Let $f \in H^{-1}(S)$, where $S \subset \mathbb{R}^n$ is some nice set in space. Can I exchange supremums and integrals over time here: $\sup_{g \in H^1(S)}\int_0^T \langle f, g \rangle_{H^{-1}(S), H^1(S)}\;dt =\int_0^T \sup_{g \in H^1(S)} \langle f, g \rangle_{H^{-1}(S), H^1(S)}\;dt$
I think so. But what if $g$ has some dependence on $t$? Or is that not allowed?