An improved Hardy-Trudinger-Moser inequality

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Let B be the unit disc in R², H be the completion of C₀∞(B) under the norm uH=(∫B|∇ u|²dx-∫Bu²/(1-|x|²)²dx)¹/²,∀ u∈ C₀∞(B). Denote λ₁(B)=u∈ H,u₂=1uH², where ·₂ stands for the L²(B)-norm. Using blow-up analysis, we prove that for any α, 0≤ α<λ₁(B), u∈H,uH²-αu₂²≤ 1∫B e⁴πᵘ²dx<+∞, and that the above supremum can be attained by some function u∈ H with uH²-αu₂²= 1. This improves an earlier result of G. Wang and D. Ye [28].