Trudinger-Moser inequality with remainder terms

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The paper gives an improvement of the Trudinger-Moser inequality, in which the constraint set is defined not by the squared gradient norm, but with the squared gradient norm minus a remainder term of the weighted Lᵖ-type. This is a two-dimensional counterpart of the Hardy-Sobolev-Mazya inequality in higher dimensions, which is a similar refinement of the limiting Sobolev inequality. In particular, we generalize two known cases of remainder terms of potential type (i.e. weighted L²-terms) found by Adimurthi and Druet and by Wang and Ye. In addition, we prove the inequality with a Lᵖ-remainder, p>2, as well as give an analogous improvement for the Onofri-Beckner inequality.