Weighted Favard and Berwald Inequalities

AbstractWeighted versions of the Favard and Benwald inequalities are proved in the class of monotone and concave (convex) functions. Some necessary majorization estimates and a double-weight characterization for a Favard-type inequality are included

Asymptotic stability of the Cauchy and Jensen functional equations

The aim of this note is to investigate the asymptotic stability behaviour of the Cauchy and Jensen functional equations. Our main results show that if these equations hold for large arguments with small error, then they are also valid everywhere with a new error term which is a constant multiple of the original error term. As consequences, we also obtain results of hyperstability character for these two functional equations

Factorization theorems for homogeneous maps on banach function spaces and approximation of compact operators

The final publication is available at Springer via http://dx.doi.org/10.1007/s00009-014-0384-3[EN] In this paper, we characterize compact linear operators from Banach function spaces to Banach spaces by means of approximations with bounded homogeneous maps. To do so, we undertake a detailed study of such maps, proving a factorization theorem and paying special attention to the equivalent strong domination property involved. Some applications to compact maximal extensions of operators are also given.The authors thank the referee for his/her careful revision and suggestions. The first author gratefully acknowledges the support of the Ministerio de Economia y Competitividad (Spain), under Project #MTM2011-22417. The second author gratefully acknowledges the support of the Ministerio de Economia y Competitividad (Spain), under Project #MTM2012-36740-c02-02.Rueda, P.; Sánchez Pérez, EA. (2015). Factorization theorems for homogeneous maps on banach function spaces and approximation of compact operators. Mediterranean Journal of Mathematics. 12(1):89-115. https://doi.org/10.1007/s00009-014-0384-3S89115121Calabuig 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)Defant, A.: Variants of the Maurey–Rosenthal theorem for quasi Köthe function spaces. Positivity 5, 153–175 (2001)Delgado, O., Sánchez Pérez, E.A.: Strong factorizations between couples of operators on Banach spaces, J. Conv. Anal. 20(3), 599–616 (2013)Diestel, J., Uhl, J.J.: Vector measures, Math. Surv. vol. 15, Amer. Math. Soc., Providence (1977)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)Ferrando I., Rodríguez J.: The weak topology on L p of a vector measure. Topol. Appl. 155(13), 1439–1444 (2008)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces, II, Springer, Berlin (1996)Maligranda L., Persson L.E.: Generalized duality of some Banach function spaces. Indag. Math. 51, 323–338 (1989)Meyer-Nieberg, P.: Banach lattices, Springer, Berlin (1991)Okada, S.: Does a compact operator admit a maximal domain for its compact linear extension? In: Vector measures, integration and related topics. Operator theory: advances and applications, Vol. 201, pp. 313–322. Birkhäuser, Basel (2009)Okada S., Ricker W.J., Rodríguez-Piazza L.: Compactness of the integration operator associated with a vector measure. Studia Math. 150(2), 133–149 (2002)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, 180. Birkhäuser, Basel (2008)Sánchez Pérez, E.A.: Compactness arguments for spaces of p-integrable functions with respect to a vector measure and factorization of operators through Lebesgue–Bochner spaces, Illinois J. Math. 45(3), 907–923 (2001

Maharam-type kernel representation for operators with a trigonometric domination

[EN] Consider a linear and continuous operator T between Banach function spaces. We prove that under certain requirements an integral inequality for T is equivalent to a factorization of T through a specific kernel operator: in other words, the operator T has what we call a Maharam-type kernel representation. In the case that the inequality provides a domination involving trigonometric functions, a special factorization through the Fourier operator is given. We apply this result to study the problem that motivates the paper: the approximation of functions in L2[0, 1] by means of trigonometric series whose Fourier coefficients are given by weighted trigonometric integrals.This research has been supported by MTM2016-77054-C2-1-P (Ministerio de Economia, Industria y Competitividad, Spain).Sánchez Pérez, EA. (2017). Maharam-type kernel representation for operators with a trigonometric domination. Aequationes Mathematicae. 91(6):1073-1091. https://doi.org/10.1007/s00010-017-0507-6S10731091916Calabuig, J.M., Delgado, O., Sánchez Pérez, E.A.: Generalized perfect spaces. Indag. Math. 19(3), 359–378 (2008)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)Delgado, O., Sánchez Pérez, E.A.: Strong factorizations between couples of operators on Banach function spaces. J. Convex Anal. 20(3), 599–616 (2013)Dodds, P.G., Huijsmans, C.B., de Pagter, B.: Characterizations of conditional expectation type operators. Pacific J. Math. 141(1), 55–77 (1990)Flores, J., Hernández, F.L., Tradacete, P.: Domination problems for strictly singular operators and other related classes. Positivity 15(4), 595–616 (2011). 2011Fremlin, D.H.: Tensor products of Banach lattices. Math. Ann. 211, 87–106 (1974)Hu, G.: Weighted norm inequalities for bilinear Fourier multiplier operators. Math. Ineq. Appl. 18(4), 1409–1425 (2015)Halmos, P., Sunder, V.: Bounded Integral Operators on $$L^2$$ L 2 Spaces. Springer, Berlin (1978)Kantorovitch, L., Vulich, B.: Sur la représentation des opérations linéaires. Compositio Math. 5, 119–165 (1938)Kolwicz, P., Leśnik, K., Maligranda, L.: Pointwise multipliers of Calderón- Lozanovskii spaces. Math. Nachr. 286, 876–907 (2013)Kolwicz, P., Leśnik, K., Maligranda, L.: Pointwise products of some Banach function spaces and factorization. J. Funct. Anal. 266(2), 616–659 (2014)Kuo, W.-C., Labuschagne, C.C.A., Watson, B.A.: Conditional expectations on Riesz spaces. J. Math. Anal. Appl. 303, 509–521 (2005)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer, Berlin (1979)Maharam, D.: The representation of abstract integrals. Trans. Am. Math. Soc. 75, 154–184 (1953)Maharam, D.: On kernel representation of linear operators. Trans. Am. Math. Soc. 79, 229–255 (1955)Maligranda, L., Persson, L.E.: Generalized duality of some Banach function spaces. Indag. Math. 51, 323–338 (1989)Neugebauer, C.J.: Weighted norm inequalities for averaging operators of monotone functions. Publ. Mat. 35, 429–447 (1991)Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal Domain and Integral Extension of Operators Acting in Function Spaces. Operator Theory: Adv. Appl., vol. 180. Birkhäuser, Basel (2008)Rota, G.C.: On the representation of averaging operators. Rend. Sem. Mat. Univ. Padova. 30, 52–64 (1960)Sánchez Pérez, E.A.: Factorization theorems for multiplication operators on Banach function spaces. Integr. Equ. Oper. Theory 80(1), 117–135 (2014)Schep, A.R.: Factorization of positive multilinear maps. Ill. J. Math. 28(4), 579–591 (1984)Schep, A.R.: Products and factors of Banach function spaces. Positivity 14(2), 301–319 (2010

Sequences of independent Walsh functions in BMO

Under examination are the sequences of independent Walsh functions in the space of functions of bounded mean oscillation. We study geometric properties of the subspaces spanned by the sequences; in particular, some necessary and sufficient conditions are found for such a subspace to be complementedValiderad; 2013; 20130429 (andbra)</p