Klainerman-Sobolev inequality ($|u|\lesssim \frac{{\sum }_{|I|\le 2}\Vert Z^{I}u{\Vert }_2}{(1+t+r)(1+|t-r|{)}^{1/2}}$)

In three spatial dimensions, with $Z$ the eleven translations, spatial rotations, boosts and scaling commutation vector fields for the wave equation, a sufficiently decaying smooth function satisfies $|u(t,x)|\le C(1+t+|x|{)}^{-1}(1+|t-|x||{)}^{-1/2}{\sum }_{|I|\le 2}\Vert Z^{I}u(t){\Vert }_2$. This weighted Sobolev inequality converts control of commuted wave energy into decay. It holds independently of any wave equation satisfied by $u$.