The Whitney extension problem for Zygmund spaces and Lipschitz selections in hyperbolic jet-spaces

TL;DR

This paper investigates a variant of the Whitney extension problem for Zygmund spaces, reformulating it as a Lipschitz selection problem in hyperbolic jet-spaces, providing a new geometric perspective on function extension issues.

Contribution

It introduces a novel geometric framework linking Zygmund spaces to Lipschitz mappings in hyperbolic metric spaces, enabling reformulation of the Whitney extension problem as a Lipschitz selection problem.

Findings

Identifies $C^k extLambda^m_ extomega(R^n)$ with Lipschitz maps into polynomial field spaces.

Reformulates the Whitney extension problem as a Lipschitz selection problem.

Provides a geometric approach to classical extension problems.

Abstract

We study a variant of the Whitney extension problem for the space $C^k\Lambda^m_{\omega}(R^n)$ of functions whose partial derivatives of order $k$ satisfy the generalized Zygmund condition. We identify $C^k\Lambda^m_{\omega}(R^n)$ with a space of Lipschitz mappings from a metric space $(R^{n+1}_+,\rho_\omega)$ supplied with a hyperbolic metric $\rho_\omega$ into a metric space $({\cal P}_{k+m-1}\times R^{n+1}_+, d_\omega)$ of polynomial fields on $R^{n+1}_+$ equipped with a hyperbolic-type metric $d_\omega$. This identification allows us to reformulate the Whitney problem for $C^k\Lambda^m_{\omega}(R^n)$ as a Lipschitz selection problem for set-valued mappings from $(R^{n+1}_+,\rho_\omega)$ into a certain family of subsets of ${\cal P}_{k+m-1}\times R^{n+1}_+$.