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\(\text {SL}(n)\) covariant vector-valued valuations on \(L^{p}\)-spaces

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Sommaire

A complete classification of continuous \(\text {SL}(n)\) covariant vector-valued valuations on \(L^{p}({\mathbb {R}}^{n},|x|dx)\) is obtained without any homogeneity assumptions. The moment vector is shown to be essentially the only such valuation.

Résumé

On obtient une classification complète des valuations vectorielles sur \(L^{p}({\mathbb {R}}^{n},|x|dx)\) qui sont continues, et \(\text {SL}(n)\) covariantes, sans aucune hypothese d’homogénéité. On montre que le vecteur moment est essentiellement la seule valuation de ce type.

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Acknowledgements

The work of the third author was supported by the Natural Science Foundation of Hunan Province (2019JJ50172).

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Authors and Affiliations

  1. School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan, 411201, People’s Republic of China

    Wei Wang & Lijuan Liu

  2. Department of Mathematics, Jiangxi University of Science and Technology, Jiangxi, 341000, People’s Republic of China

    Rigao He

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  1. Wei Wang
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  2. Rigao He
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  3. Lijuan Liu
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Correspondence to Wei Wang.

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Wang, W., He, R. & Liu, L. \(\text {SL}(n)\) covariant vector-valued valuations on \(L^{p}\)-spaces. Ann. Math. Québec 45, 465–486 (2021). https://doi.org/10.1007/s40316-020-00153-3

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