= Solution
One version of the <Klainerman-Sobolev inequality>, for a sufficiently decaying <smooth function> $\psi$ on $\mathbb R^{1+3}$, is
$$
\boxed{|\psi(t,x)|\leq
\frac{C}{(1+t+|x|)(1+|t-|x||)^{1/2}}
\sum_{|I|\leq2}\|Z^I\psi(t,\cdot)\|_{L^2(\mathbb R^3)}},\qquad t\geq0.
$$
Here $I$ denotes a word of length $|I|$ in the following eleven <commutation vector fields for the wave equation>:
$$
\begin{aligned}
&\partial_t,\ \partial_1,\ \partial_2,\ \partial_3, &&\text{spacetime translations},\\
&\Omega_{ij}=x_i\partial_j-x_j\partial_i\quad(1\leq i<j\leq3), &&\text{spatial rotations},\\
&L_i=t\partial_i+x_i\partial_t\quad(1\leq i\leq3), &&\text{Lorentz boosts},\\
&S=t\partial_t+\sum_{i=1}^3x_i\partial_i, &&\text{scaling}.
\end{aligned}
$$
The first four are <spacetime translation vector fields>; the next three are <spatial rotation vector fields>; the next three are <Lorentz boost vector fields>; and $S$ is the <scaling vector field>. The <vector field method for wave equations> uses these <vector fields> because their <commutators> with the <d'Alembert operator> obey
$$
[\Box,Z]=0\quad(Z\ne S),\qquad [\Box,S]=2\Box.
$$
In particular the <Klainerman-Sobolev inequality> implies
$$
\|\psi(t,\cdot)\|_\infty\leq\frac{C}{1+t}\sum_{|I|\leq2}\|Z^I\psi(t,\cdot)\|_2.
$$
The displayed <L2 norms> are spatial norms at fixed time; the spacetime <vector fields> can contain time derivatives. No <wave equation> assumption is needed for the <Klainerman-Sobolev inequality> itself.
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