The Trace of Jet Space JkΛωto an Arbitrary Closed Subset of Rn

The classical Whitney extension theorem describes the trace Jk|Xof the space of k-jets generated by functions from Ck(Rn) to an arbitrary closed subset$X\subset \text{R}^{n}$. It establishes existence of a bounded linear extension operator as well. In this paper we investigate a similar problem for the space CkΛω(Rn) of functions whose higher derivatives satisfy the Zygmund condition with majorant ω . The main result states that the vector function$\overset \rightarrow \to{f}=(f_{\alpha}\colon X\rightarrow \text{R})_{|\alpha|\leq k}$belongs to the corresponding trace space if the trace$\overset \rightarrow \to{f}|_{\text{Y}}$to every subset$Y\subset X$of cardinality 3· 2ℓ, where ℓ =(k+1 n+k-1), can be extended to a function fY∈ CkΛω(Rn) and$\sup _{Y}|f_{Y}|_{C^{k}\Lambda ^{\omega}}