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Fractional Sobolev norms and structure of Carnot-Carathéodory balls for Hörmander vector fields. (English) Zbl 0981.46034
Author’s summary: We study the notion of fractional \(L^p\)-differentiability of order \(s\in(0,1)\) along vector fields satisfying the Hörmander condition on \(\mathbb R^n\). We prove a modified version of the celebrated structure theorem for the Carnot-Carathéodory balls originally due to A. Nagel, E. M. Stein and S. Wainger [Acta Math. 155, 103-147 (1985; Zbl 0578.32044)]. 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 \(p\).

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35J70 Degenerate elliptic equations
46B03 Isomorphic theory (including renorming) of Banach spaces
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