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Fundamental solutions and geometry of the sum of squares of vector fields. (English) Zbl 0582.58004
Let \(X_ 1,...,X_ m\) be smooth vector fields on a smooth compact manifold M, endowed with a smooth positive measure \(\mu\). It is assumed that taking a sufficient number of commutators of \(X_ 1,...,X_ m\) they span the tangent of M at every point (Hörmander’s condition). Then the operators like \(L=\sum^{m}_{j=1}X^ 2_ j+\sum^{m}_{i,j=1}f_{ij}[[ X_ i,X_ j]]+\sum^{m}_{j=1}f_ jX_ j+f_ 0\) with smooth functions \(f_{ij}\), \(f_ j\), \(f_{ij}\) reals, are hypoelliptic and the solutions of \(Lu=f\) can be represented in the form \(u(x)=\int G(x,y)f(y)d\mu (y)\). In this paper estimates for the Green kernel G in terms of a distance \(d_ L\), canonically attached to L are found. Also an estimate for the fundamental solution K of \(\partial /\partial t-L\) is found.
Reviewer: S.Dimiev

58A30 Vector distributions (subbundles of the tangent bundles)
65H10 Numerical computation of solutions to systems of equations
58J99 Partial differential equations on manifolds; differential operators
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