Assume that $\lambda_1$ is the smallest eigenvalue of the Dirichlet Laplacean for the domain $\Omega\subset \mathbf{R}^n$ and let $0<\alpha\le 1$. Is the following statement well-known?

Let $f\in L^2(\Omega)$, $0< \alpha\le 1$ and let $$\hat f(x)=\frac{1}{(2\pi)^{n/2}}\int_{\mathbf{R}^n} f(y) e^{-i \left<x,y\right>} dy.$$ Then $$\int_{\mathbf{R}^n}\frac{|\hat f(x)|^2}{|x|^{2\alpha}}dx \le \lambda_1^{-2\alpha}\int_{\Omega}|f(x)|^2 dx .$$