# Strichartz estimates for the inhomogeneous wave equation

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$$.

As a corollary we have the following result:

Corollary 1.2. Let $$M$$ be a compact Riemannian manifold with boundary. Suppose that the triples $$(p,q,\gamma)$$ and $$(r',s',1-\gamma)$$ satisfy the condition of Theorem 1.1 Then we have the following estimate for solutions $$u$$ to (1.1) satisfy either Dirichlet or Neumann homogeneous boundary condition $$$$\|u\|_{L^{p}([-T,T];L^q(M))}\leq C(\|f\|_{H^{\gamma}(M)}+\|g\|_{H^{\gamma-1}(M)}+\|F\|_{L^r([-T,T];L^s(M))})$$$$ with $$C$$ some constant depending on $$M$$ and $$T$$.

It is easy to see that the triple $$(r',s',1-\gamma)=(\infty,2,1-1)=(\infty,2,0)$$ is in the conditions of the Corollary 1.2. so we have the estimate: $$$$\|u\|_{L^{p}([-T,T];L^q(M))}\leq C(\|f\|_{H^{1}(M)}+\|g\|_{L^2(M)}+\|F\|_{L^1([-T,T];L^2(M))}) \tag{*}$$$$ for solutions of problem $$(1.1)$$.

My question:

I would like to know if there is an estimate similar to $$(*)$$ but with $$F \in L^1([-T;T]; H^{-1}(M)).$$