Differentiability of L-p of a vector measure and applications to the Bishop-Phelps-Bollobas property

[EN] We study the properties of Gâteaux, Fréchet, uniformly Fréchet and uniformly Gâteaux smoothness of the space Lp(m) of scalar p-integrable functions with respect to a positive vector measure m with values in a Banach lattice. Applications in the setting of the Bishop-Phelps-Bollobás property (both for operators and bilinear forms) are also given.Research supported by Ministerio de Economia y Competitividad and FEDER under projects MTM2012-36740-c02-02 (L. Agud and E.A. Sanchez-Perez), MTM201453009-P (J.M. Calabuig) and MTM2014-54182-P (S. Lajara). S. Lajara was also supported by project 19275/PI/14 funded by Fundacion Seneca-Agencia de Ciencia y Tecnologia de la Region de Murcia within the framework of PCTIRM 2011-2014.Agud Albesa, L.; Calabuig, JM.; Lajara, S.; Sánchez Pérez, EA. (2017). Differentiability of L-p of a vector measure and applications to the Bishop-Phelps-Bollobas property. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 111(3):735-751. https://doi.org/10.1007/s13398-016-0327-xS7357511113Acosta, M.D., Aron, R.M., García, D., Maestre, M.: The Bishop–Phelps–Bollobás theorem for operators. J. Funct. Anal. 254(11), 2780–2799 (2008)Acosta, M.D., Becerra-Guerrero, J., Choi, Y.S., García, D., Kim, S.K., Lee, H.J., Maestre, M.: The Bishop–Phelps–Bollobás theorem for bilinear forms and polinomials. J. Math. Soc. Jpn 66(3), 957–979 (2014)Acosta, M.D., Becerra-Guerrero, J., García, D., Maestre, M.: The Bishop–Phelps–Bollobás theorem for bilinear forms. Trans. Am. Math. Soc. 11, 5911–5932 (2013)Agud, L., Calabuig, J.M., Sánchez Pérez, E.A.: On the smoothness of $$L^p$$ L p of a positive vector measure. Monatsh. Math. 178(3), 329–343 (2015)Aron, R.M., Cascales, B., Kozhushkina, O.: The Bishop–Phelps–Bollobás theorem and Asplund operators. Proc. Am. Math. Soc. 139, 3553–3560 (2011)Bishop, E., Phelps, R.R.: A proof that every Banach space is subreflexive. Bull. Am. Math. Soc. 67, 97–98 (1961)Bollobás, B.: An extension to the theorem of Bishop and Phelps. Bull. Lond. Math. Soc. 2, 181–182 (1970)Cascales, B., Guirao, A.J., Kadets, V.: A Bishop–Phelps–Bollobás theorem type theorem for uniform algebras. Adv. Math. 240, 370–382 (2013)Choi, Y.S., Song, H.G.: The Bishop–Phelps–Bollobás theorem fails for bilinear forms on $$\ell _1\times \ell _1$$ ℓ 1 × ℓ 1 . J. Math. Anal. Appl. 360, 752–753 (2009)Deville, R., Godefroy, G., Zizler, V.: Smoothness and renormings in Banach spaces. Pitman Monographs and Surveys in Pure and Appl. Math., vol. 64, Longman, Harlow (1993)Diestel, J., Uhl, J.J.: Vector Measures. Math. Surveys, vol. 15, AMS, Providence, RI (1977)Fabian, M., Godefroy, G., Montesinos, V., Zizler, V.: Inner characterizations of weakly compactly generated Banach spaces and their relatives. J. Math. Anal. Appl. 297, 419–455 (2004)Fabian, M., Godefroy, G., Zizler, V.: The structure of uniformly Gâteaux smooth Banach spaces. Israel J. Math. 124, 243–252 (2001)Fabian, M., Habala, P., Hájek, P., Montesinos, V., Zizler, V.: Banach Space Theory: The Basis for Linear and Nonlinear Analysis. CMS Books in Mathematics, Springer, New York (2011)Fabian, M., Lajara, S.: Smooth renormings of the Lebesgue–Bochner function space $$L^1(\mu, X)$$ L 1 ( μ , X ) . Stud. Math. 209(3), 247–265 (2012)Ferrando, I., Rodríguez, J.: The weak topology on $$L^p$$ L p of a vector measure. Top. Appl. 155(13), 1439–1444 (2008)Hájek, P., Johanis, M.: Smooth analysis in Banach spaces. De Gruyter Series in Nonlinear Analysis and Applications, De Gruyter (2014)Kim, S.K.: The Bishop–Phelps–Bollobás theorem for operators from $$c_0$$ c 0 to uniformly convex spaces. Israel J. Math. 197, 425–435 (2013)Kim, S.K., Lee, H.J.: The Bishop–Phelps–Bollobás theorem for operators from $$C(K)$$ C ( K ) to uniformly convex spaces. J. Math. Anal. Appl. 421(1), 51–58 (2015)Hudzik, H., Kamińska, A., Mastylo, M.: Monotonocity and rotundity properties in Banach lattices. Rock. Mount J. Math. 30(3), 933–950 (2000)Kutzarova, D., Troyanski, S.L.: On equivalent norms which are uniformly convex or uniformly differentiable in every direction in symmetric function spaces. Serdica 11, 121–134 (1985)Okada, S., Ricker, W.J., Sánchez-Pérez, E.A.: Optimal Domain and Integral Extension of Operators Acting in Function Spaces. Operator Theory: Advances and Applications, vol. 180. Birkhäuser Verlag, Basel (2008

On the importance of long-term functional assessment after stroke to improve translation from bench to bedside

Despite extensive research efforts in the field of cerebral ischemia, numerous disappointments came from the translational step. Even if experimental studies showed a large number of promising drugs, most of them failed to be efficient in clinical trials. Based on these reports, factors that play a significant role in causing outcome differences between animal experiments and clinical trials have been identified; and latest works in the field have tried to discard them in order to improve the scope of the results. Nevertheless, efforts must be maintained, especially for long-term functional evaluations. As observed in clinical practice, animals display a large degree of spontaneous recovery after stroke. The neurological impairment, assessed by basic items, typically disappears during the firsts week following stroke in rodents. On the contrary, more demanding sensorimotor and cognitive tasks underline other deficits, which are usually long-lasting. Unfortunately, studies addressing such behavioral impairments are less abundant. Because the characterization of long-term functional recovery is critical for evaluating the efficacy of potential therapeutic agents in experimental strokes, behavioral tests that proved sensitive enough to detect long-term deficits are reported here. And since the ultimate goal of any stroke therapy is the restoration of normal function, an objective appraisal of the behavioral deficits should be done