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arXiv · 2310.13173

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

Guozhen Lu·Qiaohua Yang
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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 $e^{tT_{a}}$ 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.

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Guozhen Lu, Qiaohua Yang. 2023-10-19. Trudinger-Moser and Hardy-Trudinger-Moser inequalities for the Aharonov-Bohm Magnetic field. https://arxiv.org/abs/2310.13173

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