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arXiv · 1511.04696

The Gagliardo-Nirenberg inequality on metric measure spaces

Feng Du·Jing Mao·Qiaoling Wang·Chuanxi Wu
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Abstract

In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Gagliardo-Nirenberg inequality with the same exponent $n$ $(n\geq 2)$, then it has exactly the $n$-dimensional volume growth. Besides, two interesting applications have also been given. The one is that we show that if a complete $n$-dimensional Finsler manifold of nonnegative $n$-Ricci curvature satisfies the Gagliardo-Nirenberg inequality with the sharp constant, then its flag curvature is identically zero. The other one is that we give an alternative proof to Mao's main result in [23] for smooth metric measure spaces with nonnegative weighted Ricci curvature.

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Feng Du, Jing Mao, Qiaoling Wang, Chuanxi Wu. 2015-11-15. The Gagliardo-Nirenberg inequality on metric measure spaces. https://arxiv.org/abs/1511.04696

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