Let's consider the wave equation $\Box u=F $ with $u(0)=g_0$ and $\partial_t g(0)=g_1$. Strichartz estimates tell us that $$\Vert u\Vert_{L^q_tL^r_x}+\Vert u\Vert_{C^0_t\dot{H}_x^s}+\Vert\partial_t u\Vert_{C^0_t\dot{H}_x^{s-1}}\leq C(q,r,s,n)\{\Vert (g_0,g_1)\Vert_{H^s\times H^{s-1}}+\Vert F\Vert_{L^{\tilde{q}'}_tL^{\tilde{r}'}}\}$$ for wave-admissible exponents $(q,r),(\tilde{q},\tilde{r})$. i.e. $2\leq q,\tilde{q}\leq\infty$, $2\leq r,\tilde{r}\leq\infty$ and they obey the conditions $$\frac{1}{q}+\frac{n}{r}=\frac{n}{2}-s=\frac{1}{\tilde{q}'}+\frac{n}{\tilde{r}'}-2$$ and $$\frac{1}{q}+\frac{n-1}{2r},\,\frac{1}{\tilde{q}}+\frac{n-1}{2\tilde{r}}\leq\frac{n-1}{4}$$
Can they be generalized to bound $\Vert u\Vert _{L^q_tH^{s,r}_x}$?