I am dealing with some vector-valued Sobolev spaces $H^{s}(\mathbb{R}^N,E)$ where $E$ is a Banach space.

I am looking for references about results for the scalar case $H^{s}(\mathbb{R}^N,\mathbb{C})$ that are still valid in the vector-valued case, for example it is well-known that $H^{s}(\mathbb{R}^N,\mathbb{C})$ is a Banach algebra if $s> n/2$, is this results valid for the $E$-valued? The same question for the general Galiardo-Nirenberg-Sobolev inequality on $W^{s,p}(\mathbb{R}^N,E)$ I am not sure if it still hold.

At the level of vector-valued $L^p(\mathbb{R}^N,E)$ it is clear that Holder and Minkowski's inequalities holds but I am not sure if Riesz-Thorin interpolation still holds.

Even though the books of Treves & Schaefer on TVS and distributions deal with some vector-valued spaces they do not cover this classical results.

My question is the following:

Is there some general reference about standard results for vector-valued real analysis i.e. $E$-valued $L^p$ spaces and Fourier transformations, Sobolev spaces and the classical embeddings/inequalities results etc., when $E$ is a Hilbert/Banach/ Frechet space?

Many thanks for your help


You should brows the papers of Amann and his students for this type of results. You will find a lots of interesting results in the paper


about embeddings.

In general, the classic great monograph by Diestel and Uhl is a very good start as a reference.

  • 1
    $\begingroup$ Many thanks for the reference, I had not been noticed this paper, it looks very useful. It seems that there is no "general reference" book about real analysis on vector-valued spaces, these days I have found some results spread in many different papers/books, some papers of Hans-Jurgen Schmeißer and Winfried Sickel look useful too. I realized that the extension from the scalar case to the vector valued case is quite involve and it is not as simple as to change the absolute value to the space norm. $\endgroup$
    – Coffee
    Aug 14 '15 at 17:53

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