A Novel $J_M$-Based Method for Interpolation in Metric Spaces
Roblêdo Mak's Miranda Sette
TL;DR
The paper introduces the $J_M^{\prime}$-interpolated metric space, a sequence-based interpolation construction for metric spaces that complements existing $K_M$-type methods. By leveraging bi-infinite linking sequences and a new metric $\delta_{\theta,q}$, it builds a relative completion $(\overrightarrow{X}_{\theta,q}^{J_M})^{\prime}$ inside the ambient space $(X_0\cup X_1, K_M(1))$ and establishes tight relations with the existing $J_M$ and $K_M$ metrics. Key contributions include the definition of $p_{\theta,q}$ and $\delta_{\theta,q}$, the resulting Lipschitz-interpolation framework for operators, and the demonstration of special-case collapses to the original metric; the work broadens the interpolation toolbox for nonlinear analysis and Wasserstein-type spaces. The approach preserves Lipschitz continuity across interpolated spaces and opens avenues for applications in nonlinear analysis and metric-geometry-inspired transport problems. Overall, the construction enriches metric-space interpolation by providing a robust, sequence-based alternative with well-controlled metric relations.
Abstract
We present a new interpolation method for metric spaces, termed the \emph{$J_M^{\prime}$-interpolated metric space}. Building on previous work where two interpolation methods were developed in analogy with Peetre's classical $K$ and $J$ methods, this approach introduces a distinct construction based on approximating sequences in the intersection space $X_0 \cap X_1$ equipped with the metric $J_M(1)$. The method defines a relative completion within the union space $X_0 \cup X_1$, yielding a well-defined interpolated metric space with controllable relationships between the $J_M$ and $K_M$ metrics. While we do not compare this method to the previously developed approaches nor explore its correspondence with classical interpolation functors in normed spaces, the construction enriches the set of tools available for metric-space interpolation theory. Furthermore, as expected, these methods still preserve the Lipschitz property of an operator between two interpolated spaces. Potential applications include general Nonlinear Analysis and Wasserstein-type spaces where the sequence-based nature of the method may provide practical advantages.