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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:
 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 2.2e+26 Nilpotent and solvable Lie groups
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