On the existence of optimizers for nonlinear time-frequency concentration problems: the Born--Jordan distribution
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| Format: | Preprint |
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2026
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| _version_ | 1866918494578147328 |
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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 study the $L^p$ concentration problem for the Born--Jordan distribution in dimension $d>1$, thus extending the one-dimensional analysis in [Stra-Svela-Trapasso, J. Math. Pures Appl. (2026)]. We show that the existence of concentration optimizers depends on the exponent $p$ with a critical threshold at $p_*(d)= \frac{2d}{d-2}$ for $d\geq2$ (with the understanding that $p_*(2)=\infty$). In particular, for subcritical exponents $1\leq p<p_*(d)$ we prove that the supremum is finite and is attained, whereas for supercritical exponents $p>p_*(d)$ we show that the functional is unbounded. We also provide the complete solution in the (significantly more) challenging critical regime in dimension $d=2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_10535 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the existence of optimizers for nonlinear time-frequency concentration problems: the Born--Jordan distribution Stra, Federico Svela, Erling A. T. Trapasso, S. Ivan Classical Analysis and ODEs Functional Analysis 81S30, 49Q10, 42B10, 94A12 We study the $L^p$ concentration problem for the Born--Jordan distribution in dimension $d>1$, thus extending the one-dimensional analysis in [Stra-Svela-Trapasso, J. Math. Pures Appl. (2026)]. We show that the existence of concentration optimizers depends on the exponent $p$ with a critical threshold at $p_*(d)= \frac{2d}{d-2}$ for $d\geq2$ (with the understanding that $p_*(2)=\infty$). In particular, for subcritical exponents $1\leq p<p_*(d)$ we prove that the supremum is finite and is attained, whereas for supercritical exponents $p>p_*(d)$ we show that the functional is unbounded. We also provide the complete solution in the (significantly more) challenging critical regime in dimension $d=2$. |
| title | On the existence of optimizers for nonlinear time-frequency concentration problems: the Born--Jordan distribution |
| topic | Classical Analysis and ODEs Functional Analysis 81S30, 49Q10, 42B10, 94A12 |
| url | https://arxiv.org/abs/2605.10535 |