The structure of sobolev extension operators

Charles Fefferman; Arie Israel; Garving K. Luli

doi:10.4171/rmi/787

The structure of sobolev extension operators

Charles Fefferman, Arie Israel, Garving K. Luli

Mathematics

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

Let L^m,p(ℝⁿ) denote the Sobolev space of functions whose m-th derivatives lie in L^p(ℝⁿ), and assume that p > n. For E ⊆ ℝⁿ, denote by L ^m,p(E) the space of restrictions to E of functions F ε L ^m,p(ℝⁿ). It is known that there exist bounded linear maps T : L^m,p(E) → L^m,p(ℝⁿ) such that Tf = f on E for any f ε L^m,p(E). We show that T cannot have a simple form called "bounded depth".

Original language	English (US)
Pages (from-to)	419-429
Number of pages	11
Journal	Revista Matematica Iberoamericana
Volume	30
Issue number	2
DOIs	https://doi.org/10.4171/rmi/787
State	Published - 2014

All Science Journal Classification (ASJC) codes

General Mathematics

Keywords

Linear operators
Sobolev spaces
Whitney extension problem

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title = "The structure of sobolev extension operators",

abstract = "Let Lm,p(ℝn) denote the Sobolev space of functions whose m-th derivatives lie in Lp(ℝn), and assume that p > n. For E ⊆ ℝn, denote by L m,p(E) the space of restrictions to E of functions F ε L m,p(ℝn). It is known that there exist bounded linear maps T : Lm,p(E) → Lm,p(ℝn) such that Tf = f on E for any f ε Lm,p(E). We show that T cannot have a simple form called {"}bounded depth{"}.",

keywords = "Linear operators, Sobolev spaces, Whitney extension problem",

author = "Charles Fefferman and Arie Israel and Luli, {Garving K.}",

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T1 - The structure of sobolev extension operators

AU - Fefferman, Charles

AU - Israel, Arie

AU - Luli, Garving K.

PY - 2014

Y1 - 2014

N2 - Let Lm,p(ℝn) denote the Sobolev space of functions whose m-th derivatives lie in Lp(ℝn), and assume that p > n. For E ⊆ ℝn, denote by L m,p(E) the space of restrictions to E of functions F ε L m,p(ℝn). It is known that there exist bounded linear maps T : Lm,p(E) → Lm,p(ℝn) such that Tf = f on E for any f ε Lm,p(E). We show that T cannot have a simple form called "bounded depth".

AB - Let Lm,p(ℝn) denote the Sobolev space of functions whose m-th derivatives lie in Lp(ℝn), and assume that p > n. For E ⊆ ℝn, denote by L m,p(E) the space of restrictions to E of functions F ε L m,p(ℝn). It is known that there exist bounded linear maps T : Lm,p(E) → Lm,p(ℝn) such that Tf = f on E for any f ε Lm,p(E). We show that T cannot have a simple form called "bounded depth".

KW - Linear operators

KW - Sobolev spaces

KW - Whitney extension problem

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DO - 10.4171/rmi/787

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SP - 419

EP - 429

JO - Revista Matematica Iberoamericana

JF - Revista Matematica Iberoamericana

IS - 2

ER -

The structure of sobolev extension operators

Abstract

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