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Let f be in the localized nonisotropic Sobolev space W 1,p loc (H n ) on the n-dimensional Heisenberg group H n = C n × R, where 1 ≤ p < Q and Q = 2n + 2 is the homogeneous dimension of H n . Suppose that the subelliptic gradient is gloablly L p integrable, i.e., Ê H n |∇ H n f | p du is finite. We prove a Poincaré inequality for f on the entire space H 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 (H n ) under the norm ofWe will also prove that the best constants and extremals for such Poincaré inequalities on H n are the same as those for Sobolev inequalities on H n . Using the results of Jerison and Lee on the sharp constant and extremals for L 2 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 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 H n .

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