Improved Adams-type inequalities and their extremals in dimension 2m

TL;DR

This paper establishes the existence of extremal functions for Adams-Moser-Trudinger inequalities in higher dimensions and extends these results to improved inequalities, using novel blow-up analysis and capacity estimates.

Contribution

It proves the existence of extremals for Adams-type inequalities in dimension 2m and introduces new techniques for capacity estimates and blow-up analysis.

Findings

Existence of extremal functions for Adams-Moser-Trudinger inequalities.

Extension to improved inequalities of Adimurthi-Druet type.

Development of new capacity estimate techniques on annular regions.

Abstract

In this paper we prove the existence of extremal functions for the Adams-Moser-Trudinger inequality on the Sobolev space $H^{m}(\Omega)$, where $\Omega$ is any bounded, smooth, open subset of $\mathbb{R}^{2m}$, $m\ge 1$. Moreover, we extend this result to improved versions of Adams' inequality of Adimurthi-Druet type. Our strategy is based on blow-up analysis for sequences of subcritical extremals and introduces several new techniques and constructions. The most important one is a new procedure for obtaining capacity-type estimates on annular regions.