# How to the estimate as application of Strichartz Estimates?

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 $$s_c=\frac{\gamma}{2}-1, 2<\gamma and $$\|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?