Project supported by the National Natural Science Foundation of China (No.10271107), the 973 Project of the Ministry of Science and Technology of China (No.G1999075105) and the Zhejiang Provincial Natural Science Foundation of China (No.RC97017).
Author(s):
MENG WANG Department of Mathematics, Zhejiang University, Hangzhou 310027, China School of Mathematical Sciences, Fudan University, Shanghai 200433, China
History:
Received October 10, 2003
Abstract:
Let M be an n dimensional complete Riemannian manifold satisfying the doubling volume property and an on-diagonal heat kernel estimate. The necessary-sufficient condition for the Sobolev inequality ‖f‖q ≤ Cn,,ν,p,q(‖∇ f‖p + ‖f‖p) (2 ≤ p < q < ∞) is given.
