Let be the hyperbolic space of dimension . By our previous work (Theorem 2.3 of (Yang (2012))), for any , there exists a constant depending only on and such that where , is the measure of the unit sphere in , and . In this note we shall improve the above mentioned inequality. Particularly, we show that, for any and any , the above mentioned inequality holds with the definition of replaced by . We solve this problem by gluing local uniform estimates.
"Trudinger-Moser Embedding on the Hyperbolic Space." Abstr. Appl. Anal. 2014 1 - 4, 2014. https://doi.org/10.1155/2014/908216