Strichartz estimates for the wave equation $\Box u=F $ with $u(0)=g_0$ and $\partial_t g(0)=g_1$ can be stated as $$\Vert u\Vert_{L^q_tL^r_x}+\Vert u\Vert_{C^0_t\dot{H}_x^s}+\Vert\partial_t u\Vert_{C^0_t\dot{H}_x^{s-1}}\leq C(q,r,s,n)\{\Vert (g_0,g_1)\Vert_{H^s\times H^{s-1}}+\Vert F\Vert_{L^{\tilde{q}'}_tL^{\tilde{r}'}}\}$$ for wave-admissible exponents $(q,r),(\tilde{q},\tilde{r})$. Can they be generalized to bound $\Vert u\Vert _{L^q_tH^{s,r}_x}$?
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1$\begingroup$ Yes, under various conditions on the exponents: the classic reference is Ginibre and Velo, ams.org/mathscinet-getitem?mr=1351643 , and there are many subsequent references also. $\endgroup$– Terry TaoCommented Mar 3, 2015 at 18:13
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$\begingroup$ @TerryTao Also in the case of inhomogeneous space $H^{s,r}$? $\endgroup$– SamSCommented Mar 3, 2015 at 18:44
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