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To what extend do the various different Sobolev inequalites hold if I replace the usual target space $\mathbb R$ by an arbitrary Banach space, the notion of derivative by Frechét derivative and the usual real-valued integral by the Bochner integral (correspondingly adapting the definition of weak derivative) ?

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  • $\begingroup$ Typically, Hahn-Banach helps for such generalizations. $\endgroup$ – Jochen Wengenroth Mar 22 '18 at 15:05
  • $\begingroup$ Could you give an example on how one would make use of Hahn-Banach in this situation ? I am frankly quite surprised that one can apply this Theorem here. $\endgroup$ – Berni Waterman Mar 22 '18 at 15:10
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The question you are asking (and a lot of related material) is discussed in great detail in the article "W. Arendt and M. Kreuter: Mapping theorems for Sobolev spaces of vector-valued functions" (to appear in Studia Mathematica; preprint available on arXiv). For embedding theorems, see Section 6.

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