Sharp Hardy-Littlewood-Sobolev inequality on the upper half space

Jingbo Dou; Meijun Zhu

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arXiv · 1309.2341

Sharp Hardy-Littlewood-Sobolev inequality on the upper half space

Jingbo Dou·Meijun Zhu

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Abstract

There are at least two directions concerning the extension of classical sharp Hardy-Littlewood-Sobolev inequality: (1) Extending the sharp inequality on general manifolds; (2) Extending it for the negative exponent $λ=n-α$ (that is for the case of $α>n$). In this paper we confirm the possibility for the extension along the first direction by establishing the sharp Hardy-Littlewood-Sobolev inequality on the upper half space (which is conformally equivalent to a ball). The existences of extremal functions are obtained; And for certain range of the exponent, we classify all extremal functions via the method of moving sphere.

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Jingbo Dou, Meijun Zhu. 2013-09-09. Sharp Hardy-Littlewood-Sobolev inequality on the upper half space. https://arxiv.org/abs/1309.2341

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