 Solution

One version of the Klainerman-Sobolev inequality, for a sufficiently decaying smooth function $\psi$ on ${\mathbb{R}}^{1+3}$, isHere $I$ denotes a word of length $|I|$ in the following eleven commutation vector fields for the wave equation: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 obeyIn particular the Klainerman-Sobolev inequality impliesThe 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.