The Whitney problem of existence of a linear extension operator

Remez type inequalities and Morrey-Campanato spaces on Ahlfors regular sets

“…We will prove that the new family satisfies the hypotheses of Theorem 3.5 from [BSh2]. This will complete the proof of the proposition in this case.…”

“…Hence, the only restriction is now inequality (5.3) and we should show that if a (k, ω, X)-chain satisfies condition (5.4) under restriction (5.3), then (5.4) holds for any pair Q ⊂ Q ′ from K X . Note that the necessity of conditions (5.4) and (5.5) trivially follows from that in the aforementioned Theorem 3.5 from [BSh2]. So we should only prove their sufficiency.…”

“…The restrictions (5.3) and r Q , r Q ′ ≤ 4diam X may be not satisfied for this pair. In the forthcoming derivation we explain how these restrictions can be disregarded to apply the aforementioned Theorem 3.5 of [BSH2] and in this way to complete the proof of the proposition.…”

Interpolation and extrapolation of smooth functions by linear operators

“…One of their theorems [5] includes the case σ = 0, m = 2 of our results as a special case. I am grateful to Brudnyi and Shvartsman for raising with me the issue of linear dependence of F on f above, and also to E. Bierstone and P. Milman for valuable discussions.…”

Extension of $C^{m, \omega}$-Smooth Functions by Linear Operators

“…Assume that these data satisfy conditions (SL0,...,5). We must show that there exists a linear map ξ → F ξ from Ξ into C m,ω (R n ), satisfying (SL6,7) with a constant C determined by C, m, n. We replace (7) in Section 18 of [14] by…”