In the 1994 paper [On the Maxwell-Klein-Gordon Equation with Finite Energy](https://projecteuclid.org/download/pdf_1/euclid.dmj/1077288008) of Klainerman and Machedon, the proof of Proposition 1.1 contains the following statement. For $\phi$ (the scalar field of) a classical solution of the Maxwell-Klein-Gordon system, if $$ \| \phi \|\_{L^3(\mathbb{R}^3)} \leqslant C(1+t) \left(1 +\| \phi \|\_{L^3(\mathbb{R}^3)}\right)^{1/2}, $$ then $$ \| \phi \|\_{L^3(\mathbb{R}^3)} \leqslant C(1+t). $$ I don't understand how they deduce this. For a start, it certainly doesn't follow if one ignores the fact that $\phi$ solves the Klein-Gordon equation: take, for example $\phi(t,x) = t^2 \psi(x)$, where $\psi(x)$ is any $L^3(\mathbb{R}^3)$ function with unit norm. The $(1+t)$ looks like it should come from some sort of linear wave equation estimate, but there's no indication they are using anything like that in that proof! What's going on?



Maxwell-Klein-Gordon energy estimates in Klainerman and Machedon's 1994 paper
There is a typo in the paper. Look at the bottom two lines of page 22 which I transcribe here


\begin{align}
\|\phi\|\_{L^3} & \leq \ldots \\
 & \lesssim \mathscr{I}\_0^{1/2} (1+t)^{1/2} ( \mathscr{I}\_0 + \|A\|\_{L^6} \|\phi\|\_{L^3} )^{1/2}\\
 & \lesssim \mathscr{I}\_0 \color{red}{(1+t)} (1 + \|\phi\|\_{L^3})^{1/2}
\end{align}


If we use Sobolev to control $\|A\|\_{L^6} \lesssim \|\nabla A\|\_{L^2} \lesssim \mathscr{I}\_0$, to get from the second to last line to the last line, the final line should in fact have the term that I marked in red replaced by $\color{green}{(1+t)^{1/2}}$. With this (1.5.c) follows. 


