On the existence of optimizers for nonlinear time-frequency concentration problems: the Wigner distribution
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| Format: | Preprint |
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2025
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| _version_ | 1866910255513862144 |
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| author | Stra, Federico Svela, Erling A. T. Trapasso, S. Ivan |
| author_facet | Stra, Federico Svela, Erling A. T. Trapasso, S. Ivan |
| contents | 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). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_18683 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the existence of optimizers for nonlinear time-frequency concentration problems: the Wigner distribution Stra, Federico Svela, Erling A. T. Trapasso, S. Ivan Classical Analysis and ODEs Functional Analysis 49Q10, 49R05, 42B10, 94A12, 81S30 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). |
| title | On the existence of optimizers for nonlinear time-frequency concentration problems: the Wigner distribution |
| topic | Classical Analysis and ODEs Functional Analysis 49Q10, 49R05, 42B10, 94A12, 81S30 |
| url | https://arxiv.org/abs/2510.18683 |