Suppose that we have a sequence $u_j\in C_c^\infty(B)$ that satisfies an $L^2$ bound $$ \sup_j||u_j||_2<\infty. $$ Here $B$ is the unit ball in $\mathbb R^n$.
I speculate that the Fourier transforms $\hat u_j$ have polynomial decay, i.e., there exists $s>0$ such that $$ \sup_{j,\xi}|\xi|^s|\hat u_j(\xi)|<\infty. $$ Is this true? Thank you!