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The main purpose of our paper is to prove sharp Adams-type inequalities inunbounded domains of $\mathbb{R}^{n}$ for the Sobolev space$W^{m,\frac{n}{m}}\left(\mathbb{R} ^{n}\right)$ for any positive integer $m$less than $n$. Our results complement those of Ruf and Sani \cite{RS} wheresuch inequalities are only established for even integer $m$. Our inequalitiesare also a generalization of the Adams-type inequalities in the special case$n=2m=4$ proved in \cite{Y} and stronger than those in \cite{RS} when $n=2m$for all positive integer $m$ by using different Sobolev norms.

Sharp Adams type inequalities in Sobolev spaces $W^{m,\frac{n}{m}}(\mathbb{R}^{n})$ for arbitrary integer $m$