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

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 $p_{\ast }(d)=\frac{2d}{d-2}$ for $d\ge 2$ (with the understanding that $p_{\ast }(2)=\mathrm{\infty }$). In particular, for subcritical exponents $1\le p<p_{\ast }(d)$ we prove that the supremum is finite and is attained, whereas for supercritical exponents $p>p_{\ast }(d)$ we show that the functional is unbounded. We also provide the complete solution in the (significantly more) challenging critical regime in dimension $d=2$.