Let $\Omega \subset \mathbb R^n$, $n \in \mathbb N$, be a smooth bounded domain, $T > 0$ and $p, q \in (1, \infty)$. (If it simplifies matters, I am also fine with just considering the special case $p=q$.) I am interested in properties of the space
$$W := \left\{ \varphi \in L^p((0, T); W^{2, q}(\Omega)) : \varphi_t \in L^p((0, T); L^q(\Omega))\right\}.$$
I am aware of the embedding
$$\left\{ \varphi \in L^p((0, T); W^{1, p}(\Omega)) : \varphi_t \in (L^p((0, T); W^{1, p}(\Omega)))^\star\right\} \hookrightarrow C^0([0, T]; L^p(\Omega));$$ does a similar result also hold for $W$? If $p=q \geq 2$, that's obvious, but what if, say, $p=q \leq 2$?
For instance, does $W$ at least embed into $C^0([0, T]; L^1(\Omega))$? If so, can $L^1(\Omega)$ be replaced by a smaller space?
Some context: If $f$ is bounded in $L^p((0, T); L^q(\Omega))$, bounds for solutions of the equation $u_t = \Delta u + f$ (complemented with suitable initial and boundary conditions) can be obtained by maximal Sobolev regularity. I wonder which further regularity one can obtain from these bounds.