Let $H^{s,p}(\mathbb{R}, \mathbb{C})$ be the fractional order Sobolev space of **scalar valued** functions (distributions) over the real line, where $s\in \mathbb R$ and $1<p<\infty$.

It is a theorem by E. Shamir and R. Strichartz that the indicator function of the half line $1_{\mathbb{R}_+}$ (equal to $1$ for $x\geq 0$ and equal to $0$ for $x<0$) is a pointwise multiplier on $H^{s,p}(\mathbb{R}, \mathbb{C})$ if and only if ($p'$ dual exponent)
$$- \frac{1}{p'} < s < \frac{1}{p}.$$
This means that
$$\|1_{\mathbb{R}_+} \cdot f \|_{H^{s,p}} \leq C \|f\|_{H^{s,p}}$$
for all Schwartz functions $f$, with a constant $C > 0$ independent of $f$. This result is trivial for $s = 0$ (reducing to an $L^p$-space) but non-trivial for $s\neq 0$. Strictly outside this range, because of trace considerations, the inequality cannot hold.

My question regards the case of vector-valued functions. Let $X$ be a Banach space and let $H^{s,p}(\mathbb{R}, X)$ be the Sobolev space of $X$-valued functions (distributions), defined in the same way as in the scalar valued case. We could show the multiplier property of $1_{\mathbb{R}_+}$ in the same range as in the scalar-valued case **provided the Banach space $X$ has the UMD property**. See here or here, and here, Section 4 for an elementary proof of this fact. As a rule of thumb, all reflexive standard Banach spaces have UMD. Moreover, alle UMD spaces are reflexive. Space without UMD are thus $L^1$ and $L^\infty$.

My question is as follows:

Let $X$ be a Banach space. Suppose that the inequality
$$\|1_{\mathbb{R}_+} \cdot f \|_{H^{s,p}(\mathbb{R}, X)} \leq C \|f\|_{H^{s,p}(\mathbb{R}, X)}$$ holds true for some $s\neq 0$ and some $1<p<\infty$, for all $X$-valued Schwartz functions $f$. Does this imply that $X$ has the UMD property?

I find this interesting because $X$ has the UMD property if and only if the Hilbert transform is a bounded operator on $L^p(\mathbb{R}, X)$, i.e. the signum function is a Fourier multiplier on this space. In other words,
$F^{-1} sgn F$ is a bounded operator on $L^p(\mathbb{R}, X)$ ($F$ denoting the Fourier transform).

The pointwise multiplier property is equivalent to the boundedness of
$$1_{\mathbb{R}_+} F^{-1}(1+|\cdot|^2)^{s/2} F$$ on $L^p(\mathbb{R}, X)$. So, given a positive answer the question, this would imply a new characterization of the boundedness of Hilbert transform in terms of a jump function **in the time variable** - and not in the frequency variable as in the usual definition.