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

Abstract

We prove that, for any measurable phase space subset $\hat{I}\text{c◯}\subset {\mathbb{R}}^{2d}$ with $0<|\hat{I}\text{c◯}|<\infty$ and any $1\le p<\infty$, the nonlinear concentration problem $$\underset{f\in L^2({\mathbb{R}}^d)\setminus \{0\}}{\sup }\frac{\Vert Wf{\Vert }_{L^p(\hat{I}\text{c◯})}}{\Vert f{\Vert }_{L^2}^2}$$ 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 $\hat{I}\text{c◯}$ from asymptotically separated wave packets. When $p=\infty$ we also identify the sharp constant $2^d$ and show that it is attained. We also discuss some related extensions: For $\ddot{I}„$-Wigner distributions with $\ddot{I}„\in (0,1)$ we isolate a chain phenomenon that obstructs the same strategy beyond the Wigner case ($\ddot{I}„=1/2$), while for the Born-Jordan distribution in $d=1$ we obtain weak continuity, and thus existence of concentration optimizers for all $1\le p<\infty$ (the $p=\infty$ supremum equals $\ddot{I}€$ but is not attained).

Comments: 28 pages. To appear in Journal de Mathématiques Pures et Appliquées