RecallA pair $(q,r)$ is admissible if $q\geq 2, r\geq 2$ and $\frac{2}{q} = N \left( \frac{1}{2} - \frac{1}{r} \right),(q,r,N)\neq(\infty,2,2).$

Strichartz estimates Let $\phi \in L^2(\mathbb R^N),$ $U(t)=e^{it\Delta}$ (Schrodinger propogator). Denote $$DF(t,x) = U(t)\phi(x) + \int_0^t U(t-s)F(s,x) ds.$$ Then for any time interval $I\ni0$ and admissible pairs $(q_j,r_j)$, $j=1,2,$ there exists a constant $C=C(r_1,r_2)$ such that $$ \|D(F)\|_{L^{q_1}(I,L^{r_1})} \leq C \|\phi \|_{L^2}+ C \|F\|_{L^{q'_2}(I,L^{r'_2})}, \quad\forall F \in L^{q_2'} (I, L^{r_2'}(\mathbb R^N))$$ where $q_j'$ and $ r_j'$ are H"older conjugates of $q_j$ and $r_j$ respectively.

Put $s_c=\frac{\gamma}{2}-1, 2<\gamma<N, N \geq 3$ and $\|f\|_{\dot{H}^s_r}= \left\| \hat{f}(\xi) |\xi|^s\right\|_{L_{\xi}^r}$

Question: How to use Strichartz estimates to get following estimates:

$$\|D(F))\|_{L^4(\mathbb R, \dot{H}_{\frac{2N}{N-1}}^{s_c})} \leq C \|\phi\|_{\dot{H}^{s_{c}}} + C \|F(u)\|_{L^{\frac{4}{3}}(\dot{H}_{\frac{2N}{N+1}}^{s_c})}$$

My thoughts: which pair of admissible pair I should consider to get the desired estimates?



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