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*) ?



Sobolev inequalities for Banach-valued functions
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](https://dx.doi.org/10.4064/sm8757-4-2017)" (to appear in Studia Mathematica; [preprint](https://arxiv.org/abs/1611.06161) available on arXiv). For embedding theorems, see Section 6.


