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

The main purpose of our paper is to prove sharp Adams-type inequalities in unbounded 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} where such inequalities are only established for even integer $m$. Our inequalities are 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.