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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 $$\int_{\mathbf{R}^n}\frac{|\hat f(x)|^2}{|x|^{2\alpha}}dx \le \lambda_1^{-2\alpha}\int_{\Omega}|f(x)|^2 dx .$$

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How do you define the Fourier transform if your $f$ is only defined in $\Omega$? Do you consider some kind of extension? –  András Bátkai Feb 24 '14 at 20:53
$f(x)=0$, for $x\in D^C$. –  user47005 Feb 24 '14 at 21:04
If $f$ is very close to the delta function, the Fourier transform will be very close to constant, and the LHS might not be bounded (say for $\alpha>0.5$). –  Asaf Feb 24 '14 at 21:30
I decided to roll it back to preserve here. It is not obviuos at the first sight that it is not true. –  András Bátkai Mar 6 '14 at 20:50
It seems to me that the inequality follows from Plancherel inequality and the identity $\widehat{(-\Delta)^{-\alpha}f}=|x|^{-2\alpha}\widehat{f}$. –  Gian Maria Dall'Ara Mar 6 '14 at 22:31

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