Lipschitz interpolating sequences

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 $f:X⟶\mathbb{R}$ such that $f(0)=0$. Given a set of points ${\left((x_i,y_i)\right)}_{i\in I}$ in $X^2$ with $x_i\ne y_i$ for all $i\in I$, we study the following interpolation problem: when for each bounded set ${\left(\hat{I}{\pm }_i\right)}_{i\in I}$ in $\mathbb{R}$ the algorithm $$\frac{f(x_i)-f(y_i)}{d(x_i,y_i)}=\hat{I}{\pm }_i(i\in I)$$ 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.

28 pages