Suppose $S(r)$ is set parametrised by $r \in [0,T]$. Let $\phi_t^s : H^1(S(t)) \to H^1(S(s))$ is a linear homeomorphism.
Suppose $\lVert \cdot \rVert_{H^1(S(t))}$ and $\lVert \phi_t^s(\cdot) \rVert_{H^1(S(s))}$ are equivalent norms on $H^1(S(t))$. So their dual norms are also equivalent.
The norm of $f \in H^{-1}(S(s))$ can be given as $\sup_{w \in H^1(S(s))} \frac{\langle f, w \rangle_{H^{-1}(S(s)), H^1(S(s))}}{\lVert w\rVert_{H^1(S(s))}} = \sup_{v \in H^1(S(t))} \frac{\langle (\phi_t^s)^*f, v \rangle_{H^{-1}(S(t)), H^1(S(t))}}{\lVert \phi_t^sv\rVert_{H^1(S(s))}}$ where the star denotes the adjoint.
I can derive this by just using the fact that \phi_t^s is a homeomorphism and the adjoint identity. Do I need any equivalence of dual norms at all?
Suppose that $\lVert \phi_t^sv\rVert_{H^1(S(s))} \leq C\lVert v \rVert_{H^1(S(t))}$
This, and the equivalence of $H^1$ norms implies $\lVert (\phi_t^s)^*\rVert_{\mathcal{L}(H^{-1}(S(s)), H^{-1}(S(t))} \leq C$
I still don't need any equivalence of norms to get this. Can somebody show me how to do it correctly?