On analytic microlocal hypoellipticity of linear partial differential operators of principal type

Abstract

It is shown that if P is a linear partial differential operator with analytic coefficients defined near a point x0 in Rn and if ξ0 in Rn − 0 is such that: the principal symbol pm(x, ξ) vanishes at (x0, ξ0), the differential of pm with respect to ξ is different from zero at (x0, ξ0), the Poisson bracket {pm, p̄m} is zero at (x0, ξ0) and the Poisson bracket {pm, {pm, p̄m}} is different from zero at (x0, ξ0), then P is analytic hypoelliptic at (x0, ξ0). It is also proved that P is analytic hypoelliptic under the assumption that the first non-vanishing repeated Poisson bracket of pm and p̄m is of odd length and under some additional hypothesis on the commutators of the Hamilton fields of Re pm and Im pm. © 1986, Taylor & Francis Group, LLC. All rights reserved.