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Parameter dependence of solutions of partial differential equations in spaces of real analytic functions

Authors: José Bonet and Pawel Domanski
Journal: Proc. Amer. Math. Soc. 129 (2001), 495-503
MSC (2000): Primary 35B30, 46E40, 46A63
Published electronically: August 28, 2000
MathSciNet review: 1800237
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Abstract | References | Similar Articles | Additional Information


Let $\Omega \subseteq \mathbb{R}^n$ be an open set and let $\mbox{A$(\Omega)$ }$ denote the class of real analytic functions on $\Omega$. It is proved that for every surjective linear partial differential operator $P(D,x):\mbox{A$(\Omega)$ }\to\mbox{A$(\Omega)$ }$ and every family $(f_ \lambda)\subseteq \mbox{A$(\Omega)$ }$depending holomorphically on $\lambda\in \mathbb{C}^m$ there is a solution family $(u_ \lambda)\subseteq\mbox{A$(\Omega)$ }$ depending on $\lambda$ in the same way such that \begin{equation*}P(D,x)u_ \lambda=f_ \lambda, \qquad \mbox{for } \lambda\in \mathbb{C}^m. \end{equation*} The result is a consequence of a characterization of Fréchet spaces $E$such that the class of ``weakly'' real analytic $E$-valued functions coincides with the analogous class defined via Taylor series. An example shows that the analogous assertions need not be valid if $\mathbb{C}^m$ is replaced by another set.

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  • 1. A. Alexiewicz, W. Orlicz, On analytic vector-valued functions of a real variable, Studia Math. 12 (1951), 108-111. MR 13:250b
  • 2. K. D. Bierstedt, An introduction to locally convex inductive limits, in: Functional Analysis and its Applications, H. Hogbe-Nlend (ed.), World Sci., Singapore 1988, pp. 35-133. MR 90a:46004
  • 3. J. Bochnak, J. Siciak, Analytic functions in topological vector spaces, Studia Math. 39 (1971), 77-111. MR 47:2365
  • 4. J. Bonet, P. Domanski, Real analytic curves in Fréchet spaces and their duals, Mh. Math. 126 (1998), 13-36. MR 99i:46032
  • 5. R. Braun, Surjectivity of partial differential operators on Gevrey classes, in: Functional Analysis, Proc. of the First International Workshop held at Trier University, S. Dierolf, P. Domanski, S. Dineen (eds.), Walter de Gruyter, Berlin 1996, pp. 69-80. MR 98b:35028
  • 6. R. Braun, R. Meise, D. Vogt, Applications of the projective limit functor to convolution and partial differential equations, in: Advances in the Theory of Fréchet Spaces, T. Terzioglu (ed.), Kluwer, Dordrecht 1989, pp. 29-46. MR 92b:46119
  • 7. R. Braun, R. Meise, D. Vogt, Characterization of the linear partial differential operators with constant coefficients which are surjective on non-quasianalytic classes of Roumieu type on $\mathbb{R}\sp\mathbb{N}$, Math. Nach. 168 (1994), 19-54. MR 95g:35004
  • 8. R. Braun, D. Vogt, A sufficient condition for Proj$\sp 1 X=0$, Mich. Math. J. 44 (1997), 149-156. MR 98c:46162
  • 9. E. De Giorgi, L. Cattabriga, Una dimostrazione diretta dell'esistenza di soluzioni analitiche nel piano reale di equazioni a derivate parziali a coefficienti costanti, Boll. Un. Mat. Ital. 4 (1971), 1015-1027. MR 52:3702
  • 10. B. Gramsch, Inversion von Fredholmfunktionen bei stetiger und holomorpher Abhängigkeit von Parametern, Math. Ann. 214 (1975), 95-147. MR 52:8977
  • 11. B. Gramsch, W. Kaballo, Spectral theory for Fredholm functions, in: Functional Analysis: Surveys and Recent Results II, K. D. Bierstedt, B. Fuchssteiner (eds.), North-Holland, Amsterdam 1980, pp. 319-342. MR 81d:47008
  • 12. A. Grothendieck, Sur certains espaces de functions holomorphes, I, II, J. Reine Angew. Math. 192 (1953), 35-64, 77-95.MR 15:963b
  • 13. A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955). MR 17:763c
  • 14. L. Hörmander, On the existence of real analytic solutions of partial differential equations with constant coefficients, Inventiones Math. 21 (1973), 151-182. MR 49:817
  • 15. L. Hörmander, An Introduction to Complex Analysis in Several Variables, 2nd ed., North-Holland, Amsterdam 1979.
  • 16. H. Jarchow, Locally Convex Spaces, B. G. Teubner, Stuttgart 1981. MR 83h:46008
  • 17. S. Krantz, H. R. Parks, A Primer of Real Analytic Functions, Birkhäuser, Basel 1992. MR 93j:26013
  • 18. G. Köthe, Topological Vector Spaces I and II, Springer, Berlin 1969 and 1979. MR 40:1750; MR 81g:46001
  • 19. A. Kriegl, P. W. Michor, The convenient setting for real analytic mappings, Acta Math. 165 (1990), 105-159. MR 92a:58009
  • 20. A. Kriegl, P. W. Michor, The Convenient Setting of Global Analysis, American Mathematical Society, Providence 1997. MR 98i:58015
  • 21. J. Krone, D. Vogt, The splitting relation for Köthe spaces, Math. Z. 190 (1985), 387-400. MR 86m:46009
  • 22. M. Langenbruch, Continuous linear right inverses for convolution operators in spaces of real analytic functions, Studia Math. 110 (1994), 65-82. MR 95f:46061
  • 23. M. Langenbruch, Hyperfunction fundamental solutions of surjective convolution operators on real analytic functions, J. Funct. Anal. 131 (1995), 78-93. MR 97h:35004
  • 24. J. Leiterer, Banach coherent analytic Fréchet sheaves, Math. Nachr. 85 (1978), 91-109. MR 80b:32026
  • 25. F. Mantlik, Partial differential operators depending analytically on a parameter, Ann. Inst. Fourier (Grenoble) 41 (1991), 577-599. MR 92m:35026
  • 26. F. Mantlik, Fundamental solutions or hypoelliptic differential operators depending analytically on a parameter, Trans. Amer. Math. Soc. 334 (1992), 245-257. MR 93a:35031
  • 27. A. Martineau, Sur les fonctionelles analytiques et la transformation de Fourier-Borel, J. Analyse Math. 11 (1963), 1-164.
  • 28. A. Martineau, Sur la topologie des espaces de fonctions holomorphes, Math. Ann. 163 (1966), 62-88. MR 32:8109
  • 29. R. Meise, Sequence space representations for (DFN)-algebras of entire functions modulo closed ideal, J. Reine. Angew. Math. 363 (1985), 59-95. MR 87c:46033
  • 30. R. Meise, D. Vogt, Introduction to Functional Analysis, Clarendon Press, Oxford 1997. MR 98g:46001
  • 31. P. Perez-Carreras, J. Bonet, Barrelled Locally Convex Spaces, North-Holland, Amsterdam 1987. MR 88j:46003
  • 32. H. G. Tillmann, Randverteilungen analytischer Funktionen und Distributionen, Math. Z. 59 (1953), 61-83. MR 15:211a
  • 33. D. Vogt, Charakterisierung der Unterräume eines nuklearen stabilen Potenzreihenraumes von endlichem Typ, Studia Math. 71 (1982), 251-270. MR 84d:46010
  • 34. D. Vogt, On the solvability of $P(D)f=g$ for vector valued functions, RIMS Kokyoroku 508 (1983), 168-181.
  • 35. D. Vogt, Frécheträume, zwischen denen jede stetige lineare Abbildung beschränkt ist, J. Reine Angew. Math. 345 (1983), 182-200. MR 85h:46007
  • 36. D. Vogt, Some results on continuous linear maps between Fréchet spaces, in: Functional Analysis: Surveys and Recent Results III, K. D. Bierstedt, B. Fuchssteiner (eds.), North-Holland, Amsterdam 1984, pp. 349-381. MR 86i:46075
  • 37. D. Vogt, On two classes of (F)-spaces, Arch. Math. 45 (1985), 255-266. MR 87h:46012
  • 38. D. Vogt, On the functors Ext$\sp 1$(E, F) for Fréchet spaces, Studia Math. 85 (1987), 163-197. MR 89a:46146
  • 39. D. Vogt, Lectures on projective spectra of DF-spaces, Seminar lectures, AG Funktionalanalysis, Düsseldorf/Wuppertal 1987.
  • 40. D. Vogt, Topics on projective spectra of LB-spaces, in: Advances in the Theory of Fréchet Spaces, T. Terzioglu (ed.), Kluwer, Dordrecht 1989, pp. 11-27. MR 93b:46011
  • 41. J. Wengenroth, Acyclic inductive spectra of Fréchet spaces, Studia Math. 120 (1996), 247-258. MR 97m:46006
  • 42. G. Wiechert, Dualitäts- und Strukturtheorie der Kerne linearer Differentialoperatoren, Dissertation Wuppertal (1982).
  • 43. V. P. Zaharjuta, Spaces of analytic functions and complex potential theory, in: Linear Topological Spaces and Complex Analysis 1, A. Aytuna (ed.), METU-TÜB. ITAK, Ankara 1994, pp. 74-146.

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Additional Information

José Bonet
Affiliation: Universidad Politécnica de Valencia, Departamento de Matemática Aplicada, E.T.S. Arquitectura, E-46071 Valencia, Spain

Pawel Domanski
Affiliation: Faculty of Mathematics and Computer Science, A. Mickiewicz University Poznań, Matejki 48/49, 60-769 Poznań, Poland

Keywords: Space of real analytic functions, linear partial differential operator, vector valued real analytic functions, Fr\'echet space, LB-space, surjectivity of convolution operators, parameter dependence of solutions
Received by editor(s): April 28, 1999
Published electronically: August 28, 2000
Additional Notes: The research of the first author was partially supported by DGICYT, grant no. PB 97-0333. The research of the second author was partially supported by the Committee of Scientific Research (KBN), Poland, grant 2 P03A 051 15.
Dedicated: Dedicated to V. P. Zaharjuta on the occasion of his 60th birthday
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2000 American Mathematical Society

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