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Let $u$ be a divergence-free vector field $u:\mathbb R^n \to \mathbb R^ n$. Does the following inequality hold?

$$\Big( \int_{\mathbb R^n} |u|^2 dx\Big)^2 \le C\Big(\int_{\mathbb R^n} |u|^2|x|^2 dx \Big) \Big(\int_{\mathbb R^n} |\nabla u|^2 dx \Big). $$

How can it be proved? Any reference is appreciated.

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    $\begingroup$ The inequality does not seem to scale well for functions $A\cdot u$ and $A\to 0$. $\endgroup$
    – Dirk
    Commented Sep 19, 2020 at 9:26
  • $\begingroup$ @Dirk Sorry for the typo: I wrote two extra exponents $\endgroup$
    – Riku
    Commented Sep 19, 2020 at 13:54
  • $\begingroup$ Looks pretty much like the Heisenberg uncertainty principle for the Fourier transform to me. Am I missing some subtlety here? $\endgroup$
    – fedja
    Commented Sep 21, 2020 at 4:13

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