Let $f\in L^2(\mathbb R^d)$ and $s<0.$ Assume that $ |\text{supp} \ f | \leq M^k A^d$ fix $k>0.$
Can we say that $$ \|f\|_{L^2} \leq \| (1+|x|^2)^s\|_{L^2( \{|x|\leq M^{k/d} A\})} \leq C(M) \| (1+|x|^2)^s\|_{L^2( \{|x|\leq A\})}$$? If so what is $C(M) $ constant depending on $M$?
Note. I think the above first inequality can be proved using this answer