Semiclassical Moser--Trudinger type inequalities for orthonormal families in ${\mathbb{R}}^2$

Phuoc-Tai Nguyen, Tuan-Dat Tran

For $H^1({\mathbb{R}}^2)$-orthonormal families $\{u_n{\}}_{n=1}^N$ with one-body density $\rho ={\sum }_{n=1}^N|u_n{|}^2$, we prove the whole-space endpoint Moser--Trudinger inequality $$ \int…