Look at Figure 3 in the paper by Ginibre-Velo, point $C_2$ is the estimate of $\|u\|_{L^4\dot B^{1/4}_{\infty,2}}$ in terms of the $L^1L^2$ norm of $F$, and one has $\dot B^{1/4}_{\infty,2}\hookrightarrow \dot B^{1/4}_{\infty,\infty}$.
Piero D'Ancona
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