Sharp Adams-type inequalities in $\mathbb {R}^{n}$

Abstract:

Adams’ inequality for bounded domains $\Omega \subset \mathbb {R}^4$ states that the supremum of $\int _{\Omega } e^{32 \pi ^2 u^2} dx$ over all functions $u \in W_0^{2, 2}(\Omega )$ with $\| \Delta u\|_2 \leq 1$ is bounded by a constant depending on $\Omega$ only. This bound becomes infinite for unbounded domains and in particular for $\mathbb {R}^4$.

We prove that if $\|\Delta u\|_2$ is replaced by a suitable norm, namely $\| u \|:=\|- \Delta u + u\|_2$, then the supremum of $\int _{\Omega } (e^{32 \pi ^2 u^2} -1) dx$ over all functions $u \in W_0^{2, 2}(\Omega )$ with $\|u\| \leq 1$ is bounded by a constant independent of the domain $\Omega$.

Furthermore, we generalize this result to any $W_0^{m, \frac n m}(\Omega )$ with $\Omega \subseteq \mathbb {R}^{n}$ and $m$ an even integer less than $n$.