Sharp Adams-type inequalities in ℝⁿ

Adams’ inequality for bounded domains Ω ⊂ R 4 \Omega \subset \mathbb {R}^4 states that the supremum of ∫ Ω e 32 π 2 u 2 d x \int _{\Omega } e^{32 \pi ^2 u^2} \, dx over all functions u ∈ W 0 2 , 2 ( Ω ) u \in W_0^{2, \, 2}(\Omega ) with ‖ Δ u ‖ 2 ≤ 1 \| \Delta u\|_2 \leq 1 is bounded by a constant depending on Ω \Omega only. This bound becomes infinite for unbounded domains and in particular for R 4 \mathbb {R}^4 . We prove that if ‖ Δ u ‖ 2 \|\Delta u\|_2 is replaced by a suitable norm, namely ‖ u ‖ := ‖ − Δ u + u ‖ 2 \| u \|:=\|- \Delta u + u\|_2 , then the supremum of ∫ Ω ( e 32 π 2 u 2 − 1 ) d x \int _{\Omega } (e^{32 \pi ^2 u^2} -1) \, dx over all functions u ∈ W 0 2 , 2 ( Ω ) u \in W_0^{2, \, 2}(\Omega ) with ‖ u ‖ ≤ 1 \|u\| \leq 1 is bounded by a constant independent of the domain Ω \Omega . Furthermore, we generalize this result to any W 0 m , n m ( Ω ) W_0^{m, \, \frac n m}(\Omega ) with Ω ⊆ R n \Omega \subseteq \mathbb {R}^{n} and m m an even integer less than n n .