The trace of jet space 𝐽^{𝑘}Λ^{𝜔} to an arbitrary closed subset of ℝⁿ

Abstract

The classical Whitney extension theorem describes the trace J k | X J^k|_X of the space of k k -jets generated by functions from C k ( R n ) C^k(\mathbb R^n) to an arbitrary closed subset X ⊂ R n 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 Λ C^k\Lambda of functions whose higher derivatives satisfy the Zygmund condition with majorant ω \omega . The main result states that the vector function f → = ( f α : X → R ) | α | ≤ k \vec f=(f_\alpha \colon X\to \mathbb R)_{|\alpha |\le k} belongs to the corresponding trace space if the trace f → | Y \vec f|_Y to every subset Y ⊂ X Y\subset X of cardinality 3 ⋅ 2 ℓ 3\cdot 2^\ell , where ℓ = ( n + k − 1 k + 1 ) \ell =(\begin {smallmatrix}n+k-1 k+1\end {smallmatrix}) , can be extended to a function f Y ∈ C k Λ f_Y\in C^k\Lambda and sup Y | f Y | C k Λ ω > ∞ \sup _Y|f_Y|_{C^k\Lambda ^\omega }>\infty . The number 3 ⋅ 2 l 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 ⊂ X Y\subset X . The approach is based on the theory of local polynomial approximations and a result on Lipschitz selections of multivalued mappings.