Suppose $\Omega$ is a bounded domain and $\omega \subset \Omega$. Suppose we have the following estimate: $$ \|u\|_{H^1((0,T) \times\Omega)} \leq C (\|u\|_{H^1((0,T) \times \omega)} + \|\Box u\|_{L^2((0,T) \times \Omega)})$$ for all $u \in H^1((0,T) \times\Omega) \cap L^2((0,T);H^1_0(\Omega))$ and $\Box u \in L^2((0,T) \times \Omega)$.
Can one automatically deduce that the following estimate holds? $$\|u\|_{L^2((0,T) \times\Omega)} \leq C (\|u\|_{L^2((0,T) \times \omega)} + \|\Box u\|_{H^{-1}((0,T) \times \Omega)})$$
