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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References
Arendt, W., Batty, C., Hieber, M., Neubrander, F.: Vector-valued Laplace transforms and Cauchy problems, 2nd edn. Monographs in Mathematics, 96. Birkhäuser/Springer Basel AG, Basel (2011)
Arendt, W., Kreuter, M.: Mapping theorems for Sobolev spaces of vector-valued functions. Stud. Math. 240, 275–299 (2018)
Benyamini, Y., Lindenstrauss, J.: Geometric nonlinear functional analysis. Vol. 1. American Mathematical Society Colloquium Publications, 48. American Mathematical Society, Providence, RI (2000)
Diestel, J., Uhl, J.: Vector measures. With a foreword by B. J. Pettis. Mathematical Surveys, No. 15. American Mathematical Society, Providence, R.I. (1977)
Hajłasz, P.: Sobolev spaces on metric-measure spaces. Contemp. Math. 338, 173–218 (2003)
Hajłasz, P., Tyson, J.T.: Sobolev peano cubes. Mich. Math. J. 56, 687–702 (2008)
Heinonen, J., Koskela, P., Shanmugalingam, N., Tyson, J.T.: Sobolev classes of Banach space-valued functions and quasiconformal mappings. J. Anal. Math. 85, 87–139 (2001)
Heinonen, J., Koskela, P., Shanmugalingam, N., Tyson, J.: Sobolev spaces on metric measure spaces. An approach based on upper gradients. New Mathematical Monographs, 27. Cambridge University Press, Cambridge (2015)
Kreuter, M.: Sobolev spaces of vector-valued functions. Master Thesis, Ulm University (2015)
Reshetnyak, YuG: Sobolev classes of functions with values in a metric space. Sib. Math. J. 38, 567–583 (1997)
Shanmugalingam, N.: Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoam. 16, 243–279 (2000)
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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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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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DOI: https://doi.org/10.1007/s13398-020-00959-4