Abstract
Results of Hörmander on evolution operators together with a characterization of the present authors [Ann. Inst. Fourier, Grenoble 40, 619–655 (1990)] are used to prove the following: Let P ∈ ℂ[z1,...,z n ] and denote by P m its principal part. If P − Pm is dominated by P m then the following assertions for the partial differential operators P(D) and P m(D) are equivalent for N ∈ S n−1:
-
(1)
P(D) and/or Pm D)admit a continuous linear right inverse on C ∞(H +(N)).
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(2)
P(D) admits a continuous linear right inverse on C ∞ (ℝn) and a fundamental solution E ∈ C ∞(ℝn) satisfying Supp \(E \subset \overline {H - (N)} \)
where H +(N) := {x ∈ ℝn :±(x,N) τ; 0}.
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Meise, R., Taylor, B.A. & Vogt, D. Right inverses for linear, constant coefficient partial differential operators on distributions over open half spaces. Arch. Math. 68, 311–319 (1997). https://doi.org/10.1007/s000130050061
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DOI: https://doi.org/10.1007/s000130050061