Global Poincaré Inequalities on the Heisenberg Group and Applications

et f be in the localized nonisotropic Sobolev space W1,ploc(Hn) on the n-dimensional Heisenberg group .n = .n . ., where 1 = p < Q and Q = 2n + 2 is the homogeneous dimension of .n. Suppose that the subelliptic gradient is gloablly L p integrable, i.e., .Hn|.Hnf|pdu is finite. We prove a Poincaré inequality for f on the entire space .n. Using this inequality we prove that the function f subtracting a certain constant is in the nonisotropic Sobolev space formed by the completion of C.0(Hn) under the norm of (.Hn|f|QpQ.p)Q.pQp+(.Hn|.Hnf|p)1p. We will also prove that the best constants and extremals for such Poincaré inequalities on .n are the same as those for Sobolev inequalities on .n. Using the results of Jerison and Lee on the sharp constant and extremals for L 2 to L2QQ.2 Sobolev inequality on the Heisenberg group, we thus arrive at the explicit best constant for the aforementioned Poincaré inequality on .n when p = 2. We also derive the lower bound of the best constants for local Poincaré inequalities over metric balls on the Heisenberg group .n