Let $I$ be a compact interval of $\mathbb{R}$ and $U$ be a bounded subset of $\mathbb{R}^2$.

If $f \in L^\infty(I, W^{1,2}(U))$, what (non-trivial) condition ($L^p$-estimate on $f$ or decay-like estimate on the $W^{1,2}$-norm of $f$) is sufficient to obtain that $f \in L^\infty(I \times \Omega)$?