= Peetre weight inequality
{c}
{title2=$\langle\xi\rangle^s\le C_s\langle\eta\rangle^s\langle\xi-\eta\rangle^{|s|}$}
= Peetre's inequality
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{synonym}
For real $s$ and vectors $\xi,\eta$, the displayed bound follows from $\langle a+b\rangle\le\sqrt2\langle a\rangle\langle b\rangle$, applied in the opposite direction when $s<0$. One can take $C_s=2^{|s|/2}$. Combined with <Young's convolution inequality>, it proves <Sobolev multiplication by a smooth cutoff> for all real orders: the rapidly decreasing transform of the cutoff absorbs the extra weight. The negative-order case is particularly useful when a <distribution> is not initially a function.
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