In this paper, we prove the existence of an extremal function for the Adams–Moser–Trudinger inequality on the Sobolev space H0m(Ω), where Ω is any bounded, smooth, open subset of ℝ2m, m ≥ 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.

中文翻译：

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改进的 Adams 型不等式及其在 2m 维上的极值

在本文中，我们证明了 Sobolev 空间上 Adams-Moser-Trudinger 不等式的极值函数的存在H0米(Ω)， 在哪里Ω是任何有界的、光滑的、开的子集ℝ2米,米 ≥ 1. 此外，我们将此结果扩展到 Adimurthi-Druet 类型的 Adams 不等式的改进版本。我们的策略基于对亚临界极值序列的爆炸分析，并引入了几种新技术和结构。最重要的是获得环状区域容量类型估计的新程序。