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Though there have been extensive works on best constants for Moser-Trudinger inequalities in Euclidean spaces, Heisenberg groups or compact Riemannian manifolds, much less is known for sharp constants for the Moser-Trudinger inequalities on hyperbolic spaces. Earlier works only include the sharp constant for the Moser-Trudinger inequality on the twodimensional hyperbolic disc. In this paper, we establish best constants for several types of Moser-Trudinger inequalities on high dimensional hyperbolic spaces ℍ n  (n ≥ 2). These include sharp constants for the Moser-Trudinger inequalities on both bounded and unbounded domains of the hyperbolic space ℍ n  (see Theorems 1.1 and 1.2), sharp constants for the singular Moser-Trudinger inequality on unbounded domains when we impose restrictions only on the gradient norms (Theorem 1.3) or on the full hyperbolic Sobolev norms (Theorem 1.4). Our results are surprisingly general and extend most results in Euclidean spaces to hyperbolic spaces of any dimension. In particular, we have used a rearrangement-free argument in the hyperbolic spaces to establish Theorems 1.3 and 1.4 where symmetrization argument does not work to prove such sharp singular Moser-Trudinger inequalities on the entire hyperbolic space.

Best Constants for Moser-Trudinger Inequalities on High Dimensional Hyperbolic Spaces