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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 2020 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 PDF
Trans. Amer. Math. Soc. 345 (1994), 729-752 Request permission

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.
References
  • Eric Bedford, The operator $(dd^{c})^{n}$ on complex spaces, Seminar Pierre Lelong-Henri Skoda (Analysis), 1980/1981, and Colloquium at Wimereux, May 1981, Lecture Notes in Math., vol. 919, Springer, Berlin-New York, 1982, pp. 294–323. MR 658889
  • Eric Bedford and B. A. Taylor, The Dirichlet problem for a complex Monge-AmpĂšre equation, Invent. Math. 37 (1976), no. 1, 1–44. MR 445006, DOI 10.1007/BF01418826
  • Eric Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), no. 1-2, 1–40. MR 674165, DOI 10.1007/BF02392348
  • C. A. Berenstein and B. A. Taylor, Interpolation problems in $\textbf {C}^{n}$ with applications to harmonic analysis, J. Analyse Math. 38 (1980), 188–254. MR 600786, DOI 10.1007/BF03033881
  • Carlos A. Berenstein and B. A. Taylor, On the geometry of interpolating varieties, Seminar Pierre Lelong-Henri Skoda (Analysis), 1980/1981, and Colloquium at Wimereux, May 1981, Lecture Notes in Math., vol. 919, Springer, Berlin-New York, 1982, pp. 1–25. MR 658877
  • E. M. Chirka, Complex analytic sets, Mathematics and its Applications (Soviet Series), vol. 46, Kluwer Academic Publishers Group, Dordrecht, 1989. Translated from the Russian by R. A. M. Hoksbergen. MR 1111477, DOI 10.1007/978-94-009-2366-9
  • J.-P. Demailly, Scindage holomorphe d’un morphisme de fibrĂ©s vectoriels semi-positifs avec estimations $L^{2}$, Seminar Pierre Lelong-Henri Skoda (Analysis), 1980/1981, and Colloquium at Wimereux, May 1981, Lecture Notes in Math., vol. 919, Springer, Berlin-New York, 1982, pp. 77–107 (French). MR 658880
  • Sönke Hansen, On the “fundamental principle” of L. Ehrenpreis, Partial differential equations (Warsaw, 1978) Banach Center Publ., vol. 10, PWN, Warsaw, 1983, pp. 185–201. MR 737222
  • Lars Hörmander, On the range of convolution operators, Ann. of Math. (2) 76 (1962), 148–170. MR 141984, DOI 10.2307/1970269
    • —, An introduction to complex analysis in several variables, Princeton Univ. Press, 1967.
  • Lars Hörmander, On the existence of real analytic solutions of partial differential equations with constant coefficients, Invent. Math. 21 (1973), 151–182. MR 336041, DOI 10.1007/BF01390194
    • —, The analysis of linear partial differential operators. II, Springer, 1983. G. M. Khenkin and V. S. Mityagin, Linear problems of complex analysis, Russian Math. Surveys 26 (1971), 99-164.
  • C. O. Kiselman, Existence and approximation theorems for solutions of complex analogues of boundary problems, Ark. Mat. 6 (1966), 193–207 (1966). MR 194721, DOI 10.1007/BF02592029
  • Christer O. Kiselman, The partial Legendre transformation for plurisubharmonic functions, Invent. Math. 49 (1978), no. 2, 137–148. MR 511187, DOI 10.1007/BF01403083
  • Maciej Klimek, Pluripotential theory, London Mathematical Society Monographs. New Series, vol. 6, The Clarendon Press, Oxford University Press, New York, 1991. Oxford Science Publications. MR 1150978
  • Michael Langenbruch, Solution operators for partial differential equations in weighted Gevrey spaces, Michigan Math. J. 37 (1990), no. 1, 3–24. MR 1042511, DOI 10.1307/mmj/1029004063
  • Bernard Malgrange, Existence et approximation des solutions des Ă©quations aux dĂ©rivĂ©es partielles et des Ă©quations de convolution, Ann. Inst. Fourier (Grenoble) 6 (1955/56), 271–355 (French). MR 86990
  • AndrĂ© Martineau, Indicatrices de croissance des fonctions entiĂšres de $N$-variables. Corrections et complĂ©ments, Invent. Math. 3 (1967), 16–19 (French). MR 220966, DOI 10.1007/BF01425488
  • Reinhold Meise and B. Alan Taylor, Each nonzero convolution operator on the entire functions admits a continuous linear right inverse, Math. Z. 197 (1988), no. 1, 139–152. MR 917855, DOI 10.1007/BF01161635
  • R. Meise, B. A. Taylor, and D. Vogt, Characterization of the linear partial differential operators with constant coefficients that admit a continuous linear right inverse, Ann. Inst. Fourier (Grenoble) 40 (1990), no. 3, 619–655 (English, with French summary). MR 1091835
    • —, Equivalence of analytic and plurisubharmonic PhragmĂ©n-Lindelöf principles on algebraic varieties, Proc. Sympos. Pure Math., vol. 52, Part 3, Amer. Math. Soc., Providence, RI, 1991, pp. 287-308.
  • Reinhold Meise and Dietmar Vogt, EinfĂŒhrung in die Funktionalanalysis, Vieweg Studium: Aufbaukurs Mathematik [Vieweg Studies: Mathematics Course], vol. 62, Friedr. Vieweg & Sohn, Braunschweig, 1992 (German). MR 1195130, DOI 10.1007/978-3-322-80310-8
  • Siegfried Momm, Convex univalent functions and continuous linear right inverses, J. Funct. Anal. 103 (1992), no. 1, 85–103. MR 1144684, DOI 10.1016/0022-1236(92)90136-7
    • —, A PhragmĂ©n-Lindelöf theorem for plurisubharmonic functions on cones in ${\mathbb {C}^N}$, Indiana Univ. Math. J. 41 (1992), 861-867.
  • Siegfried Momm, A division problem in the space of entire functions of exponential type, Ark. Mat. 32 (1994), no. 1, 213–236. MR 1277926, DOI 10.1007/BF02559529
  • Siegfried Momm, Boundary behavior of extremal plurisubharmonic functions, Acta Math. 172 (1994), no. 1, 51–75. MR 1263997, DOI 10.1007/BF02392790
  • Siegfried Momm, A critical growth rate for the pluricomplex Green function, Duke Math. J. 72 (1993), no. 2, 487–502. MR 1248682, DOI 10.1215/S0012-7094-93-07218-3
  • D. E. Papush and A. M. RussakovskiÄ­, Interpolation on plane sets in $\textbf {C}^2$, Ann. Fac. Sci. Toulouse Math. (6) 1 (1992), no. 3, 337–362 (English, with English and French summaries). MR 1225664
  • Markus Poppenberg and Dietmar Vogt, A tame splitting theorem for exact sequences of FrĂ©chet spaces, Math. Z. 219 (1995), no. 1, 141–161. MR 1340854, DOI 10.1007/BF02572355
    • A. Sadullaev, An estimate for polynomials on analytic sets, Math. USSR-Izv. 20 (1983), 493-502.
  • JĂłzef Siciak, Extremal plurisubharmonic functions in $\textbf {C}^{n}$, Ann. Polon. Math. 39 (1981), 175–211. MR 617459, DOI 10.4064/ap-39-1-175-211
    • F. TrĂšves, Linear partial differential equations with constant coefficients, Gordon and Breach, 1966.
  • Dietmar Vogt, Eine Charakterisierung der PotenzreihenrĂ€ume von endlichem Typ und ihre Folgerungen, Manuscripta Math. 37 (1982), no. 3, 269–301 (German, with English summary). MR 657522, DOI 10.1007/BF01166224
    • —, Operators between FrĂ©chet spaces, preprint.
  • Max-Josef Wagner, QuotientenrĂ€ume von stabilen PotenzreihenrĂ€umen endlichen Typs, Manuscripta Math. 31 (1980), no. 1-3, 97–109 (German, with English summary). MR 576492, DOI 10.1007/BF01303269
  • V. P. Zaharjuta, Extremal plurisubharmonic functions, Hilbert scales, and the isomorphism of spaces of analytic functions of several variables. I, II, Teor. FunkciÄ­ Funkcional. Anal. i PriloĆŸen. Vyp. 19 (1974), 133–157, 161; ibid. Vyp. 21 (1974), 65–83, 127 (Russian). MR 0447632
  • Ahmed ZĂ©riahi, Fonction de Green pluricomplexe Ă  pĂŽle Ă  l’infini sur un espace de Stein parabolique et applications, Math. Scand. 69 (1991), no. 1, 89–126 (French). MR 1143476, DOI 10.7146/math.scand.a-12371
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Additional 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