Power-aggregation of pseudometrics and the McShane-Whitney extension theorem for Lipschitz p-convace maps

Abstract

[EN] Given a countable set of families {Dk:k¿N} of pseudometrics over the same set D, we study the power-aggregations of this class, that are defined as convex combinations of integral averages of powers of the elements of ¿kDk. We prove that a Lipschitz function f is dominated by such a power-aggregation if and only if a certain property of super-additivity involving the powers of the elements of ¿kDk is fulfilled by f. In particular, we show that a pseudo-metric is equivalent to a power-aggregation of other pseudometrics if this kind of domination holds. When the super-additivity property involves a p-power domination, we say that the elements of Dk are p-concave. As an application of our results, we prove under this requirement a new extension result of McShane-Whitney type for Lipschitz p-concave real valued maps.Both authors gratefully acknowledge the support of the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigaciones and FEDER under each grants MTM2015-64373-P (MINECO/FEDER, UE) and MTM2016-77054-C2-1-P.Rodríguez López, J.; Sánchez Pérez, EA. (2020). Power-aggregation of pseudometrics and the McShane-Whitney extension theorem for Lipschitz p-convace maps. Acta Applicandae Mathematicae. 170:611-629. https://doi.org/10.1007/s10440-020-00349-3S611629170Dahia, E., Achour, D., Rueda, P., Sánchez Pérez, E.A., Yahi, R.: Factorization of Lipschitz operators on Banach function spaces. Math. Inequal. Appl. 21(4), 1091–1104 (2018)Beliakov, G.: Optimization and aggregation functions. In: Lodwick, W.A., Kacprzyk, J. (eds.) Fuzzy Optimization: Recent Advances and Applications, pp. 77–108. Springer, Berlin (2010)Beliakov, G., Bustince Sola, H., Calvo Sánchez, T.: A Practical Guide to Averaging Functions, vol. 329. Springer, Heidelberg (2016)Botelho, G., Pellegrino, D., Rueda, P.: A unified Pietsch domination theorem. J. Math. Anal. Appl. 365(1), 269–276 (2010)Botelho, G., Pellegrino, D., Rueda, P.: On Pietsch measures for summing operators and dominated polynomials. Linear Multilinear Algebra 62(7), 860–874 (2014)Chávez-Domínguez, J.A.: Duality for Lipschitz p-summing operators. J. Funct. Anal. 261, 387–407 (2011)Chávez-Domínguez, J.A.: Lipschitz $p$-convex and $q$-concave maps (2014). arXiv:1406.6357Defant, A.: Variants of the Maurey-Rosenthal theorem for quasi Köthe function spaces. Positivity 5, 153–175 (2001)Defant, A., Floret, K.: Tensor Norms and Operator Ideals. Elsevier, Amsterdam (1992)Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge University Press, Cambridge (1995)Dugundji, J.: Topology. Allyn and Bacon, Needham Heights (1966)Farmer, J., Johnson, W.: Lipschitz $p$-summing operators. Proc. Am. Math. Soc. 137(9), 2989–2995 (2009)Juutinen, P.: Absolutely minimizing Lipschitz extensions on a metric space. Ann. Acad. Sci. Fenn., Math. 27, 57–67 (2002)Marler, T.R., Arora, J.S.: The weighted sum method for multi-objective optimization: new insights. Struct. Multidiscip. Optim. 41(6), 853–862 (2010)Mustata, C.: Extensions of semi-Lipschitz functions on quasi-metric spaces. Rev. Anal. Numér. Théor. Approx. 30(1), 61–67 (2001)Mustata, C.: On the extremal semi-Lipschitz functions. Rev. Anal. Numér. Théor. Approx. 31(1), 103–108 (2002)Pellegrino, D., Santos, J.: Absolutely summing multilinear operators: a panorama. Quaest. Math. 34(4), 447–478 (2011)Romaguera, S., Sanchis, M.: Semi-Lipschitz functions and best approximation in quasi-metric spaces. J. Approx. Theory 103, 292–301 (2000)Willard, S.: General Topology. Addison-Wesley, Reading (1970)Yahi, R., Achour, D., Rueda, P.: Absolutely summing Lipschitz conjugates. Mediterr. J. Math. 13(4), 1949–1961 (2016