Trudinger-Moser Embedding on the Hyperbolic Space

Abstract

Let (ℍn,g) be the hyperbolic space of dimension n. By our previous work (Theorem 2.3 of (Yang (2012))), for any 00 depending only on n and α such that supu∈W1,n(ℍn),∥u∥1,τ≤1∫ℍn(eαun/(n-1)-∑k=0n-2αk|u|nk/(n-1)/k!)dvg0, the above mentioned inequality holds with the definition of u1,τ replaced by (∫ℍn‍(|∇gu|n+τ|u|n)dvg)1/n. We solve this problem by gluing local uniform estimates.