# The Structure of Sobolev Extension Operators

@article{Fefferman2012TheSO,
title={The Structure of Sobolev Extension Operators},
author={Charles Fefferman and Arie Israel and Garving K Luli},
journal={arXiv: Classical Analysis and ODEs},
year={2012}
}
• Published 9 June 2012
• Mathematics
• arXiv: Classical Analysis and ODEs
Let $L^{m,p}(\R^n)$ denote the Sobolev space of functions whose $m$-th derivatives lie in $L^p(\R^n)$, and assume that $p>n$. For $E \subset \R^n$, denote by $L^{m,p}(E)$ the space of restrictions to $E$ of functions $F \in L^{m,p}(\R^n)$. It is known that there exist bounded linear maps $T : L^{m,p}(E) \rightarrow L^{m,p}(\R^n)$ such that $Tf = f$ on $E$ for any $f \in L^{m,p}(E)$. We show that $T$ cannot have a simple form called "bounded depth."
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## References

SHOWING 1-10 OF 12 REFERENCES
A Bounded Linear Extension Operator for $L^{2,p}(\R^2)$
For a finite $E \subset \R^2$, $f:E \rightarrow \R$, and $p>2$, we produce a continuous $F:\R^2 \rightarrow \R$ depending linearly on $f$, taking the same values as $f$ on $E$, and with
THE CHARACTERIZATION OF FUNCTIONS ARISING AS POTENTIALS
The symbol "^ denotes the Fourier transform. The integral Ia is a well-known Riesz potential, while the integral Ja is a modification of it, the so-called "Bessel potential." The local behavior of Ia
Sobolev extension by linear operators
• Mathematics
• 2012
Let L(R) be the Sobolev space of functions with mth derivatives lying in L(R). Assume that n < p < ∞. For E ⊂ R, let L(E) denote the space of restrictions to E of functions in L(R). We show that
The characterization of functions arising as potentials. II
1. Statement of result. We continue our study of the function spaces Z£, begun in [7]. We recall that f^Ll(En) when f=Ka*<l>, where <££!/ (£„) . Ka is the Bessel kernel, characterized by its Fourier
The Structure of Linear Extension Operators for $C^m$
For any subset E ⊂ Rn, let Cm(E) denote the Banach space of restrictions to E of functions F ∈ Cm(Rn). It is known that there exist bounded linear maps T : Cm(E) −→ Cm(Rn) such that Tf = f on E for
Extension of C-Smooth Functions by Linear Operators
Let Cm,ω(Rn) be the space of functions on Rn whose mth derivatives have modulus of continuity ω. For E ⊂ Rn, let Cm,ω(E) be the space of all restrictions to E of functions in Cm,ω(Rn). We show that
C m,ω extension by bounded-depth linear operators