Consider a sequence of functions $f_n: [0,T]\times [0,T] \times \Omega \rightarrow \mathbb{R}^3 $, $\Omega \subset \mathbb{R}^3$, such that
\begin{align} t &\rightarrow f_n(t,s,x) \ \ \ \text{is bounded in } W^{1,\infty}([0,T]), \text{ uniformly wrt } n,s,x; \\ s &\rightarrow f_n(t,s,x) \ \ \ \text{is bounded in } W^{1,\infty}([0,T]), \text{ uniformly wrt } n,t,x; \\ x &\rightarrow f_n(t,s,x) \ \ \ \text{is bounded in } C^2(\Omega), \text{ uniformly wrt } n,t,s. \\ \end{align}
Now the claim is that
\begin{equation} f_n \rightarrow f,\ \ \ \text{in } C^{0,\alpha}([0,T]\times [0,T];C^1(\Omega)),\ \ \ \alpha < 1; \end{equation}
but I can't see why this is the case. I know that from the Morrey inequality we have the compact embeddings $W^{1,\infty}(0,T) \subset C^{0,\alpha}([0,T])$ and $C^2(\Omega) \subset C^1(\Omega)$ but this does not seem to be enough. Those embeddings probably give us the desired convergence for functions bounded in $W^{1,\infty}([0,T]\times[0,T];C^2(\Omega))$ but such a bound does not seem to follow from the conditions on $f_n$.
Another option might be Aubin-Lions, but here I don't see how it can be applied since instead of a time interval we have $[0,T]\times[0,T]$ and also, at best, it can give us $C^0$-convergence in time instead of $C^{0,\alpha}$.
Does anyone have an idea here?