Chapter

Phase Space Analysis of Partial Differential Equations

Volume 69 of the series Progress in Nonlinear Differential Equations and Their Applications pp 315-329

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

  • François TrevesAffiliated withMathematics Department, Rutgers University

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Abstract

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 = X 1 2 + ... + X r 2 , i.e., a sum-of-squares of real-analytic, real vector fields X i 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 = fC ω. 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