Suppose you have a function $u$ that lies in the (fractional) Sobolev space $H^s$, with $0<s<1$. Take a smooth cutoff $\varphi$ such that $\varphi(x)=0$ if $x>R+1$. What can be said about the norm $\(1\varphi)u\_{H^s}^2$? Does it have a precise decay, as $R \to +\infty$?
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It would not have a decay rate, since the support of fourier transform of $u$ may lies in the infinity, the cut off would not work. 

