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New types of Lipschitz summing maps between metric spaces. (English) Zbl 1367.26013
Summary: Building upon the results of M. C. Matos and extending previous work of J. D. Farmer, W. B. Johnson and J. A. Chávez-Domínguez we define a Lipschitz mixed summable sequence as the pointwise product of a strongly summable sequence and a weakly Lipschitz summable one. Then we introduce classes of Lipschitz maps satisfying inequalities between Lipschitz mixed summable sequence and strongly summable sequences analogously to the linear case. These classes generalize the classes of Lipschitz summable maps considered earlier in the literature. We use standard techniques to establish several basic properties, showing that these classes of maps are ideals and some relationships between them. We establish various composition and inclusion theorems between different classes of Lipschitz summing maps and several characterizations. Furthermore, we prove that the classes of Lipschitz $$p$$-summing maps coincide and the nonlinear “Pietsch Domination Theorem” for the case $$0<p<1$$. We also identify cases where all Lipschitz maps are in the aforementioned classes of Lipschitz maps and discuss a sufficient condition for a Lipschitz composition formula as in the linear case.

##### MSC:
 26A16 Lipschitz (Hölder) classes 47H99 Nonlinear operators and their properties 47J99 Equations and inequalities involving nonlinear operators 47L20 Operator ideals 46T99 Nonlinear functional analysis
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##### References:
 [1] Bourgain, On Lipschitz embedding of finite metric spaces in Hilbert space, Israel J. Math 52 pp 46– (1985) · Zbl 0657.46013 [2] Chávez-Domínguez, Duality for Lipschitz p-summing operators, J. Funct. Anal 261 pp 387– (2011) · Zbl 1234.46009 [3] Chávez-Domínguez, Lipschitz (q;p)-mixing operators, Proc. Amer. Math. Soc 140 pp 3101– (2012) · Zbl 1281.46065 [4] Chen, Remarks on Lipschitz p-summing operators, Proc. Amer. Math. Soc 139 pp 2891– (2011) · Zbl 1227.46018 [5] Defant, North-Holland Mathematics Studies (1995) [6] Diestel, Absolutely Summing Operators, Cambridge Studies in Advanced Mathematics (1995) · Zbl 0855.47016 [7] Farmer, Lipschitz p-summing operators, Proc. Amer. Math. Soc 137 pp 2989– (2009) · Zbl 1183.46020 [8] Gohberg, Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space, Tranlations of Mathematical Monographs (1969) · Zbl 0181.13504 [9] Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques, Resenhas do Instituto de Matemática e Estatística da Universidade de São Paulo 2 pp 401– (1996) [10] A. Hinrichs M. A. S. Saleh Computations of Lipschitz p -summing norms and applications to composition formulas, in preparation [11] Johnson, Diamond graphs and super-reflexivity, J. Topol. Anal 1 pp 177– (2009) · Zbl 1183.46022 [12] Lévy, Sur une application de la dérivée d’ordre non entier au calcul des probabilités (1923) [13] Matos, Mappings between Banach spaces that send mixed summable sequences into absolutely summable sequences, J. Math. Anal. Appl 297 pp 833– (2004) · Zbl 1067.47029 [14] M. C. Matos http://www.ime.unicamp.br/conteudo/Absolutely summing mappings-nuclear mappings and convolution equations 2007 [15] Maurey, Démonstration d’une conjecture de Pietsch et applications (1972) · Zbl 0246.47038 [16] Maurey, Théorèmes de factorisation pour les opérateurs linéaires à valeurs dans les espaces Lp (1974) [17] Pietsch, Absolut p-summierende Abbildungen in normierten Räumen, Studia Math 28 (1966/1967) · Zbl 0156.37903 [18] Pietsch, Operator Ideals (1980) [19] Sawashima, Methods of duals in nonlinear analysis, Lecture Notes in Econom. and Math. Systems 419 pp 247– (1995) · Zbl 0942.47052 [20] Schatten, Norm Ideals of Completely Continuous Operators (1960)
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