Moser-Trudinger and logarithmic HLS inequalities for systems

Abstract

We prove several optimal Moser-Trudinger and logarithmic Hardy-Littlewood-Sobolev inequalities for systems in two dimensions. These include inequalities on the sphere $S2$, on a bounded domain $\Omega \subset \mathbb{R}2$ and on all of $\mathbb{R}2$. In some cases we also address the question of existence of minimizers.