Integrating the Wigner Distribution on subsets of the phase space, a Survey

We review several properties of integrals of the Wigner distribution on subsets of the phase space. Along our way, we provide a theoretical proof of the invalidity of Flandrin's conjecture, a fact already proven via numerical arguments in our joint paper [MR4054880] with B.Delourme and T.Duyckaerts. We use also the J.G.Wood and A.J.Bracken paper [MR2131219], for which we offer a mathematical perspective. We review thoroughly the case of subsets of the plane whose boundary is a conic curve and show that Mehler's formula can be helpful in the analysis of these cases, including for the higher dimensional case investigated in the paper [MR2761287] by E.Lieb and Y.Ostrover. Using the Feichtinger algebra, we show 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. We study as well 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.