A sharpened form of Adams-type inequalities on higher-order Sobolev spaces W^{m, n/m} (ℝⁿ): A simple approach

In this paper, we develop an extremely simple method to establish the sharpened Adams-n type inequalities on higher-order Sobolev spaces W^{m, n/m} (ℝⁿ) in the entire space ℝⁿ, which can be stated as follows: Given (Equation presented) and the Adams sharp constant β_{n, m}. Then, (Equation presented) for any 0 < α < 1. Furthermore, we construct a proper test function sequence to derive the sharpness of the exponent α of the above Adams inequalities. Namely, we will show that if α ≥ 1, then the above supremum is infinite. Our argument avoids applying the complicated blow-up analysis often used in the literature to deal with such sharpened inequalities.