Peetre weight inequality ($⟨\xi {⟩}^s\le C_s⟨\eta {⟩}^s⟨\xi -\eta {⟩}^{|s|}$)

For real $s$ and vectors $\xi ,\eta$, the displayed bound follows from $⟨a+b⟩\le \sqrt{2}⟨a⟩⟨b⟩$, 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.