On page 4 of [this paper](http://www.math.uvic.ca/~ibrahim/Ibra-Maj-Mas-cpam.pdf) of Ibrahim, Majdoub, and Masmoudi, the authors claim in Proposition 2 that solutions to 


$\left\{\begin{array}{ll}\square u=F(t,x)\\ u(0,x)=f(x), \partial\_tu(0,x)=g(x)\end{array}\right.$


enjoy the estimate 


$||u||\_{L\_T^4(C^{1/4}(\mathbb{R}^2))}\le C[||g||\_{L^2(\mathbb{R}^2)}+||f||\_{H^1(\mathbb{R}^2)}+||\square u+u||\_{L\_T^1(L^2(\mathbb{R}^2))}]$


They reference a paper of [Ginibre and Velo](http://www.sciencedirect.com/science/article/pii/S0022123685711196/pdf?md5=8325dad4b9dc787c6e3e4cfdf5a6399e&pid=1-s2.0-S0022123685711196-main.pdf), which present a series of estimates which do not resemble the estimate cited above. I'm a bit confused on how the authors derived their estimate from Ginibre and Velo, though I recognize they could be using the fact that the $\dot{B}\_{\infty,\infty}^{\alpha}$ and $C^{\alpha}$ norms coincide. I've only encountered these estimates in their Sobolev form, and can't see where they're getting the $\square u+u$ term on the right, so I hope you'll forgive me if I'm not seeing the forest for the trees. 



Besov Characterization of Strichartz Estimate.
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 $\dot B^{1/4}\_{\infty,2}\hookrightarrow \dot B^{1/4}\_{\infty,\infty}$.


