Trudinger-Moser Embedding on the Hyperbolic Space. (18th February 2014)

• Record Type: Journal Article
• Authors: Yang, Yunyan; Zhu, Xiaobao
• Other Names: Rossi Julio Academic Editor.
• Is Part Of: Abstract and applied analysis. Volume 2014(2014)
• Publication Date: 2014-02-18
• Subjects: Mathematical analysis -- Periodicals; Mathematical analysis; Applied Mathematics; Mathematical Analysis; Periodicals; 515.05
• DOI: 10.1155/2014/908216
• Languages: English
• ISSNs: 1085-3375

Abstract

Abstract : Let ( ℍ n, g ) be the hyperbolic space of dimension n . By our previous work (Theorem 2.3 of (Yang (2012))), for any 0 < α < α n, there exists a constant τ > 0 depending only on n and α such that s u p u ∈ W 1, n ( ℍ n ), ∥ u ∥ 1, τ ≤ 1 ∫ ℍ n ( e α u n / ( n - 1 ) - ∑ k = 0 n - 2 α k | u | n k / ( n - 1 ) / k ! ) d v g < ∞, where α n = n ω n - 1 1 / ( n - 1 ), ω n - 1 is the measure of the unit sphere in ℝ n, and u 1, τ = ∇ g u L n ( ℍ n ) + τ u L n ( ℍ n ) . In this note we shall improve the above mentioned inequality. Particularly, we show that, for any 0 < α < α n and any τ > 0, the above mentioned inequality holds with the definition of u 1, τ replaced by ( ∫ ℍ n ‍ ( | ∇ g u | n + τ | u | n ) d v g ) 1 / n . We solve this problem by gluing local uniform estimates.