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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> ${\mathbb{R}}^{2}$ for the Aharonov–Bohm magnetic fields. The second purpose is to derive the singular Hardy–Trudinger–Moser inequality in the unit ball <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">B</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> ${\mathbb{B}}^{2}$ with Aharonov–Bohm magnetic potential. Moreover, we will show the constant <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mn>4</m:mn> <m:mi>π</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>−</m:mo> <m:mfrac> <m:mrow> <m:mi>β</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:mfrac> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> $4\pi \left(1-\frac{\beta }{2}\right)$ 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.