Global Poincaré Inequalities on the Heisenberg Group and Applications
نویسندگان
چکیده
Let f be in the localized nonisotropic Sobolev space W 1,p loc (H ) on the n-dimensional Heisenberg group H = C × R, where 1 ≤ p < Q and Q = 2n + 2 is the homogeneous dimension of H. Suppose that the subelliptic gradient is gloablly L integrable, i.e., Hn |∇Hnf |pdu is finite. We prove a Poincaré inequality for f on the entire space H. 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 (H ) under the norm of Hn |f | Qp Q−p Q−p Qp + Hn |∇Hnf | 1 p . We will also prove that the best constants and extremals for such Poincaré inequalities on H are the same as those for Sobolev inequalities on H. Using the results of Jerison and Lee on the sharp constant and extremals for L to L 2Q Q−2 Sobolev inequality on the Heisenberg group, we thus arrive at the explicit best constant for the aforementioned Poincaré inequality on H when p = 2. We also derive the lower bound of the best constants for local Poincaré inequalities over metric balls on the Heisenberg group H.
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