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$\Omega$isaboundeddomainin$$\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 $W_0^{m,\frac{n}{m}}(\mathrm{\Omega })$ 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 $\check{\mathrm{C}}$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.