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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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On the dependence of analytic solutions of partial differential equations on the right-hand side
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by Siegfried Momm
Trans. Amer. Math. Soc. 345 (1994), 729-752
DOI: https://doi.org/10.1090/S0002-9947-1994-1254192-7
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Abstract:

Given a nonzero polynomial $P(z) = \sum \nolimits _{|\alpha | \leq m} {{a_\alpha }{z^\alpha }}$ on ${\mathbb {C}^N}$, Martineau proved in the 1960s that for each convex domain G of ${\mathbb {C}^N}$ the partial differential operator $P(D)f = \sum \nolimits _{|\alpha | \leq m} {{a_\alpha }{f^{(\alpha )}}}$ acting on the Fréchet space $A(G)$ of all analytic functions on G is surjective. In the present paper it is investigated whether solutions f of the equation $P(D)f = g$ can be chosen as $f = R(g)$ with a continuous linear operator $R:A(G) \to A(G)$. For bounded G we give a necessary and sufficient condition for the existence of such an R.
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Bibliographic Information
  • © Copyright 1994 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 345 (1994), 729-752
  • MSC: Primary 46E10; Secondary 32F05, 35B30, 35E10
  • DOI: https://doi.org/10.1090/S0002-9947-1994-1254192-7
  • MathSciNet review: 1254192