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On the dependence of analytic solutions of partial differential equations on the right-hand side


Author: Siegfried Momm
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
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Abstract: Given a nonzero polynomial $ P(z) = \sum\nolimits_{\vert\alpha \vert \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_{\vert\alpha \vert \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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  • [1] E. Bedford, The operator $ {(d{d^c})^n}$ on complex spaces, Seminaire Lelong-Skoda, Lecture Notes in Math., vol. 919, Springer, 1982, pp. 294-323. MR 658889 (83i:32025)
  • [2] E. Bedford and B. A. Taylor, The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math. 37 (1976), 1-44. MR 0445006 (56:3351)
  • [3] -, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), 1-40. MR 674165 (84d:32024)
  • [4] C. A. Berenstein and B. A. Taylor, Interpolation problems in $ {\mathbb{C}^N}$ with application to harmonic analysis, J. Analyse Math. 38 (1980), 188-254. MR 600786 (82h:32002)
  • [5] -, On the geometry of interpolating varieties, Seminaire Lelong-Skoda, Lecture Notes in Math., vol. 919, Springer, 1982, pp. 1-25. MR 658877 (83k:32004)
  • [6] E. M. Chirka, Complex analytic sets, Kluwer, 1989. MR 1111477 (92b:32016)
  • [7] J.-P. Demailly, Scindage holomorphe d'un morphisme de fibrés vectoriels semi-positifs avec estimations $ {L^2}$, Seminaire Lelong-Skoda, Lecture Notes in Math., vol. 919, Springer, 1982, pp. 77-107. MR 658880 (83j:32019)
  • [8] S. Hansen, On the "fundamental principle" of L. Ehrenpreis, Partial Differential Equations, Banach Center Publ., vol. 10, PWN, 1983, pp. 185-201. MR 737222 (85h:35054)
  • [9] L. Hörmander, On the range of convolution operators, Ann. of Math. (2) 76 (1962), 148-170. MR 0141984 (25:5379)
  • [10] -, An introduction to complex analysis in several variables, Princeton Univ. Press, 1967.
  • [11] -, On the existence of real analytic solutions of partial differential equations with constant coefficients, Invent. Math. 21 (1973), 151-182. MR 0336041 (49:817)
  • [12] -, The analysis of linear partial differential operators. II, Springer, 1983.
  • [13] G. M. Khenkin and V. S. Mityagin, Linear problems of complex analysis, Russian Math. Surveys 26 (1971), 99-164.
  • [14] C. O. Kiselman, Existence and approximation theorems for solutions of complex analogues of boundary problems, Ark. Mat. 6 (1965), 193-207. MR 0194721 (33:2927)
  • [15] -, The partial Legendre transform for plurisubharmonic functions, Invent. Math. 49 (1978), 137-148. MR 511187 (80d:32014)
  • [16] M. Klimek, Pluripotential theory, Oxford Univ. Press, 1991. MR 1150978 (93h:32021)
  • [17] M. Langenbruch, Solution operators for partial differential equations in weighted Gevrey spaces, Michigan Math. J. 37 (1990), 3-24. MR 1042511 (91g:35063)
  • [18] B. Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier (Grenoble) 6 (1955-56), 271-355. MR 0086990 (19:280a)
  • [19] A. Martineau, Équations différentielles d'ordre infini, Bull. Soc. Math. France 95 (1967), 109-154. MR 0220966 (36:4018)
  • [20] R. Meise and B. A. Taylor, Each non-zero convolution operator on the entire functions admits a continuous linear right inverse, Math. Z. 197 (1988), 139-152. MR 917855 (89g:47036)
  • [21] 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), 619-655. MR 1091835 (92e:46083)
  • [22] -, Equivalence of analytic and plurisubharmonic Phragmén-Lindelöf principles on algebraic varieties, Proc. Sympos. Pure Math., vol. 52, Part 3, Amer. Math. Soc., Providence, RI, 1991, pp. 287-308.
  • [23] R. Meise and D. Vogt, Einführung in die Funktionanalysis, Vieweg, 1992. MR 1195130 (94f:46003)
  • [24] S. Momm, Convex univalent functions and continuous linear right inverses, J. Fund. Anal. 103 (1992), 85-103. MR 1144684 (93m:30057)
  • [25] -, A Phragmén-Lindelöf theorem for plurisubharmonic functions on cones in $ {\mathbb{C}^N}$, Indiana Univ. Math. J. 41 (1992), 861-867.
  • [26] -, A division problem in the space of entire functions of exponential type, Ark. Mat. 32 (to appear). MR 1277926 (95b:32003)
  • [27] -, Boundary behavior of extremal plurisubharmonic functions, Acta Math. 172 (1994), 51-75. MR 1263997 (95a:32022)
  • [28] -, A critical growth rate of the pluricomplex Green function, Duke Math. J. 72 (1993), 487-502. MR 1248682 (94i:32019)
  • [29] D. E. Papush and A. M. Russakowskii, Interpolation on plane sets in $ {\mathbb{C}^2}$, Ann. Fac. Sci. Toulouse 1 (1992), 337-362. MR 1225664 (94k:32015)
  • [30] M. Poppenberg and D. Vogt, A tame splitting theorem for exact sequences of Fréchet spaces, Math. Z. (to appear). MR 1340854 (96h:46109)
  • [31] A. Sadullaev, An estimate for polynomials on analytic sets, Math. USSR-Izv. 20 (1983), 493-502.
  • [32] J. Siciak, Extremal plurisubharmonic functions on $ {\mathbb{C}^N}$, Ann. Polon. Math. 39 (1981), 175-211. MR 617459 (83e:32018)
  • [33] F. Trèves, Linear partial differential equations with constant coefficients, Gordon and Breach, 1966.
  • [34] D. Vogt, Eine Charakterisierung der Potenzreihenräume von endlichem Typ und ihre Folgerungen, Manuscripta Math. 37 (1982), 269-301. MR 657522 (83i:46017)
  • [35] -, Operators between Fréchet spaces, preprint.
  • [36] M. J. Wagner, Quotientenräume von stabilen Potenzreihenräumen endlichen Typs, Manuscripta Math. 31 (1980), 97-109. MR 576492 (81m:46021)
  • [37] V. P. Zaharjuta, Extremal plurisubharmonic functions, Hilbert scales and isomorphisms of spaces of analytic functions, I, II, Theor. Funciĭ, Funktsional Anal. i Prilozhen. vyps. 19,21 (1974), 133-157, 65-83. (Russian) MR 0447632 (56:5942)
  • [38] A. Zeriahi, Fonction de Green pluricomplexe à pôle à l'infini sur un espace de Stein parabolique et applications, Math. Scand. 69 (1991), 89-126. MR 1143476 (93a:32021)

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DOI: https://doi.org/10.1090/S0002-9947-1994-1254192-7
Article copyright: © Copyright 1994 American Mathematical Society

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