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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

##### MSC:
 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
##### Keywords:
hypoelliptic operator; Green kernel; vector fields
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##### References:
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