A new approach to sharp Moser–Trudinger and Adams type inequalities: A rearrangement-free argument

Abstract

The main purpose of this paper is two-fold. On the one hand, we will develop a new approach to establish sharp singular Moser–Trudinger and Adams type inequalities in unbounded domains of Euclidean spaces without using the standard symmetrization. On the other hand, we will prove the sharp singular Adams type inequality on high order Sobolev spaces Wm,nm(Rn) of arbitrary integer order m (Theorem 1.1) which improves the results of Ruf and Sani (2013) [48] where sharp Adams inequalities were established for even m and those of the authors (Lam and Lu, 2012 [28,29]) for odd m but with different and more restricted norms. We first establish the sharp local singular Adams inequality on domains Ω in Rn of finite measure (Theorem 1.4). We take a perspective that any function in the high order Sobolev spaces Wm,nm(Rn) can be represented as a Bessel potential. Thus, we can fully use the tools from harmonic analysis and the kernel properties of the polyharmonic operators (τI−Δ)m2. Once we have established this sharp local Adams inequality, then we can adapt the rearrangement-free method we will develop in this paper to derive a global sharp Adams inequality from a local one. Our argument substantially simplifies those in Ruf and Sani (2013) [48] and Lam and Lu (2012) [28,29] and avoids the use of rather deep and complicated comparison principle of solutions to polyharmonic operators used in Ruf and Sani (2013) [48], Lam and Lu (2012) [28,29]. Moreover, our theorem holds on Sobolev spaces Wα,nα(Rn) of any positive fractional order α<n (Theorems 1.6 and 1.7). As consequences, we can also establish sharp Adams inequalities with less restrictions on the Sobolev norms (see Theorems 1.2, 1.3 and 1.5). Our approach is surprisingly simple and general and can be easily applied to scenarios such as Riemannian and sub-Riemannian manifolds where symmetrization argument does not work (see, e.g., on the Heisenberg group Lam and Lu (2012) [30]).