Suppose that $\Psi$ is a function on $\mathbb{R}^n$ which is absolutely integrable, continuously differentiable, and has integral zero. Also, assume $$ |\Psi(x)|+|\nabla \Psi(x)|\leq (1+|x|)^{-(n+1)}$$ Does Fourier inversion hold for $\Psi$?
I think the answer is yes, and since $\Psi$ is in $L^1$, it suffices to show that its Fourier transform is absolutely integrable. I tried using the inequality $$1\leq C(1+|x|)\sum\limits_{|\beta|\leq 1}|x^{\beta}|$$ which eventually led to the pointwise inequality $$|\hat{\Psi}(x)|\leq C(1+|x|)\left|[(1+|\cdot|)^{-(n+1)}]^{\land}(x)\right|$$ and don't see how to proceed further. Any help is appreciated.