Trudinger-Moser inequality with remainder terms

TL;DR

This paper improves the Trudinger-Moser inequality by incorporating a remainder term involving weighted L^p norms, extending previous results and providing new inequalities in two-dimensional analysis.

Contribution

It introduces a generalized form of the Trudinger-Moser inequality with remainder terms of various types, including weighted L^p norms, and extends known cases to broader settings.

Findings

Generalized Trudinger-Moser inequality with weighted L^p remainder

Extended previous potential-type remainder results to new cases

Provided analogous improvements for the Onofri-Beckner inequality

Abstract

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 L^p-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 L^2-terms) found by Adimurthi and Druet and by Wang and Ye. In addition, we prove the inequality with a L^p-remainder, p>2, as well as give an analogous improvement for the Onofri-Beckner inequality.