On the existence of optimizers for nonlinear time-frequency concentration problems: the Born-Jordan distribution
Abstract
We study the Lp 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)= 2d/(d−2) for d≥2 (with the understanding that p*(2)=∞). In particular, for subcritical exponents 1≤ 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. For the critical case in dimension d=2 and the supercritical case in dimension d>2 we also show that L2 functions whose Born-Jordan distribution does not belong to Lp are generic in the Baire sense. For the critical exponent in dimension d>2, for which Wang has recently shown that the supremum is finite, we prove that the functional is not sequentially weakly upper semicontinuous, showing that the still open question of attainment is nontrivial.
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BibTeX
@misc{stra2026existence,
title = {{On the existence of optimizers for nonlinear time-frequency concentration problems: the Born-Jordan distribution}},
author = {Federico Stra and Erling A. T. Svela and S. Ivan Trapasso},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2605.10535},
eprint = {2605.10535},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.10535},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the existence of optimizers for nonlinear time-frequency concentration problems: the Born-Jordan distribution
AU - Federico Stra
AU - Erling A. T. Svela
AU - S. Ivan Trapasso
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2605.10535
UR - https://arxiv.org/abs/2605.10535
AB - 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)= 2d/(d−2) for d≥2 (with the understanding that p_*(2)=∞). In particular, for subcritical exponents 1≤ 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. For the critical case in dimension d=2 and the supercritical case in dimension d>2 we also show that L^2 functions whose Born-Jordan distribution does not belong to L^p are generic in the Baire sense. For the critical exponent in dimension d>2, for which Wang has recently shown that the supremum is finite, we prove that the functional is not sequentially weakly upper semicontinuous, showing that the still open question of attainment is nontrivial.
ER -