. FA ] 2 7 Ja n 20 06 Local approximations and intrinsic characterizations of spaces of smooth functions on regular subsets

We give an intrinsic characterization of the restrictions of Sobolev W k p (R n), Triebel-Lizorkin F s pq (R n) and Besov B s pq (R n) spaces to regular subsets of R n via sharp maximal functions and local approximations. The purpose of this paper is to study the problem of extendability of functions defined on measurable subsets of R n to functions defined on the whole space and satisfying certain smoothness conditions. We will consider three kinds of spaces of smooth functions defined on R n. First we deal with classical Sobolev spaces, see e.g. Maz'ja [27]. We recall that, given an open set Ω ⊂ R n , k ∈ N and p ∈ [1, ∞], the Sobolev space W k p (Ω) consists of all functions f ∈ L 1, loc (Ω) whose distributional partial derivatives on Ω of all orders up to k belong to L p (Ω). W k p (Ω) is normed by f W k p (Ω) := {D α f Lp(Ω) : |α| ≤ k}. There is an extensive literature devoted to describing the restrictions of Sobolev functions to different classes of subsets of R n. and references therein for numerous results and techniques in this direction.) Let us recall some of these results. Calderón [14] showed that, if Ω is a Lipschitz domain in R n and 1 < p < ∞, then W k p (R n)| Ω = W k p (Ω). Stein [38] extended this result for p = 1, ∞ and Jones [24] showed that the same isomorphism also holds for every (ε, δ)−domain and 1 ≤ p ≤ ∞. Here as usual, for any Banach space (A, · A) of measurable functions defined on R n and a measurable set S ⊂ R n of positive Lebesgue measure, we let A| S denote the Math Subject Classification 46E35