The Gagliardo-Nirenberg inequality on metric measure spaces

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 $n$ $(n\ge 2)$, 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.

18 pages. For the case of Hardy type inequalities, similar results have been shown in [Feng Du, Jing Mao, Qiao-Ling Wang and Chuan-Xi Wu, The Hardy type inequality on metric measure spaces, submitted]. arXiv admin note: text overlap with arXiv:1409.5741