Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$, $X=W_{0}^{1,p}(\Omega)$ and $Y=W_{0}^{s,p}(\Omega)$ be the classical and fractional Sobolev spaces. Both are uniformly convex Banach spaces. Let $Z=X\cap Y$ be the space under the norm
$$
\|z\|_Z=\|z\|_X+\|z\|_Y.
$$
Then it can be easily seen that $Z$ is a Banach space.
But I am unable to predict whether $Z$ is uniformly convex under the above norm, which would also give the reflexivity since uniform convex spaces are reflexive.
Best Answer
In fact, it is enough to notice that, for any $s\geq 0$ we have $$ X\cap Y = W^{k,p}_0(\Omega), \quad \hbox{where}\quad k=\max\{1,s\}. $$ Then, since $W^{k,p}_0$ is uniformly convex (as you said on your post), we deduce the uniform convexity of the intersection.