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Sobolev spaces of vector-valued functions

Abstract

We are concerned here with Sobolev-type spaces of vector-valued functions. For an open subset \(\Omega \subset {\mathbb {R}}^N\) and a Banach space V, we compare the classical Sobolev space \(W^{1,p}(\Omega , V)\) with the so-called Sobolev–Reshetnyak space \(R^{1,p}(\Omega , V)\). We see that, in general, \(W^{1,p}(\Omega , V)\) is a closed subspace of \(R^{1,p}(\Omega , V)\). As a main result, we obtain that \(W^{1,p}(\Omega , V)=R^{1,p}(\Omega , V)\) if, and only if, the Banach space V has the Radon–Nikodým property

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Acknowledgements

We would like to thank the referee for his/her interesting comments and suggestions, which helped to improve the quality of the paper.

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Correspondence to Jesús Á. Jaramillo.

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Research supported in part by Grant PGC2018-097286-B-I00 (Spain).

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Caamaño, I., Jaramillo, J.Á., Prieto, Á. et al. Sobolev spaces of vector-valued functions. RACSAM 115, 19 (2021). https://doi.org/10.1007/s13398-020-00959-4

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Keywords

  • Sobolev spaces
  • Vector-valued functions

Mathematics Subject Classification

  • 46E35
  • 46E40
  • 46B22