New types of Lipschitz summing maps between metric spaces

Abstract

Building upon the results of M. C. Matos and extending previous work of J. D. Farmer, W. B. Johnson and J. A. Chavez-Dominguez 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.