A Hardy–Moser–Trudinger inequality

In this paper we obtain an inequality on the unit disk B   in R2R2, which improves the classical Moser–Trudinger inequality and the classical Hardy inequality at the same time. Namely, there exists a constant C0>0C0>0 such that∫Be4πu2H(u)dx⩽C0<∞,∀u∈C0∞(B)∖{0}, whereH(u):=∫B|∇u|2dx−∫Bu2(1−|x|2)2dx. This inequality is a two-dimensional analog of the Hardy–Sobolev–Mazʼya inequality in higher dimensions, which has been intensively studied recently. We also prove that the supremum is achieved in a suitable function space, which is an analog of the celebrated result of Carleson–Chang for the Moser–Trudinger inequality.