Abstract
Abstract In this paper, we establish the following sharp Adams inequality with exact growth condition in the entire space ℝn (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 < βn, there exists a constant C(α, n) > 0 such that for all satisfying , we have . Moreover, if α ≥ βn then the inequality cannot hold with a uniform constant C(α, n).
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