Sobolev extension by linear operators

Abstract

Let L m,p (R n ) be the Sobolev space of functions with m th derivatives lying in L p (R n ). Assume that n < p < ∞. For E ⊂ R n , let L m,p (E) denote the space of restrictions to E of functions in L m,p (R n ). We show that there exists a bounded linear map T : L m,p (E) → L m,p (R n ) such that, for any f ∈ L m,p (E), we have Tf = f on E. We also give

Explicit traces of functions from Sobolev spaces and quasi-optimal linear interpolators

“…Shvartsman has obtained in [15,16] results for the non-homogeneous Sobolev space W 1 p , both in R n and in metric spaces. In a recent paper [8], Fefferman, Israel and Luli extend Israel's result to L r p (R n ). They show that a linear extension operator can be constructed such that it has assisted bounded depth.…”

On planar SobolevLpm-extension domains

“…Finally, we notice that the existence of a bounded linear extension operator from the trace space W m p (R n )| E into W m p (R n ) whenever E ⊂ R n is an arbitrary closed set and p > n has been proven in papers [33] (m = 1, p ∈ (n, ∞)), [20], [36] (n = 2, m = 2, p > 2), and [11] (arbitrary m, n, p > n). 7.2.…”

The structure of Sobolev extension operators

“…There exists Z ≥ 1 depending only on p and ǫ, such that the following holds. Assume (4). Then for any G ∈ L 2,p (R 2 ) with…”

“…For X = C m (R n ) or C m,s (R n ) and E ⊆ R n arbitrary, there exists an extension operator whose norm depends only on m, n. Similarly, for X = L m,p (R n ) and E arbitrary, there exists an extension operator whose norm depends only on m, n, p. See [1,2,4].…”