Consider the uniformly elliptic equation $(\partial_t^2 + L)u = 0$ on $(0, \infty) \times \Omega$, where $\Omega \subset \mathbb{R}^n$ is an open bounded domain with smooth boundary, and $L$ is a second order negative definite elliptic operator (not necessarily of divergence type) with smooth coefficients. The domain of $L$ satisfies $\mathcal{D}(L) \subset H^2(\Omega),$ and is defined by the regular elliptic boundary conditions $B(x, \partial_x)u = 0$ on $\partial\Omega$. I am trying to find references to the following kinds of elliptic estimates: $$ \Vert Lu (1, .) \Vert_{L^\infty (\{1\} \times B\cap\Omega)}\leq C\Vert u (., .)\Vert_{L^2((\frac{1}{2}, \frac{3}{2}) \times B^{'}\cap\Omega)},$$ where $B \subset B^{'} \subset \mathbb{R}^n$ are open balls with centers inside $\Omega$. I would highly appreciate a reference. This is a bit similar to the following MO question.

Edit: It might suffice to obtain bounds like $\Vert u\Vert_{W^{k, 2}(I\times B\cap\Omega)} \lesssim \Vert u\Vert_{L^2(I^{'} \times B^{'}\cap\Omega)}$ for high enough $k$, where $I \subset I^{'}$ are open intervals in $(0, \infty)$, but I am not sure what happens at the boundary.