Trudinger-Moser inequality with remainder terms

The paper gives the following improvement of the Trudinger-Moser inequality:(0.1)sup∫Ω|∇u|2dx−ψ(u)⩽1,u∈C0∞(Ω)∫Ωe4πu2dx<∞,Ω∈R2, related to the Hardy-Sobolev-Mazya inequality in higher dimensions. We show (0.1) with ψ(u)=∫ΩV(x)u2dx for a class of V>0 that includesV(r)=14r2(log1r)2max{log1r,1}, which refines two previously known cases of (0.1) proved by Adimurthi and Druet [2] and by Wang and Ye [23]. In addition, we verify (0.1) for ψ(u)=λ‖u‖p2, as well as give an analogous improvement for the Onofri-Beckner inequality for the unit disk (Beckner [6]).