Transactions of the American Mathematical Society

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

Author(s): Rüdiger W. Braun; Reinhold Meise; B. A. Taylor
Journal: Trans. Amer. Math. Soc. 356 (2004), 1315-1383.
MSC (2000): Primary 32C25; Secondary 32U05, 35E10
Posted: October 21, 2003
Retrieve article in: PDF DVI PostScript

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 $\mathbb{R} ^4$ .

1.: Andersson K.G.: Propagation of analyticity of solutions of partial differential equations with constant coefficients, Ark. Mat. 8 (1971), 277-302. MR 45:8986
2.: Andreotti, A., Nacinovich, M.: Analytic convexity and the principle of Phragmén-Lindelöf, Scuola Norm. Sup. Pisa, (1980). MR 83h:32018
3.: Bochnak, J., Coste, M., and Roy, M.-F.: Géometrie algébrique réelle, Ergebnisse Math. Grenzgebiete, 3. Folge, Band 12, Springer, Berlin 1987.
4.: Boiti, C., Nacinovich, M.: The overdetermined Cauchy problem, Ann. Inst. Fourier (Grenoble) 47 (1997), 155-199. MR 98a:35095
5.: Braun, R. W.: Hörmander's Phragmén-Lindelöf principle and irreducible singularities of codimension 1, Boll. Un. Mat. Ital. (7) 6-A (1992), 339-348. MR 94b:35012
6.: Braun, R. W.: The surjectivity of a constant coefficient homogeneous differential operator on the real analytic functions and the geometry of its symbol, Ann. Inst. Fourier (Grenoble), 45 (1995), 223-249. MR 96e:35025
7.: Braun, R.W., Meise, R., Taylor, B.A.: Algebraic varieties on which the classical Phragmén-Lindelöf estimates hold for plurisubharmonic function, Math. Z. 232 (1999), 103-135. MR 2001d:32048
8.: Braun, R.W., Meise, R., Taylor, B.A.: Local radial Phragmén-Lindelöf estimates for plurisubharmonic functions on analytic varieties, Proc. Amer. Math. Soc. 131 (2003), 2423-2433.
9.: Braun, R.W., Meise, R., Taylor, B.A.: Higher order tangents to analytic varieties along curves, Canad. J. Math. 55 (2003), 64-90.
10.: Braun, R.W., Meise, R., Taylor, B.A.: Surjectivity of constant coefficient partial differential operators on $\mathcal{A} (\mathbb{R} ^4)$ and Whitney's -cone, Bull. Soc. R. Sci. Liège 70 (2001), 195-206. MR 2003f:35039
11.: Braun, R.W., Meise, R., Taylor, B.A.: The algebraic surfaces on which the classical Phragmén-Lindelöf theorem holds, manuscript.
12.: Braun, R.W., Meise, R., Vogt, D.: Characterization of the linear partial differential operators with constant coefficients which are surjective on non-quasianalytic classes of Roumieu type on $\mathbb{R} ^N$ , Math. Nachrichten 168 (1994), 19-54. MR 95g:35004
13.: Cartan, H.: Variétés analytiques réelles et variétés analytiques complexes. Bull. Soc. Math. France 85 (1957), 77-99. MR 20:1339
14.: Chirka, E.M.: Complex Analytic sets, Kluwer, Dordrecht, 1989. MR 92b:32016
15.: Franken, U., Meise, R.: Extension and lacunas of solutions of linear partial differential equations, Ann. Inst. Fourier (Grenoble) 46 (1996), 429-464. MR 97h:35005
16.: Franken, U., Meise, R.: Extension and lacunas of solutions of linear partial differential equations on ultradifferentiable functions of Beurling type, in `Structure of Solutions of Differential Equations', M. Morimoto, T. Kawai (Eds.), World Scientific (1996), 103-127. MR 98g:35035
17.: Hörmander, L.: On the existence of real analytic solutions of partial differential equations with constant coefficients, Invent. Math. 21 (1973), 151-183. MR 49:817
18.: Kaneko, A.: Hartogs type extension theorem of real analytic solutions of linear partial differential equations with constant coefficients, in Advances in the Theory of Fréchet Spaces, T. Terzioglu (Ed.) NATO ASI Series C, Vol. 289 (Kluwer), 1989, 63-72.
19.: Klimek, M.: Pluripotential Theory, Oxford Science Publications, 1991. MR 93h:32021
20.: Langenbruch, M.: Surjective partial differential operators on real analytic functions defined on open convex sets, Manuscr. Math. 103 (2000), 241-263. MR 2001k:35050
21.: Langenbruch, M.: Characterization of surjective partial differential operators on spaces of real analytic functions, preprint.
22.: Meise, R., and Taylor, B.A.: Phragmén-Lindelöf conditions for graph varieties, Result. Math. 36 (1999), 121-148. MR 2000j:32057
23.: Meise, R., Taylor, B.A., Vogt, D.: 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 92e:46083
24.: Meise, R., Taylor, B.A., Vogt, D.: Extremal plurisubharmonic functions of linear growth on algebraic varieties, Math. Z. 219 (1995), 515-537. MR 96j:32019
25.: Meise, R., Taylor, B.A., Vogt, D.: Continuous linear right inverses for partial differential operators on non-quasianalytic classes and on ultradistributions, Math. Nachr. 180 (1996), 213-242. MR 97j:46036
26.: Meise, R., Taylor, B.A., Vogt, D.: Phragmén-Lindelöf principles on algebraic varieties, J. Amer. Math. Soc. 11 (1998), 1-39. MR 98j:32007
27.: Momm, S.: On the dependence of analytic solutions of partial differential equations on the right-hand side, Trans. Amer. Math. Soc. 345 (1994), 729-752. MR 95a:46036
28.: Nevanlinna, R.: Eindeutige analytische Funktionen, Springer, Berlin, Heidelberg, New York 1974. MR 49:9165
29.: Palamodov, V.P.: A criterion for splitness of differential complexes with constant coefficients, in `Geometrical and Algebraical Aspects in Several Complex Variables', C.A. Berenstein and D.C. Struppa (Eds.), EditEl (1991), pp. 265-290. MR 94d:58137
30.: Siciak, J.: Extremal plurisubharmonic functions in $\mathbb{C} ^n$ , Ann. Polon. Math. 39 (1981), 177-211. MR 83e:32018
31.: Stutz, J.: Analytic sets as branched coverings, Trans. Amer. Math. Soc. 166 (1972), 241-259. MR 48:2420
32.: Whitney, H.: Complex analytic varieties, Addison-Wesley Pub. Co., 1972. MR 52:8473
33.: Zampieri, G.: Propagation of singularity and existence of real analytic solutions of locally hyperbolic equations, J. Fac. Sci. Univ. Tokyo, Sect. I A, Math. 31 (1984), 373-390. MR 86c:58142
34.: Zampieri, G.: An application of the fundamental principle of Ehrenpreis to the existence of global solutions of linear partial differential equations, Boll. U. M. I. 6 (1986), 361-392. MR 88a:35044
35.: Zariski, O.: Studies in equisingularity I, equivalent singularities of plane algebroid curves, Amer. J. Math. 97 (1965), 507-536. MR 31:2243
36.: Zeriahi, A.: Fonction de Green pluricomplexe à pole à l'infini sur un espace de Stein parabolique et applications, Math. Scand. 69 (1991), 89-126. MR 93a:32021

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 32C25, 32U05, 35E10

Rüdiger W. Braun
Affiliation: Mathematisches Institut, Heinrich-Heine-Universität, Universitätsstraße 1, 40225 Düsseldorf, Germany
Email: Ruediger.Braun@uni-duesseldorf.de

Reinhold Meise
Affiliation: Mathematisches Institut, Heinrich-Heine-Universität, Universitätsstraße 1, 40225 Düsseldorf, Germany
Email: meise@cs.uni-duesseldorf.de

B. A. Taylor
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: taylor@umich.edu

DOI: 10.1090/S0002-9947-03-03448-2
PII: S 0002-9947(03)03448-2
Received by editor(s): July 12, 2002
Posted: October 21, 2003
Additional Notes: The authors gratefully acknowledge support of DAAD and NSF under the program ``Projektbezogene Förderung des Wissenschaftleraustausch mit den USA in Zusammenarbeit mit der National Science Foundation'' and of the Volkswagen-Stiftung (RiP-program in Oberwolfach). The research of the third-named author was supported in part by the National Science Foundation under grant number DMS 0070725.
Copyright of article: Copyright 2003, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS