Mityagin's extension problem. Progress report

Abstract

Abstract. Given a compact set K ⊂ R d , let E(K) denote the space of Whitney jets on K. The compact set K is said to have the extension property if there exists a continuous linear extension operator W :In 1961 B. S. Mityagin posed a problem to give a characterization of the extension property in geometric terms. We show that there is no such complete description in terms of densities of Hausdorff contents or related characteristics. Also the extension property cannot be characterized in terms of growth of Markov's factors for the set.

Bases in some spaces of Whitney functions

“…As in [5], we consider γ = (γ s ) ∞ s=1 with 0 < γ s ≤ 1/32 and ∞ s=1 γ s < ∞. Let r 0 = 1, let P 2 (x) = x(x − 1), and let r s = γ s r 2 s−1 , P 2 s+1 = P 2 s (P 2 s + r s ) where s ∈ N. Then E s := {x ∈ R : P 2 s+1 (x) ≤ 0} = 2 s j=1 I j,s , where the sth level basic intervals I j,s are disjoint.…”

“…As in [5], we use B k = 2 −k−1 • log 1 δ k . By Theorem 5.3 in [5], K(γ) has the extension property if and only if B n+s / n+s k=s B k → 0 as n → ∞ uniformly with respect to s. This condition can be written as…”

“…In Section 5, by means of extension of basis elements, we construct an extension operator W for the spaces E(K(γ)), provided K(γ) has EP. In the case when the space has a Faber basis, the operator W coincides with the operator presented in [5].…”

“…This article is supplementary to [5], where the extension problem is discussed for equilibrium Cantor sets K(γ) introduced in [4]. The set K(γ) is defined by means of a sequence of parameters γ = (γ s ) ∞ s=0 and can be considered as a generalization of the classical quadratic Julia set, but as opposed to Julia sets, it is more flexible with respect to its features.…”

“…Following [10] (see also [1] and [2]), we say that a compact set K ⊂ R d has the extension property (EP) if, for the space E(K) of Whitney jets on K, there exists a linear continuous extension operator W : E(K) → C ∞ (R d ). In [5], we present a characterization of EP for K(γ) and, by means of local Newton interpolations, construct an operator W , when it exists. This approach goes back to [8] (see also [9]), so we can say that W is a local version of the Paw lucki-Pleśniak operator.…”