Negative Sobolev regularity of a compactly supported distribution ($u\in H^s,s<-M-n/2$)

A compactly supported distribution of order of a distribution at most $M$ has a smooth function as its Fourier transform, bounded by $C⟨\xi {⟩}^M$. It therefore belongs to every Sobolev space $H^s$ with $s<-M-n/2$.