On the analyticity of solutions of sums of squares of vector fields

  • François Treves
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 69)


The note describes, in simple analytic and geometric terms, the global Poisson stratification of the characteristic variety Char L of a second-order linear differential operator −L = X12 + ... + Xr2, i.e., a sum-of-squares of real-analytic, real vector fields Xi on an analytic manifold Ω. It is conjectured that the leaves in the bicharacteristic foliation of each Poisson stratum of Char L propagate the analytic singularities of the solutions of the equation Lu = fCω. Closely related conjectures of necessary and sufficient conditions for local, germ and global analytic hypoellipticity, respectively, are stated. It is an open question whether the new conjecture regarding local analytic hypoellipticity is equivalent to that put forward by the author in earlier articles.

Key words

Stratification symplectic sums of squares of vector fields analytic hypoellipticity 


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Copyright information

© Birkhäuser Boston 2006

Authors and Affiliations

  • François Treves
    • 1
  1. 1.Mathematics DepartmentRutgers UniversityNew BrunswickUSA

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