Trudinger-Moser inequality with remainder terms

The paper gives the following improvement of the Trudinger-Moser inequality: [GRAPHICS] related to the Hardy-Sobolev-Mazya inequality in higher dimensions. We show (0.1) with psi(u) = integral(Omega) V(x)u(2) dx for a class of V > 0 that includes [GRAPHICS] 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 psi(u) = lambda parallel to u parallel to(2)(P), as well as give an analogous improvement for the Onofri-Beckner inequality for the unit disk (Beckner [6]).