A generalization of Lipschitz mappings
Anil Kumar Karn, Arindam Mandal
TL;DR
This work generalizes Lipschitz mappings by introducing extensively bounded mappings through the modulus of continuity at a designated point, formulating a metric that induces a norm when the codomain is normed. It constructs a Lipschitz-free–like framework via $F_e(M)$ and a norm-preserving dilation $\delta_M^e$, enabling a universal nonlinear linearization $\hat T:F_e(M)\to Y$ for any $T\in E(M,Y)$. A nonlinear operator-algebraic structure is developed, including a dilation-based multiplicative framework and a dual-embedding of $X^*$ into $X^e$, to study compositions and adjoints of extensively bounded maps. The paper then defines extensively bounded operator ideals, including finite rank and compact variants, and proves their equivalence with linearization to corresponding linear operator ideals, establishing Banach-space–level ideal properties for this nonlinear setting. Together, these results yield a broad, robust category of mappings extending Lipschitz analysis with potential applications to nonlinear operator theory and nonlinear functional analysis.
Abstract
Using the notion of modulus of continuity at a point of a mapping between metric spaces, we introduce the notion of extensively bounded mappings generalizing that of Lipschitz mappings. We also introduce a metric on it which becomes a norm if the codomain is a normed linear space. We study its basic properties. We also discuss a linearization of an extensively bounded mapping into a bounded linear mapping. As an application, we introduce the notion of extensively bounded operator ideals. We also discuss extensively bounded finite rank and extensively bounded compact mappings and their corresponding operator ideals.