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

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We study the notion of fractional $L^p$-differentiability of order $s∈(0,1)$ 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 $W^{1,p} ⊂ W^{s,q}$, where q is a suitable exponent greater than p.
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  • Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40127 Bologna, Italy
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