### Abstract

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 ⊆ ℝ^{n}, denote by L ^{m,p}(E) the space of restrictions to E of functions F ε L ^{m,p}(ℝ^{n}). It is known that there exist bounded linear maps T : L^{m,p}(E) → L^{m,p}(ℝ^{n}) 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) |
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Pages (from-to) | 419-429 |

Number of pages | 11 |

Journal | Revista Matematica Iberoamericana |

Volume | 30 |

Issue number | 2 |

DOIs | |

State | Published - 2014 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Keywords

- Linear operators
- Sobolev spaces
- Whitney extension problem

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## Cite this

Fefferman, C., Israel, A., & Luli, G. K. (2014). The structure of sobolev extension operators.

*Revista Matematica Iberoamericana*,*30*(2), 419-429. https://doi.org/10.4171/rmi/787