More precisely, the question is formulated as follows. Let $F$ be an arbitrary Banach space, especially not having the UMD−property. Let $N\in\mathbb N$ and $s\in\mathbb R$ and $1\le p\le +\infty$ . The Bessel potential space $H={\rm H\,}s=H^{s,p}(\mathbb R^N,F)$ of tempered distributions on $\mathbb R^N$ with values in $F$ is then defined so that a linear homeomorphism $H\to L^p(\mathbb R^N,F)\hookrightarrow\mathscr S'=\mathscr S'(\mathbb R^N,F)$ is defined by $T\mapsto\mathcal F^{-1}(\chi^s\mathcal F\,T)$ when $\mathcal F$ denotes the Fourier transform of tempered distributions and $\chi^s$ is the function $\mathbb R^N\to\mathbb R$ given by $\xi\mapsto(1+|\xi|^2)^{\frac 12\,s}$ . Then the distributional partial derivative $\partial^\alpha$ defines a (continuous) linear map $\mathscr S'\to\mathscr S'$ , and the question is whether it restricts to a map ${\rm H\,}s\to{\rm H\,}(s-|\alpha|)$ .

Some comments. (1) It seems to be "well-known" by referring to articles of McConnell (On Fourier multiplier transformations of Banach-valued functions, Trans. AMS, 285(2) 1984) and Zimmermann (On vector-valued Fourier multiplier theorems, Studia Math. 1989) treating vector valued Fourier multipliers $L^p\to L^p$ that the vector valued Bessel potential spaces ${\rm H\,}m$ and the Sobolev spaces $W^{m,p}(\mathbb R^N,F)$ for positive integer $m$ coincide if and only if $F$ is UMD. However, these two articles do not provide any kind of proof of this. Of course, it should be easy to see that the UMD property quaranteeing the validity of the Mihlin-Hörmander type multiplier condition is sufficient for these spaces to be identical. At least to me, it is not at all clear that it is also necessary.

(2) For a scale $s\mapsto {\rm S}^s(\mathbb R^N,F)$ of spaces of distributions on $\mathbb R^N$ with values in $F$ where the index $s$ is intended to be some kind of order of differentiabily, a certain kind of minimal reasonability requirement is a positive answer to the above question. For the Sobolev spaces, this follows almost trivially from the definition. For the Besov and Lizorkin-Triebel spaces $B^s_{p,q}$ and $F^s_{p,q}$ , a positive answer follows from a certain modified multiplier condition (see Proposition 2.4, p. 6, Proposition 3.10, p. 9 here) which does not require $F$ to be UMD.


Generally, the answer is no. I have known this already for some years but have not bothered to write the answer here since the question seems not having raised much interest. Now having some spare time, I put it here in case someone possibly be interested. The assertion follows from (1) in Theorem 5.6.12 on page 456 in

T. Hytönen, J. van Neerven, M. Veraar and L. Weis: Analysis in Banach Spaces, Vol. I: Martingales and Littlewood–Paley Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, A Series of Modern Surveys in Mathematics 63, Springer 2016.


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