The geometry of analytic varieties satisfying the local Phragmén-Lindelöf condition and a geometric characterization of the partial differential operators that are surjective on {𝒜}({ℝ}⁴)

Braun, Rüdiger; Meise, Reinhold; Taylor, B.

doi:10.1090/S0002-9947-03-03448-2

The geometry of analytic varieties satisfying the local Phragmén-Lindelöf condition and a geometric characterization of the partial differential operators that are surjective on $\mathcal \{A\}(\mathbb \{R\}^4)$
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by Rüdiger W. Braun, Reinhold Meise and B. A. Taylor

Trans. Amer. Math. Soc. 356 (2004), 1315-1383

DOI: https://doi.org/10.1090/S0002-9947-03-03448-2

Published electronically: October 21, 2003

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Abstract:

The local Phragmén-Lindelöf condition for analytic subvarieties of $\mathbb {C}^n$ at real points plays a crucial role in complex analysis and in the theory of constant coefficient partial differential operators, as Hörmander has shown. Here, necessary geometric conditions for this Phragmén-Lindelöf condition are derived. They are shown to be sufficient in the case of curves in arbitrary dimension and of surfaces in $\mathbb {C}^3$. The latter result leads to a geometric characterization of those constant coefficient partial differential operators which are surjective on the space of all real analytic functions on $\mathbb {R}^4$.

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