Localizations of partial differential operators and surjectivity on real analytic functions

Volume 140 / 2000

Michael Langenbruch Studia Mathematica 140 (2000), 15-40 DOI: 10.4064/sm-140-1-15-40


Let P(D) be a partial differential operator with constant coefficients which is surjective on the space A(Ω) of real analytic functions on an open set $ Ω ⊂ ℝ^n$. Then P(D) admits shifted (generalized) elementary solutions which are real analytic on an arbitrary relatively compact open set ω ⊂ ⊂ Ω. This implies that any localization $P_{m,Θ}$ of the principal part $P_m$ is hyperbolic w.r.t. any normal vector N of ∂Ω which is noncharacteristic for $P_{m,Θ}$. Under additional assumptions $P_m$ must be locally hyperbolic.


  • Michael Langenbruch

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