# Real analysis on vector-valued spaces, $L^{p}(\mathbb{R}^N,E)$ ,$H^{s}(\mathbb{R}^N,E)$

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?