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The sharp Poincaré inequality for free vector fields: An endpoint result. (English) Zbl 0860.35006
The main purpose of this paper is to show a Poincaré type inequality of the following form: $\left( {1 \over |B(r)|}\int_{B(r)} |f-f_{B(r)} |^q\right)^{1/q} \leq Cr \left({1 \over|B(r) |}\int_{B(r)} \left(\sum^m_{i=1} |X_if |\right)^p \right)^{1/p}$ for all $$f\in C^\infty (\overline {B(r)})$$ with $$q=pQ/(Q-p)$$, $$1<p<Q$$, where $$Q$$ is a positive integer, and $$X_1, \dots, X_m$$ are $$C^\infty$$ vector fields on $$\mathbb{R}^d$$ satisfying Hörmander’s condition. $$B(r)$$ denotes a metric ball of radius $$r$$ associated to the natural metric induced by the vector fields, and $f_{B(r)} = {1\over|B(r)|} \int_{B(r)}f.$

##### MSC:
 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 26D10 Inequalities involving derivatives and differential and integral operators 42B25 Maximal functions, Littlewood-Paley theory
##### Keywords:
Poincaré type inequality; Hörmander’s condition
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