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

Abstract

We study a variant of the Whitney extension problem for the space $C^k{\Lambda }_{\omega }^m(R^n)$ of functions whose partial derivatives of order $k$ satisfy the generalized Zygmund condition. We identify $C^k{\Lambda }_{\omega }^m(R^n)$ 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 $({\mathcal{P}}_{k+m-1}\times R_{+}^{n+1},d_{\omega })$ 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 $C^k{\Lambda }_{\omega }^m(R^n)$ as a Lipschitz selection problem for set-valued mappings from $(R_{+}^{n+1},{\rho }_{\omega })$ into a certain family of subsets of ${\mathcal{P}}_{k+m-1}\times R_{+}^{n+1}$.

43 pages