Abstract
In this paper we study the question when a linear partial differential operator P(D) with constant coefficients admits a continuous linear right inverse in the space A(ℝn) of real analytic functions on ℝn (or, more generally, in A(Ω) where Ω is a open subset of ℝn). To obtain a necessary condition, we investigate when P(D) admits solvability “with real analytic parameter” in A(Ω) and solve it completely for convex Ω, using a different approach from the one used in Domański (Funct. Approx. 44:79–109, 2011). To obtain a sufficient condition, we show that the global real analytic Cauchy problem is solvable if and only if the principal part of P(D) is hyperbolic. In this way we get a complete solution of our main problem for A(ℝ2) and, in the homogeneous case, for A(Ω) where Ω is the open unit ball in ℝn.
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Dedicated to the memory of Leon Ehrenpreis.
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Vogt, D. (2012). Right Inverses for P(D) in Spaces of Real Analytic Functions. In: Sabadini, I., Struppa, D. (eds) The Mathematical Legacy of Leon Ehrenpreis. Springer Proceedings in Mathematics, vol 16. Springer, Milano. https://doi.org/10.1007/978-88-470-1947-8_22
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DOI: https://doi.org/10.1007/978-88-470-1947-8_22
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