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

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