Trudinger-Moser and Hardy-Trudinger-Moser inequalities for the Aharonov-Bohm Magnetic field

Abstract

The main results of this paper concern sharp constant of the Trudinger-Moser inequality in ${\mathbb{R}}^2$ for Aharonov-Bohm magnetic fields. This is a borderline case of the Hardy type inequalities for Aharonov-Bohm magnetic fields in ${\mathbb{R}}^2$ studied by A. Laptev and T. Weidl. As an application, we obtain the exact asymptotic estimates on best constants of magnetic Hardy-Sobolev inequalities. In order to achieve our goal, we introduce a new operator $T_a$ on the unit circle ${\mathbb{S}}^1$ and give the asymptotic estimates of the heat kernel $e^{t{T}_a}$ via the Poisson summation formula. Finally, we show that such Trudinger-Moser inequalities in the unit ball ${\mathbb{B}}^2$ can be improved via subtraction of an additional Hardy term to derive a Hardy-Trudinger-Moser inequality.

25 pages