On the Banach lattice structure of L-w(1) of a vector measure on a delta-ring

We study some Banach lattice properties of the space L-w(1)(v) of weakly integrable functions with respect to a vector measure v defined on a delta-ring. Namely, we analyze order continuity, order density and Fatou type properties. We will see that the behavior of L-w(1)(v) differs from the case in which is defined on a sigma-algebra whenever does not satisfy certain local sigma-finiteness property.J. M. Calabuig and M. A. Juan were supported by the Ministerio de Economia y Competitividad (project MTM2008-04594). O. Delgado was supported by the Ministerio de Economia y Competitividad (project MTM2009-12740-C03-02). E. A. Sanchez Perez was supported by the Ministerio de Economia y Competitividad (project MTM2009-14483-C02-02).Calabuig Rodriguez, JM.; Delgado Garrido, O.; Juan Blanco, MA.; Sánchez Pérez, EA. (2014). On the Banach lattice structure of L-w(1) of a vector measure on a delta-ring. Collectanea Mathematica. 65(1):67-85. doi:10.1007/s13348-013-0081-8S6785651Brooks, J.K., Dinculeanu, N.: Strong additivity, absolute continuity and compactness in spaces of measures. J. Math. Anal. Appl. 45, 156–175 (1974)Calabuig, J.M., Delgado, O., Sánchez Pérez, E.A.: Factorizing operators on Banach function spaces through spaces of multiplication operators. J. Math. Anal. Appl. 364, 88–103 (2010)Calabuig, J.M., Juan, M.A., Sánchez Pérez, E.A.: Spaces of $$p$$ -integrable functions with respect to a vector measure defined on a $$\delta$$ -ring. Oper. Matrices 6, 241–262 (2012)Curbera, G.P.: El espacio de funciones integrables respecto de una medida vectorial. Ph. D. thesis, University of Sevilla, Sevilla (1992)Curbera, G.P.: Operators into $$L^1$$ of a vector measure and applications to Banach lattices. Math. Ann. 293, 317–330 (1992)Curbera, G.P., Ricker, W.J.: Banach lattices with the Fatou property and optimal domains of kernel operators. Indag. Math. (N.S.) 17, 187–204 (2006)G. P. Curbera and W. J. Ricker, Vector measures, integration and applications. In: Positivity (in Trends Math.), Birkhäuser, Basel, pp. 127–160 (2007)Curbera, G.P., Ricker, W.J.: The Fatou property in $$p$$ -convex Banach lattices. J. Math. Anal. Appl. 328, 287–294 (2007)Delgado, O.: $$L^1$$ -spaces of vector measures defined on $$\delta$$ -rings. Arch. Math. 84, 432–443 (2005)Delgado, O.: Optimal domains for kernel operators on $$[0,\infty )\times [0,\infty )$$ . Studia Math. 174, 131–145 (2006)Delgado, O., Soria, J.: Optimal domain for the Hardy operator. J. Funct. Anal. 244, 119–133 (2007)Delgado, O., Juan, M.A.: Representation of Banach lattices as $$L_w^1$$ spaces of a vector measure defined on a $$\delta$$ -ring. Bull. Belg. Math. Soc. Simon Stevin 19(2), 239–256 (2012)Diestel, J., Uhl, J.J.: Vector measures (Am. Math. Soc. surveys 15). American Mathematical Society, Providence (1997)Dinculeanu, N.: Vector measures, Hochschulbcher fr Mathematik, vol. 64. VEB Deutscher Verlag der Wissenschaften, Berlin (1966)Fernández, A., Mayoral, F., Naranjo, F., Sáez, C., Sánchez Pérez, E.A.: Spaces of $$p$$ -integrable functions with respect to a vector measure. Positivity 10, 1–16 (2006)Fremlin, D.H.: Measure theory, broad foundations, vol. 2. Torres Fremlin, Colchester (2001)Jiménez Fernández, E., Juan, M.A., Sánchez Pérez, E.A.: A Komlós theorem for abstract Banach lattices of measurable functions. J. Math. Anal. Appl. 383, 130–136 (2011)Lewis, D.R.: On integrability and summability in vector spaces. Ill. J. Math. 16, 294–307 (1972)Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces II. Springer, Berlin (1979)Luxemburg, W.A.J., Zaanen, A.C.: Riesz spaces I. North-Holland, Amsterdam (1971)Masani, P.R., Niemi, H.: The integration theory of Banach space valued measures and the Tonelli-Fubini theorems. I. Scalar-valued measures on $$\delta$$ -rings. Adv. Math. 73, 204–241 (1989)Masani, P.R., Niemi, H.: The integration theory of Banach space valued measures and the Tonelli-Fubini theorems. II. Pettis integration. Adv. Math. 75, 121–167 (1989)Thomas, E.G.F.: Vector integration (unpublished) (2013)Turpin, Ph.: Intégration par rapport à une mesure à valeurs dans un espace vectoriel topologique non supposé localement convexe, Intègration vectorielle et multivoque, (Colloq., University Caen, Caen, 1975), experiment no. 8, Dèp. Math., UER Sci., University Caen, Caen (1975)Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal domain and integral extension of operators acting in function spaces (Oper. Theory Adv. Appl.), vol. 180. Birkhäuser, Basel (2008)Zaanen, A.C.: Riesz spaces II. North-Holland, Amsterdam (1983

Maurey-Rosenthal domination for abstract Banach lattices

We extend the Maurey-Rosenthal theorem on integral domination and factorization of p-concave operators from a p-convex Banach function space through Lp-spaces for the case of operators on abstract p-convex Banach lattices satisfying some essential lattice requirements - mainly order density of its order continuous part - that are shown to be necessary. We prove that these geometric properties can be characterized by means of an integral inequality giving a domination of the pointwise evaluation of the operator for a suitable weight also in the case of abstract Banach lattices. We obtain in this way what in a sense can be considered the most general factorization theorem of operators through Lp-spaces. In order to do this, we prove a new representation theorem for abstract p-convex Banach lattices with the Fatou property as spaces of p-integrable functions with respect to a vector measure.The authors are supported by grants MTM2011-23164 and MTM2012-36740-C02-02 of the Ministerio de Economia y Competitividad (Spain).Juan Blanco, MA.; Sánchez Pérez, EA. (2013). Maurey-Rosenthal domination for abstract Banach lattices. Journal of Inequalities and Applications. (213). https://doi.org/10.1186/1029-242X-2013-213S213Defant A: Variants of the Maurey-Rosenthal theorem for quasi Köthe function spaces. Positivity 2001, 5: 153–175. 10.1023/A:1011466509838Defant A, Sánchez Pérez EA: Maurey-Rosenthal factorization of positive operators and convexity. J. Math. Anal. Appl. 2004, 297: 771–790. 10.1016/j.jmaa.2004.04.047Defant A, Sánchez Pérez EA: Domination of operators on function spaces. Math. Proc. Camb. Philos. Soc. 2009, 146: 57–66. 10.1017/S0305004108001734Fernández A, Mayoral F, Naranjo F, Sáez C, Sánchez-Pérez EA: Vector measure Maurey-Rosenthal type factorizations and l -sums of L 1 -spaces. J. Funct. Anal. 2005, 220: 460–485. 10.1016/j.jfa.2004.06.010Palazuelos C, Sánchez Pérez EA, Tradacete P: Maurey-Rosenthal factorization for p -summing operators and Dodds-Fremlin domination. J. Oper. Theory 2012, 68(1):205–222.Luxemburg WAJ, Zaanen AC: Riesz Spaces I. North-Holland, Amsterdam; 1971.Zaanen AC: Riesz Spaces II. North-Holland, Amsterdam; 1983.Lindenstrauss J, Tzafriri L: Classical Banach Spaces II. Springer, Berlin; 1979.Aliprantis CD, Burkinshaw O: Positive Operators. Academic Press, New York; 1985.Curbera GP, Ricker WJ: Vector measures, integration and applications. Trends Math. In Positivity. Birkhäuser, Basel; 2007:127–160.Okada S, Ricker WJ, Sánchez Pérez EA: Optimal domains and integral extensions of operators acting in function spaces. 180. In Operator Theory Advances and Applications. Birkhäuser, Basel; 2008.Delgado O: L 1 -spaces of vector measures defined on δ -rings. Arch. Math. 2005, 84: 432–443. 10.1007/s00013-005-1128-1Calabuig, JM, Delgado, O, Juan, MA, Sánchez Pérez, EA: On the Banach lattice structure of L w 1 of a vector measure on a δ-ring. Collect. Math. doi:10.1007/s13348–013–0081–8Calabuig JM, Delgado O, Sánchez Pérez EA: Factorizing operators on Banach function spaces through spaces of multiplication operators. J. Math. Anal. Appl. 2010, 364: 88–103. 10.1016/j.jmaa.2009.10.034Delgado O:Optimal domains for kernel operators on [ 0 , ∞ ) × [ 0 , ∞ ) .Stud. Math. 2006, 174: 131–145. 10.4064/sm174-2-2Delgado O, Soria J: Optimal domain for the Hardy operator. J. Funct. Anal. 2007, 244: 119–133. 10.1016/j.jfa.2006.12.011Jiménez Fernández E, Juan MA, Sánchez Pérez EA: A Komlós theorem for abstract Banach lattices of measurable functions. J. Math. Anal. Appl. 2011, 383: 130–136. 10.1016/j.jmaa.2011.05.010Curbera, GP: El espacio de funciones integrables respecto de una medida vectorial. PhD thesis, Univ. of Sevilla (1992)Sánchez Pérez EA: Compactness arguments for spaces of p -integrable functions with respect to a vector measure and factorization of operators through Lebesgue-Bochner spaces. Ill. J. Math. 2001, 45(3):907–923.Fernández A, Mayoral F, Naranjo F, Sáez C, Sánchez-Pérez EA: Spaces of p -integrable functions with respect to a vector measure. Positivity 2006, 10: 1–16. 10.1007/s11117-005-0016-zCalabuig JM, Juan MA, Sánchez Pérez EA: Spaces of p -integrable functions with respect to a vector measure defined on a δ -ring. Oper. Matrices 2012, 6: 241–262.Lewis DR: On integrability and summability in vector spaces. Ill. J. Math. 1972, 16: 294–307.Masani PR, Niemi H: The integration theory of Banach space valued measures and the Tonelli-Fubini theorems. I. Scalar-valued measures on δ -rings. Adv. Math. 1989, 73: 204–241. 10.1016/0001-8708(89)90069-8Masani PR, Niemi H: The integration theory of Banach space valued measures and the Tonelli-Fubini theorems. II. Pettis integration. Adv. Math. 1989, 75: 121–167. 10.1016/0001-8708(89)90035-2Brooks JK, Dinculeanu N: Strong additivity, absolute continuity and compactness in spaces of measures. J. Math. Anal. Appl. 1974, 45: 156–175. 10.1016/0022-247X(74)90130-9Curbera GP:Operators into L 1 of a vector measure and applications to Banach lattices.Math. Ann. 1992, 293: 317–330. 10.1007/BF01444717Delgado O, Juan MA: Representation of Banach lattices as L w 1 spaces of a vector measure defined on a δ -ring. Bull. Belg. Math. Soc. Simon Stevin 2012, 19: 239–256.Curbera GP, Ricker WJ: Banach lattices with the Fatou property and optimal domains of kernel operators. Indag. Math. 2006, 17: 187–204. 10.1016/S0019-3577(06)80015-7Curbera GP, Ricker WJ: The Fatou property in p -convex Banach lattices. J. Math. Anal. Appl. 2007, 328: 287–294. 10.1016/j.jmaa.2006.04.086Aliprantis CD, Border KC: Infinite Dimensional Analysis. Springer, Berlin; 1999.Delgado, O: Optimal extension for positive order continuous operators on Banach function spaces. Glasg. Math. J. (to appear

Elementary calculus

Elementary Calculu