A Hardy-Moser-Trudinger inequality

In this paper we obtain an inequality on the unit disk B in R-2, which improves the classical Moser-Trudinger inequality and the classical Hardy inequality at the same time. Namely, there exists a constant C-0 > 0 such that integral(B) e(4 pi u2/H(u)) dx <= C-0 < infinity, for all u is an element of C-0(infinity) (B) \{0}, where H(u) := integral(B) vertical bar del u vertical bar(2) dx - integral(B) u(2)/(1 - vertical bar x vertical bar(2))(2)dx. This inequality is a two-dimensional analog of the Hardy-Sobolev-Maz'ya 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. (C) 2011 Elsevier Inc. All rights reserved.