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

TL;DR

This paper proves the existence of optimizers for nonlinear time-frequency concentration problems involving the Wigner distribution, addressing covariance issues and establishing sharp constants for certain cases.

Contribution

It introduces a new approach using concentration compactness to prove the existence of optimizers and identifies sharp constants for the supremum cases.

Findings

Optimizers exist for the nonlinear concentration problem with the Wigner distribution.

The sharp constant for the supremum when p=∞ is 2^d and is attained.

Extensions to τ-Wigner and Born-Jordan distributions reveal obstructions and partial results.

Abstract

We prove that, for any measurable phase space subset $\Omega\subset\mathbb{R}^{2d}$ with $0<|\Omega|<\infty$ and any $1\le p < \infty$, the nonlinear concentration problem $$ \sup_{f \in L^2(\mathbb{R}^d)\setminus\{0\}}\frac{\|Wf\|_{L^p(\Omega)}}{\|f\|_{L^2}^2}$$ 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 $\Omega$ 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 $\tau$-Wigner distributions with $\tau \in (0,1)$ we isolate a chain phenomenon that obstructs the same strategy beyond the Wigner case ($\tau=1/2$), 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 $\pi$ but is not attained).