Trudinger-Moser Embedding on the Hyperbolic Space

Let (H-n, g) be the hyperbolic space of dimension n. By our previous work (Theorem 2.3 of (Yang (2012))), for any 0 < alpha < alpha(n), there exists a constant tau > 0 depending only on n and alpha such that sup(u is an element of Wi,n(Hn),parallel to u parallel to 1,tau <= 1) integral(Hn)(e(alpha vertical bar u vertical bar n/n-1)) - Sigma(n-2)(k=0)alpha(k)vertical bar u vertical bar(nk/(n-1))/k!)dv(g) < infinity, where alpha(n) = n omega(1/(n-1))(n-1), omega(n-1) is the measure of the unit sphere in R-n, and parallel to u parallel to(1,tau) = parallel to del gu parallel to(Ln(Hn)) + tau parallel to u parallel to(Ln(Hn)). In this note we shall improve the above mentioned inequality. Particularly, we show that, for any 0 < alpha < alpha(n) and any tau > 0, the above mentioned inequality holds with the definition of parallel to u parallel to(1,tau) replaced by (integral(Hn)(vertical bar del(g)u vertical bar(n) + tau vertical bar u vertical bar(n))dv(g))(1/n). We solve this problem by gluing local uniform estimates.