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We prove an improvement for the sharp Adams inequality in $W^{m,\fracnm}_0(\Omega)$ where $\Omega$ is a bounded domain in $\mathbb{R}^n$ inspired byLions Concentration--Compactness principle for the sharp Moser--Trudingerinequality. Our method gives an alternative approach to aConcentration--Compactness principle in $W^{m,\frac nm}_0(\Omega)$ recentlyestablished by do \'O and Macedo. Moreover, when $m$ is odd, we obtain animprovement for their result by finding the best exponent in this principle.Our approach also is successfully applied to whole space $\mathbb{R}^n$ toestablish an improvement for the sharp Adams inequalities in $W^{m,\fracnm}(\mathbb{R}^n)$ due to Ruf, Sani, Lam, Lu, Fontana and Morpurgo. This typeof improvement is still unknown, in general, except the special case $m=1$ dueto do \'O, de Souza, de Medeiros and Severo. Our method is a furtherdevelopment for the method of $\check{\rm C}$erny, Cianchi and Hencl combiningwith some estimates for the decreasing rearrangement of a function in terms ofthe one of its higher order derivatives.

An improvement for the sharp Adams inequalities in bounded domains and whole space $\mathbb{R}^n$