An improvement for the sharp Adams inequalities in bounded domains and whole space Rⁿ

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We prove an improvement for the sharp Adams inequality in Wm, nm₀(Ω) where Ω is a bounded domain in Rⁿ inspired by Lions Concentration--Compactness principle for the sharp Moser--Trudinger inequality. Our method gives an alternative approach to a Concentration--Compactness principle in Wm, nm₀(Ω) recently established by do Ó 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 Rⁿ to establish an improvement for the sharp Adams inequalities in Wm, nm(Rⁿ) 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 Ó, de Souza, de Medeiros and Severo. Our method is a further development for the method of Cerny, Cianchi and Hencl combining with some estimates for the decreasing rearrangement of a function in terms of the one of its higher order derivatives.