Best Constants for Adams’ Inequalities with the Exact Growth Condition in ℝ n
Guozhen Lu, Hanli Tang, Maochun Zhu · Advanced Nonlinear Studies · 2015
Abstract In this paper, we establish the following sharp Adams inequality with exact growth condition in the entire space ℝ n (n ≥ 3): There exists a constant C(n) > 0 such that for all with , , where . This extends the main result in [27] when n = 4 to all dimensions n ≥ 3. A crucial technical lemma we need is Lemma 4.2 for all p > 1 (corresponding to the Adams inequality for all n ≥ 3) whose proof is quite involved. As an application, we obtain the best constant for Ozawa’s inequality of Adams type in the Sobolev space : For any α 0 such that for all satisfying , we have . Moreover, if α ≥ β n then the inequality cannot hold with a uniform constant C(α, n).