# Is $\partial^\alpha$ a map $H^{s,p}(\mathbb R^N,F)\to H^{s-|\alpha|,p}(\mathbb R^N,F)$?

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.