Integrating the Wigner Distribution on Subsets of the Phase Space, a Survey

The author reviews several properties of integrals of the Wigner distribution on subsets of the phase space. Along our way, the author provides a theoretical proof of the invalidity of Flandrin's conjecture, a fact already proven via numerical arguments in his joint paper [J. Fourier Anal. Appl. 26 (2020), no. 1, article no. 6 with B. Delourme and T. Duyckaerts]. The author also uses the J. G. Wood and A. J. Bracken paper [J. Math. Phys. 46 (2005), no. 4, article no. 042103], for which he offers a mathematical perspective. The author thoroughly reviews the case of subsets of the plane whose boundary is a conic curve and shows that Mehler's formula can be helpful in the analysis of these cases, including for the higher dimensional case investigated in the paper [J. Math. Phys. 51 (2010), no. 10, article no. 102101] by E. Lieb and Y. Ostrover. Using the Feichtinger algebra, the author shows that, generically in the Baire sense, the Wigner distribution of a pulse in $L^2(\mathbb{R}^{n})$ does not belong to $L^{1}(\mathbb{R}^{2n})$, providing as a byproduct a large class of examples of subsets of the phase space $\mathbb{R}^{2n}$ on which the integral of the Wigner distribution is infinite. The author also studies the case of convex polygons of the plane, with a rather weak estimate depending on the number of vertices, but independent of the area of the polygon.

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