The Hardy type inequality on metric measure spaces
The Hardy type inequality on metric measure spaces
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发表时间：
2018
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通讯作者：
Feng Du;Jing Mao;Qiaoling Wang;Chuanxi Wu
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作者：
Feng Du;Jing Mao;Qiaoling Wang;Chuanxi Wu
.本文证明了：如果度量测度空间萨蒂斯体积加倍条件和具有相同指数n（n ≥ 3）的哈代型不等式，则它恰好具有n维体积增长.此外，还给出了这一事实的三个有趣的应用。第一个是证明了具有非负加权Ricci曲线的完备非紧光滑度量测度空间与同维欧氏空间等距，且在该度量空间上哈代型不等式成立且具有最佳常数。第二个是证明了如果一个完备的n维Finsler流形的n-Ricci曲率非负且满足具有最佳常数的哈代型不等式，则它的曲率恒为零。最后一个结果是一个有趣的刚性结果，即证明了：如果一个完备的n维Berwald空间的n-Ricci曲率非负，且满足具有最佳常数的哈代型不等式，则它与n维Minkowski空间等距.
. In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Hardy type inequality with the same exponent n ( n ≥ 3), then it has exactly the n -dimensional volume growth. Besides, three interesting applications of this fact have also been given. The first one is that we prove that complete noncompact smooth metric measure space with non-negative weighted Ricci curva- ture on which the Hardy type inequality holds with the best constant are isometric to the Euclidean space with the same dimension. The second one is that we show that if a complete n -dimensional Finsler manifold of nonnegative n -Ricci curvature satisfies the Hardy type inequality with the best constant, then its flag curvature is identically zero. The last one is an interesting rigidity result, that is, we prove that if a complete n -dimensional Berwald space of non-negative n -Ricci curvature satisfies the Hardy type inequality with the best constant, then it is isometric to the Minkowski space of dimension n .