Construction of Sobolev spaces of fractional order with sub-Riemannian vector fields. (English) Zbl 1143.46021

Authors’ abstract: Given a smooth family of vector fields satisfying Chow-Hörmander’s condition of step 2 and a regularity assumption, we prove that the Sobolev spaces of fractional order constructed by the standard functional analysis can actually be “computed” with a simple formula involving the sub-Riemannian distance. Our approach relies on a microlocal analysis of translation operators in an anisotropic context. It also involves classical estimates of the heat kernel associated to the sub-elliptic Laplacian.


46E40 Spaces of vector- and operator-valued functions
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E50 Spaces of differentiable or holomorphic functions on infinite-dimensional spaces
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
26A33 Fractional derivatives and integrals
47G30 Pseudodifferential operators
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