Lipschitz interpolating sequences
A. Jiménez-Vargas, Abraham Rueda Zoca
TL;DR
This work addresses when a Lipschitz interpolation problem in the real Lipschitz space $\mathrm{Lip}_0(X)$, defined by pairs $(x_i,y_i)$ in a pointed metric space, is solvable, i.e., when the associated operator $T:\mathrm{Lip}_0(X)\to\ell_\infty(I)$ is onto. Using the Lipschitz-free space $\mathcal{F}(X)$ and elementary molecules $m_{x,y}$, the authors connect surjectivity to the injectivity of a canonical map $S:\ell_1(I)\to\mathcal{F}(X)$ and to Beurling sets of Lipschitz functions, providing several equivalent characterizations involving liftings, projections, and dualities. They establish that Beurling sets exist (and imply linear interpolation) whenever the Lipschitz interpolation constant $M$ equals $1$, prove necessary separation and stability properties in terms of the Lipschitz-molecular metric, and extend the theory to compact spaces where subsequences and little Lipschitz spaces play a key role. The vector-valued extension via Banach space targets $E$ is developed through tensor products, yielding a comprehensive framework: $\mathrm{Lip}_0(X,E)$ interpolation is equivalent to Beurling-set existence and to a Beurling-type lifting, with a precise tensor-product description $S\otimes_\pi\mathrm{Id}_E$ governing the constants. Overall, the paper provides a unifying operator-theoretic approach to Lipschitz interpolation, with clear criteria, stability results, and vector-valued generalizations grounded in Lipschitz-free space theory.
Abstract
Let $X$ be a metric space with a base point $0$, and let $\mathrm{Lip}_0(X)$ be the Banach space of all Lipschitz functions $f:X\longrightarrow \mathbb R$ such that $f(0)=0$. Given a set of points $\left((x_i,y_i)\right)_{i\in I}$ in $X^2$ with $x_i\neq y_i$ for all $i\in I$, we study the following interpolation problem: when for each bounded set $\left(α_i\right)_{i\in I}$ in $\mathbb{R}$ the algorithm $$ \frac{f(x_i)-f(y_i)}{d(x_i,y_i)}=α_i\qquad (i\in I) $$ can be implemented by a function $f\in\mathrm{Lip}_0(X)$? Our approach involves the concept of a Beurling set of functions in $\mathrm{Lip}_0(X)$ for $\left((x_i,y_i)\right)_{i\in I}$ which has shown to be useful in the so-called transportation problem.