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

TL;DR

This paper investigates the existence of optimizers for the $L^p$ concentration problem related to the Born--Jordan distribution in higher dimensions, identifying critical thresholds and providing solutions in complex regimes.

Contribution

It extends the analysis of $L^p$ concentration problems for the Born--Jordan distribution to higher dimensions and critical cases, establishing existence and unboundedness results.

Findings

Existence of optimizers depends on the exponent $p$ with a critical threshold at $p_*(d)= rac{2d}{d-2}$ for $d eq2$.

For $1 extless p extless p_*(d)$, the supremum is finite and attained.

For $p extgreater p_*(d)$, the functional is unbounded.

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 $1\leq p<p_*(d)$ 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 $d=2$.