The trace of jet space $J^k\Lambda ^\omega$ to an arbitrary closed subset of $\mathbb {R}^n$

Abstract:

The classical Whitney extension theorem describes the trace $J^k|_X$ of the space of $k$-jets generated by functions from $C^k(\mathbb R^n)$ to an arbitrary closed subset $X\subset \mathbb R^n$. It establishes existence of a bounded linear extension operator as well. In this paper we investigate a similar problem for the space $C^k\Lambda$ of functions whose higher derivatives satisfy the Zygmund condition with majorant $\omega$. The main result states that the vector function $\vec f=(f_\alpha \colon X\to \mathbb R)_{|\alpha |\le k}$ belongs to the corresponding trace space if the trace $\vec f|_Y$ to every subset $Y\subset X$ of cardinality $3\cdot 2^\ell$, where $\ell =(\begin {smallmatrix}n+k-1 k+1\end {smallmatrix})$, can be extended to a function $f_Y\in C^k\Lambda$ and $\sup _Y|f_Y|_{C^k\Lambda ^\omega }<\infty$. The number $3\cdot 2^l$ generally speaking cannot be reduced. The Whitney theorem can be reformulated in this way as well, but with a two-pointed subset $Y\subset X$. The approach is based on the theory of local polynomial approximations and a result on Lipschitz selections of multivalued mappings.