= Klainerman-Sobolev inequality
{c}
{title2=$|u|\lesssim\frac{\sum_{|I|\leq2}\|Z^Iu\|_2}{(1+t+r)(1+|t-r|)^{1/2}}$}
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 $|u(t,x)|\leq C(1+t+|x|)^{-1}(1+|t-|x||)^{-1/2}\sum_{|I|\leq2}\|Z^Iu(t)\|_2$. This weighted <Sobolev inequality> converts control of <commuted wave energy> into decay. It holds independently of any <wave equation> satisfied by $u$.
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