterative thematic analysis consolidated comparable interventions into a shortlist. In phase 3, the shortlist was co-prioritized based on patient and clinician views on acceptability, feasibility, and safety. In phase 4, ranked lists of interventions were presented by their relevance to high-income countries and low–middle-income countries. Results In phase 1, 43 interventions were identified, which had low uptake in practice according to 3042 professionals globally. In phase 2, a shortlist of 15 intervention domains was generated. In phase 3, interventions were deemed acceptable for more than 90 per cent of patients except for reducing general anaesthesia (84 per cent) and re-sterilization of ‘single-use’ consumables (86 per cent). In phase 4, the top three shortlisted interventions for high-income countries were: introducing recycling; reducing use of anaesthetic gases; and appropriate clinical waste processing. In phase 4, the top three shortlisted interventions for low–middle-income countries were: introducing reusable surgical devices; reducing use of consumables; and reducing the use of general anaesthesia. Conclusion This is a step toward environmentally sustainable operating environments with actionable interventions applicable to both high– and low–middle–income countries

Sobre MM-tonelación y dual local completitud en espacios de funciones contínuas con valores vectoriales provistos de la topología de la convergencia puntual

In this article we give conditions on $C_s(X)$ and $E[\mathfrak{j}]$ for $C(X,E)[\mathfrak{j}_s]$ to be $m$-barrelled (dual locally complete) and we obtain the $m$-barrelled topology associated to $\mathfrak{j}_s$ in terms of the $m$-barrelled topology associated to $\mathfrak{j}$ when $C_s(X)$ is a barrelled space

Una nota sobre espacios MM-tonelados y espacios dual localmente completos

In this paper we give some results on topological direct sums of locally convex spaces, we deduce some consequences from the fact that the space \omega\thinspace\otimes\thinspace_\pi\thinspace\phi is not dual locally complete, and we prove new results in codimension of subspaces of certain locally convex spaces

Una nota sobre los espacios de Sobolev anisotropos vectoriales LΓp(E)∗L^p_\Gamma(E)\ast

Let $E$ be a Fréchet space and let $\Gamma$ be a finite non-empty subset of $\mathbb{N}^n$ such that if $(\alpha_j)\in\Gamma$ then $(\beta_j)\in\Gamma$ whenever $0\leq\beta_j\leq\alpha_j$ for $j=1,\cdots,n$. In this note we prove that the vector-valued anisotropic Sobolev spaces $L^p_\Gamma(E):=\{f\in L^p(E):D^\alpha f\in L^p(E)$ for $\alpha\in\Gamma$\},1\leq p < \infty, have the approximation property if $E$ has this property

Sobre los espacios de Hörmander vectoriales Bp,wB_{p,w}(E)

We extend some of the basic results on Hörmander spaces $B_{p,w}$ to the vector-valued case. Some new results on the $B_{p,w}$ are also given. Finally, we study locally convex properties of the vector-valued Hörmander local spaces