Trudinger-Moser Embedding on the Hyperbolic Space

Abstract EN

Let (ℍn,g) be the hyperbolic space of dimension n.

By our previous work (Theorem 2.3 of (Yang (2012))), for any 0<α<αn, there exists a constant τ>0 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!)dvg<∞, where αn=nωn-11/(n-1), ωn-1 is the measure of the unit sphere in ℝn, and u1,τ=∇guLn(ℍn)+τuLn(ℍn).

In this note we shall improve the above mentioned inequality.

Particularly, we show that, for any 0<α<αn and any τ>0, 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.