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

TL;DR

The paper proves the existence of optimizers for nonlinear time-frequency concentration of the Wigner distribution on any finite-measure phase-space domain $Ω$, for all $1≤p<∞$, and identifies the sharp constant $2^d$ with attainment in the $p=∞$ case. The primary obstacle is the covariance of $Wf$ under time-frequency shifts, which breaks weak upper semicontinuity; to overcome this, the authors deploy concentration-compactness with dislocations and derive a novel asymptotic formula quantifying the limiting cross-interference from antipodally separated wave packets. They also analyze extensions to other covariant time-frequency representations: for $τ$-Wigner distributions, a chain phenomenon obstructs the same strategy in general; for Born–Jordan in $d=1$ they obtain weak continuity and thus existence of optimizers for all finite $p$, though the $p=∞$ bound is not attained. The work advances understanding of phase-space concentration, providing rigorous existence results and clarifying how interference patterns govern optimization in time-frequency analysis, with implications for Cohen-class representations and potential extensions to higher dimensions and subspace restrictions.

Abstract

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

Theorems & Definitions (30)

• Theorem 1.1
• Lemma 2.1: CNT_19
• Theorem 2.2: CT_20
• Lemma 2.3
• Theorem 3.1
• proof : Proof of Theorem \ref{['thm-avgbd']}
• Corollary 3.2
• Corollary 3.3
• proof
• Corollary 3.4