Let M be a -smooth n-dimensional manifold and ν₁, ..., νₙ be -smooth vector fields on M which span the tangent space at each point x ∈ M. The vector fields ν₁, ..., νₙ may have nonzero commutators. We construct a calculus of pseudodifferential operators (ψDOs) which act on sections of vector bundles over M and have symbols belonging to anisotropic analogues of the Hörmander classes , and apply it to semi-elliptic operators generated by ν₁,...,νₙ. The results obtained include the formula expressing the symbol of a ψDO in terms of its amplitude, the formula for the symbol of the adjoint ψDO, the theorem on composition of ψDOs, the L₂-boundedness of ψDOs with symbols from , 0 ≤ δ < ϱ ≤ 1, and the -boundedness, 1 < p < ∞, of ψDOs with symbols from . We prove that a semi-elliptic ψDO A is Fredholm if M is compact and obtain analogues of the well known “elliptic” results concerning the resolvent and complex powers of A and the exponential . We also prove an asymptotic formula for the spectral function of A with a remainder estimate and more precise, in particular two-term, asymptotic formulae for the Riesz means of the spectral function.
@book{bwmeta1.element.bwnjournal-rm-doi-10_4064-dm409-0-1, author = {E. Shargorodsky}, title = {Semi-elliptic operators generated by vector fields}, series = {GDML\_Books}, year = {2002}, zbl = {1018.58016}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-rm-doi-10_4064-dm409-0-1} }
E. Shargorodsky. Semi-elliptic operators generated by vector fields. GDML_Books (2002), http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-rm-doi-10_4064-dm409-0-1/