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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
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