Singular Trudinger–Moser inequalities for the Aharonov–Bohm magnetic field

Abstract

The first purpose of this paper is to establish the singular Trudinger–Moser inequality in R2 ${\mathbb{R}}^{2}$ for the Aharonov–Bohm magnetic fields. The second purpose is to derive the singular Hardy–Trudinger–Moser inequality in the unit ball B2 ${\mathbb{B}}^{2}$ with Aharonov–Bohm magnetic potential. Moreover, we will show the constant 4π(1−β2) $4\pi \left(1-\frac{\beta }{2}\right)$ is sharp in these two inequalities. The main skills include providing the asymptotic estimates of the related heat kernel and adapting the level set to derive a global Trudinger–Moser inequality from a local one. These results extend the inequalities established by Lu and Yang [Calc. Var. Partial Differ. Equ., 63 (2024)] into the weighted version.

Keywords

• singular trudinger–moser inequalities
• hardy–trudinger–moser inequality
• aharonov–bohm magnetic field
• sharp constant
• primary 35j20
• 46e35