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Fractional Sobolev norms and structure of Carnot-Carathéodory balls for Hörmander vector fields

Daniele Morbidelli — 2000

Studia Mathematica

We study the notion of fractional $L^{p}$ -differentiability of order $s \in (0, 1)$ along vector fields satisfying the Hörmander condition on $ℝ^{n}$ . We prove a modified version of the celebrated structure theorem for the Carnot-Carathéodory balls originally due to Nagel, Stein and Wainger. This result enables us to demonstrate that different $W^{s, p}$ -norms are equivalent. We also prove a local embedding $W^{1, p} \subset W^{s, q}$ , where q is a suitable exponent greater than p.

Spazi frazionari di tipo Sobolev per campi vettoriali e operatori di evoluzione di tipo Kolmogorov-Fokker-Planck

Daniele Morbidelli — 1999

Bollettino dell'Unione Matematica Italiana

Balls defined by nonsmooth vector fields and the Poincaré inequality

Annamaria Montanari; Daniele Morbidelli — 2004

Annales de l’institut Fourier

We provide a structure theorem for Carnot-Carathéodory balls defined by a family of Lipschitz continuous vector fields. From this result a proof of Poincaré inequality follows.

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Currently displaying 1 – 3 of 3

Fractional Sobolev norms and structure of Carnot-Carathéodory balls for Hörmander vector fields

Spazi frazionari di tipo Sobolev per campi vettoriali e operatori di evoluzione di tipo Kolmogorov-Fokker-Planck

Balls defined by nonsmooth vector fields and the Poincaré inequality