A Singular Trudinger-Moser Inequality in Hyperbolic Space

In this paper, we establish a singular Trudinger-Moser inequality for the whole hyperbolic space $$H^n: sup_{u∈W^{1,n}(H^n),∫_{H^n}|∇_H^nu|^ndμ ≤ 1}∫_{H^n}frac{e^{α|u|frac{n}{n-1}}-Σ^{n-2}_{k=0}frac{α^k|u|^frac{nk}{n-1}}{k!}}{ρ^β}dμ‹∞ ⇔ frac{α}{α_n}+frac{β}{n} ≤ 1,$$ where α>0,α ∈ [0,n), ρ and dμ are the distance function and volume element of $H^n$ respectively.