# Strichartz estimates

In the Blair, Smith and Sogge's paper Strichartz estimates for the wave equation on manifolds with boundary, the authors study integrability estimates for solution of the following problem:

$$$$\partial_t^2 u(t,x)-\Delta_g u(t,x)=0, ~ u(0,x)=f(x), ~ \partial_tu(0,x)=g(x) \tag{1.1}$$$$ where $$\Delta_g$$ denotes the Laplace–Beltrami operator on $$(M,g)$$ with $$(M, g)$$ a Riemannian manifold of dimension $$n \geq 2$$.

They establish the following result:

Theorem 1.1. Let $$M$$ be a compact Riemannian manifold with boundary. Suppose $$2, $$2 \leq q < \infty$$ and $$(p,q,\gamma)$$ is a triple satisfying $$$$\frac{1}{q} + \frac{n}{q}= \frac{n}{2}-\gamma \hspace{ 2cm}\begin{cases} \frac{3}{p}+\frac{n-1}{q} \leq \frac{n-1}{2}, n \leq 4 \\ \frac{1}{p}+\frac{1}{q} \leq \frac{1}{2}, n \geq 4 \end{cases}$$$$ Then we have the following estimates for solutions $$u$$ to (1.1) satisfying either Dirichlet or Neumann homogeneous boundary conditions $$$$\|u\|_{L^{p}([-T,T];L^q(M))}\leq C(\|f\|_{H^{\gamma}(M)}+\|g\|_{H^{\gamma-1}(M)})$$$$ with $$C$$ some constant depending on $$M$$ and $$T$$.

My question: Does anyone know of any place that has a didactic proof (with details) of this result, showing the dependency of the constant $$C$$ on $$T$$ and $$M$$? Containing the prerequisites and tools necessary to understand the proof.

The proof can be done in a bounded domain of $$\mathbb{R}^{n}$$.