Lipschitz interpolative nonlinear ideal procedure

TL;DR

This paper develops a comprehensive nonlinear ideal theory for Lipschitz operators, introducing new classes of Lipschitz interpolative nonlinear ideals and their properties, with applications to metric and Banach spaces.

Contribution

It extends linear ideal concepts to nonlinear Lipschitz operators, defining new nonlinear ideals and establishing their fundamental properties and characterizations.

Findings

Lipschitz interpolative nonlinear ideal is an injective Banach nonlinear ideal.

Defined Lipschitz ,p,, q,  operators and characterized them.

Presented counterexamples illustrating the properties of these nonlinear ideals.

Abstract

We treat the general theory of nonlinear ideals and extend as many notions as possible from the linear theory to the nonlinear theory. We define nonlinear ideals with special properties which associate new non-linear ideals to given ones and establish several properties and characterizations of them. Building upon the results of U. Matter we define a Lipschitz interpolative nonlinear ideal procedure between metric spaces and Banach spaces and establish this class of Lipschitz operators is an injective Banach nonlinear ideal and show several standard basic properties for such class. Extending the work of J. A. L\'{o}pez Molina and E. A. S\'{a}nchez P\'{e}rez we define a Lipschitz $\left(p,\theta, q, \nu\right)$-dominated operators for $1\leq p, q <\infty$; $0\leq \theta, \nu< 1$ and establish several characterizations. Afterwards we generalize a notion of Lipschitz interpolative nonlinear ideal procedure between arbitrary metric spaces and prove its a nonlinear ideal. Finally, we present certain basic counter examples of Lipschitz interpolative nonlinear ideal procedure between arbitrary metric spaces.