## Abstract

We study operators T from C^{1}(R^{n},R^{n}) to C(R^{n},L(R^{n},R^{n})) satisfying the "chain rule". T(f·g)(x)=((Tf)·g)(x)(Tg)(x);f,g∈C^{1}(R^{n},R^{n}),x∈R^{n}. Assuming a local surjectivity and non-degeneracy condition, we show that for n≥2 the operator T is of the form. (Tf)(x)=|detf'(x)|pH(f(x))f'(x)H(x)-1 for a suitable p≥0 and H∈C(R^{n},GL(n)). For even n there might be an additional factor sgn(detf'(x)). This is the multidimensional extension of our results (Artstein-Avidan et al., 2010 [3]) for n=1. In this setting the non-commutativity of the linear operators L(R^{n},R^{n}) from R^{n} to R^{n} creates additional difficulties but also clarifies and enriches the understanding of the problem.

Original language | English |
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Pages (from-to) | 861-875 |

Number of pages | 15 |

Journal | Journal of Functional Analysis |

Volume | 261 |

Issue number | 4 |

DOIs | |

State | Published - 15 Aug 2011 |

### Funding

Funders | Funder number |
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Fields Institute | |

Alexander von Humboldt-Stiftung | |

United States-Israel Binational Science Foundation | 200 6079 |

Israel Science Foundation | 387/09 |

## Keywords

- Automorphisms of GL(n)
- Chain rule in R
- Functional equation

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