Adams' inequality for bounded domains Ω ⊂ R 4 states that the supremum of Ω e 32π 2 u 2 dx over all functions u ∈ W 2, 2 0 (Ω) with Δu 2 ≤ 1 is bounded by a constant depending on Ω only. This bound becomes infinite for unbounded domains and in particular for R 4. We prove that if Δu 2 is replaced by a suitable norm, namely u := − Δu + u 2 , then the supremum of Ω (e 32π 2 u 2 − 1) dx over all functions u ∈ W 2, 2 0 (Ω) with u ≤ 1 is bounded by a constant independent of the domain Ω. Furthermore, we generalize this result to any W m, n m 0 (Ω) with Ω ⊆ R n and m an even integer less than n.

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Adams' inequality is an extension of the Trudinger‐Moser inequality to the case when the Sobolev space considered has more than one derivative. The goal of this paper is to give the optimal growth rate of the exponential‐type function in Adams' inequality when the problem is considered in the whole space ℝ4. © 2014 Wiley Periodicals, Inc.

We first investigate concentration and vanishing phenomena concerning Moser type inequalities in the whole plane which involve complete and reduced Sobolev norms. In particular we show that the critical Ruf inequality is equivalent to an improved version of the subcritical Adachi-Tanaka inequality which we prove to be attained. Then, we consider the limiting space D 1,2 (R 2 ), completion of smooth compactly supported functions with respect to the Dirichlet norm ∇ ⋅ 2 , and we prove an optimal Lorentz-Zygmund type inequality with explicit extremals and from which can be derived classical inequalities in H 1 (R 2 ) such as the Adachi-Tanaka inequality and a version of Ruf's inequality.

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