A Hardy-Moser-Trudinger inequality

By Guofang Wang, D. Ye. 2010

90 citations recorded by the source corpus.

Abstract 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 ∫ B e 4 π u 2 H ( u ) d x ⩽ C 0 ∞ , ∀ u ∈ C 0 ∞ ( B ) ∖ { 0 } , where H ( u ) : = ∫ B | ∇ u | 2 d x − ∫ B u 2 ( 1 − | x | 2 ) 2 d x . This inequality is a two-dimensional analog of the Hardy–Sobolev–Mazʼya 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f283cc958fff9d8305e254781289f1448ef4586d

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In Applied Mathematics

1. Nguyen Huong Lam
2. G. Lu
3. Y. Yang
4. Qiaohua Yang
5. Guozhen Lu
6. Xiaobao Zhu
7. Caifeng Zhang
8. Hanli Tang

In Computational Theory and Mathematics

1. G. Lu
2. Nguyen Huong Lam
3. Y. Yang
4. Xiaobao Zhu
5. Caifeng Zhang
6. Guozhen Lu
7. Qiaohua Yang
8. Federica Sani

In Mathematical Physics

1. G. Lu
2. Nguyen Huong Lam
3. Xiaobao Zhu
4. Y. Yang
5. Qiaohua Yang
6. Guozhen Lu
7. Hanli Tang
8. Caifeng Zhang

Citations in the corpus

• A sharp Trudinger–Moser inequality on any bounded and convex planar domain (2015, methodology): “Wang and Ye conjectured in [22]: Let Ω be a regular, bounded and convex domain in R(2).”
• Blow-up analysis concerning singular Trudinger–Moser inequalities in dimension two (2016, methodology): “Our method of proving Theorems 1-4 can be applied to establish singular versions of other kind of Trudinger-Moser inequalities, say the Hardy-Trudinger-Moser inequality [27, 34] and the Trudinger-Moser inequality involving the Gaussian curvature [33].”
• Extremal functions for Trudinger-Moser inequalities of Adimurthi-Druet type in dimension two (2015, methodology): “For earlier works involving this method, we refer the reader to [23, 40, 24, 26, 38].”
• Paneitz operators on hyperbolic spaces and high order Hardy-Sobolev-Maz'ya inequalities on half spaces (2019, background): “Furthermore, there holds some first order Trudinger-Moser type inequalities on hyperbolic spaces [42, 43] and HardyTrudinger-Moser inequality on H2 (see [49, 44, 58]) and Hardy-Adams inequality on H4 and Hn for all even dimensions n≥ 4 [45, 38].”
• Sharp singular Adams inequalities in high order Sobolev spaces (2011, background): “Very recently, Wang and Ye [56] proved an interesting Hardy-Moser-Trudinger inequality on the unit disk in R(2), which improves the classical Moser-Trudinger inequality and the classical Hardy inequality at the same time.”
• A new approach to sharp Moser–Trudinger and Adams type inequalities: A rearrangement-free argument (2013, background): “and in [55]: there exists a constant C0 > 0 such that ∫”