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Construction of Sobolev spaces of fractional order with sub-riemannian vector fields

Sami Mustapha, François Vigneron (2007)

Annales de l’institut Fourier

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.

Cutting the loss of derivatives for solvability under condition ( Ψ )

Nicolas Lerner (2006)

Bulletin de la Société Mathématique de France

For a principal type pseudodifferential operator, we prove that condition  ( ψ ) implies local solvability with a loss of 3/2 derivatives. We use many elements of Dencker’s paper on the proof of the Nirenberg-Treves conjecture and we provide some improvements of the key energy estimates which allows us to cut the loss of derivatives from ϵ + 3 / 2 for any ϵ > 0 (Dencker’s most recent result) to 3/2 (the present paper). It is already known that condition  ( ψ ) doesnotimply local solvability with a loss of 1 derivative,...

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