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Global Poincaré inequalities on the Heisenberg group and applications. (English) Zbl 1131.46024
Let \({\mathbb H}^n={\mathbb C}^n\times{\mathbb R}\) be the Heisenberg group, whose structure is given by \[ (z,t)\cdot (z',t')=(z+z', t+t'+2\operatorname {Im}(z{\bar z}')), \] for any two points \((z,t), (z',t')\in {\mathbb H}^n\), \(Q=2n+2\) is the homogeneous dimension of \({\mathbb H}^n\). The main result of the paper is the following Theorem: Let \(1\leq p< Q\), \(\frac1{p^*}=\frac1p-\frac1Q\). Suppose that \(f\in L^p_{\text{loc}}({\mathbb H}^n)\) and \(| \nabla_{{\mathbb H}^n}f| \in L^p({\mathbb H}^n)\), where \(\nabla_{{\mathbb H}^n}f\) is the subelliptic gradient of \(f\). Then there is a unique finite constant \(f_\infty\) such that the following Poincaré inequality holds: \[ \| f-f_\infty\| _{p^*}\leq C(p, Q)\,\| \nabla_{{\mathbb H}^n}f\| _p, \] where \(C(p, Q)\) is a constant independent of \(f\). The authors also prove that the best constants and extremals for such Poincaré inequalities on \({\mathbb H}^n\) are the same as those for Sobolev inequalities on \({\mathbb H}^n\).

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
22E25 Nilpotent and solvable Lie groups
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