On the existence of optimizers for nonlinear time-frequency concentration problems: the Born-Jordan distribution
Year: 2026
Type: preprint
Abstract: 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 $1\leq pp_*(d)$ 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.
Source: arXiv (Cornell University)
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7160955814