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Let $C$ be a convex subset of $\mathbb R^{2n}$ and $\mathbf 1_C$ be the characteristic function of $C$. Is it true that $$\forall u\in\mathscr S(\mathbb R^n),\quad \langle\mathbf 1_C^{Weyl}u,u\rangle\le \Vert u\Vert_{L^2(\mathbb R^n)}^2,\tag F $$ or more explicitly, $ \iint_{\mathbb R^n\times\mathbb R^n}\mathbf 1_C(x,\xi)\mathscr H(u,u)(x,\xi) dx d\xi\le \Vert u\Vert_{L^2(\mathbb R^n)}^2, $ where $\mathscr H(u,u)$ is the Wigner function given by $$ \mathscr H(u,u)(x,\xi)=\int_{\mathbb R^n}e^{-2i\pi z\cdot \xi} u(x+\frac{z}{2})\bar u(x-\frac{z}{2}) dz. \tag W$$ Since $(W)$ is real-valued, the lhs of $(F)$ is also real. The property is easy for $C$ equal to a half-space, since, due to Segal identity on symplectic transformations of Weyl symbols, the operator with Weyl symbol $\mathbf 1_C$ is, in that case, unitarily equivalent to the multiplication by $H(x_1)$.

Inequality (F) seems to have received the name of Flandrin's conjecture.

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  • $\begingroup$ +1, but... What is intuition behind ? Can you write it in terms of Weyl-Moyl product ? May be it can generalized to arbitrary star-product on symplectic manifold, if yes, then how ? What are references ? $\endgroup$ – Alexander Chervov Sep 13 '12 at 17:36
  • $\begingroup$ Intuition: think about the probability measure with density $\mathbf 1_C/\vert C\vert$. Then the expectation of the Wigner function should be smaller than 1 for a normalized $u$. Note that the Wigner function is real-valued but not positive in general and with integral 1. References: it seems to me that Philippe Jaming spoke about these problems, somewhat linked to his work on the uncertainty principle. I think that Flandrin is an engineer, specialized in signal business. $\endgroup$ – Bazin Sep 13 '12 at 19:02
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    $\begingroup$ Here is a reference : arxiv.org/pdf/1007.1796.pdf $\endgroup$ – jjcale Sep 13 '12 at 19:14
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I know this may be too late, but in 2019 appeared a paper containing a counterexample to Flandrin's conjecture:

https://link.springer.com/article/10.1007/s00041-019-09722-9

by Bérangère Delourme, Thomas Duyckaerts & Nicolas Lerner.

Here's the arXiv version.

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