I was wondering whether the following continuous embedding theorem for parabolic Sobolev space is correct?


Let $I=[0,T]$ and $\Omega$ be a sufficiently smooth domain in $\mathbb{R}^n$, we consider the space 
$$W=L^2(I;H^2(\Omega))\cap H^1(I;L^2(\Omega))=\{u\in L^2(I;H^2(\Omega))\mid u\_t\in L^2(I;L^2(\Omega))\}. $$
Then is it possible to find some $r>2$, such that 
$W\hookrightarrow L^r(I;W^{1,r}(\Omega))$ continuously?



Continuous embedding between parabolic Sobolev spaces
This should work according to the embedding $$W \hookrightarrow L^{\frac2{1-2s}}\bigl((L^2(\Omega),H^2(\Omega))\_{\theta,1}\bigr)$$ where $0 < s < \frac12$ and $0 \leq \theta < 1-s$ as in [Amann: Linear parabolic problems involving measures, Theorem 3](http://www.rac.es/ficheros/doc/00047.pdf). 


To elaborate a bit more: By a relation between real and complex interpolation spaces, interpolation identitites for Bessel potential spaces and the supposed smoothness of $\Omega$, $$\bigl(L^2(\Omega),H^2(\Omega)\bigr)\_{\theta,1} \hookrightarrow \bigl[L^2(\Omega),H^2(\Omega)\bigr]\_\theta = H^{2\theta}(\Omega),$$
and $H^{2\theta}(\Omega) \hookrightarrow W^{1,r}(\Omega)$ where $2\theta - \frac{n}2 \geq 1-\frac{n}{r}$. (The standard textbooks by [Triebel](https://zbmath.org/?q=an:0830.46028), there Thms. 1.10.3.1 and 4.3.1.2 and 4.6.2, and [Bergh/Löfström](https://zbmath.org/?q=an:0344.46071) contain this.) Now it remains to connect $r$ and the integrability $2/(1-2s)$ from the above embedding.


(My calculations gave that your desired embedding is correct for all $r \in [2,2+\frac4n)$ which corresponds to $s < \frac1{2+n}$ and $\theta = \frac12(ns+1)$, but of course you should double-check that..)


