On the existence of optimizers for nonlinear time-frequency concentration problems: the Wigner distribution

We prove that, for any measurable phase space subset $\mathrm{\Omega }\subset {\mathbb{R}}^{2d}$ with $0<\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }<\mathrm{\infty }$ and any $1\le p<\mathrm{\infty }$, the nonlinear concentration problem $$\underset{f\in L^2({\mathbb{R}}^d)\setminus \{0\}}{\sup }\frac{\mathrm{\parallel }Wf{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}}{\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2}^2}$$ admits an optimizer, where $Wf$ is the Wigner distribution of $f$. The main obstruction is that $Wf$ is covariant (not invariant) under time-frequency shifts, which impedes weak upper semicontinuity, so the effects of constructive interference must be taken into account. We close this compactness gap via concentration compactness for Heisenberg-type dislocations, together with a new asymptotic formula that quantifies the limiting contribution to concentration over $\mathrm{\Omega }$ from asymptotically separated wave packets. When $p=\mathrm{\infty }$ we also identify the sharp constant $2^d$ and show that it is attained. We also discuss some related extensions: For $\tau$-Wigner distributions with $\tau \in (0,1)$ we isolate a chain phenomenon that obstructs the same strategy beyond the Wigner case ($\tau =1\mathrm{/}2$), while for the Born-Jordan distribution in $d=1$ we obtain weak continuity, and thus existence of concentration optimizers for all $1\le p<\mathrm{\infty }$ (the $p=\mathrm{\infty }$ supremum equals $\pi$ but is not attained).