An example such that $||x \hat{f}||_{L^2(\mathbb{R}^n)} = \infty$

Question

Let $\hat{}$ denote the Fourier transform.

I want to know an example such that

$||xf||_{L^2(\mathbb{R}^n)} < \infty$ and

$||x \hat{f}||_{L^2(\mathbb{R}^n)} = \infty$.

Answer 1

Let $n =1$. Take $f = \chi_{[-1, 1]}$, then \begin{align} \hat f(\xi) = \int^\infty_{-\infty}e^{-2\pi i x\xi}f(x)\ dx = \int^1_{-1}e^{-2\pi i x\xi}\ dx = \frac{\sin 2\pi \xi}{\pi \xi} \end{align} which means \begin{align} \|\xi \hat f\|_{L^2(d\xi)}^2 = \frac{1}{\pi^2}\int^\infty_{-\infty} |\sin 2\pi \xi|^2\ d\xi = \infty. \end{align}