Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 

 

Phragmén-Lindelöf principles
on algebraic varieties


Authors: R. Meise, B. A. Taylor and D. Vogt
Journal: J. Amer. Math. Soc. 11 (1998), 1-39
MSC (1991): Primary 32F05; Secondary 46J99
MathSciNet review: 1458816
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Abstract: Estimates of Phragmén-Lindelöf (PL) type for plurisubharmonic functions on algebraic varieties in $\mathbb{C}^n$ have been of interest for a number of years because of their equivalence with certain properties of constant coefficient partial differential operators; e.g. surjectivity, continuation properties of solutions and existence of continuous linear right inverses. Besides intrinsic interest, their importance lies in the fact that, in many cases, verification of the relevant PL-condition is the only method to check whether a given operator has the property in question.

In the present paper the property $\textup{PL}({\mathbb R}^n,\omega)$ which characterizes the existence of continuous linear right inverses is investigated. It is also the one closest in spirit to the classical Phragmén-Lindelöf Theorem as various equivalent formulations for homogeneous varieties show. These also clarify the relation between $\textup{PL}({\mathbb R}^n,\omega)$ and the PL-condition used by Hörmander to characterize the surjectivity of differential operators on real-analytic functions.

We prove the property $\textup{PL}({\mathbb R}^n,\omega)$ for an algebraic variety $V$ implies that $V_h$, the tangent cone of $V$ at infinity, also has this property. The converse implication fails in general. However, if $V_h$ is a manifold outside the origin, then $V$ satisfies $\textup{PL}({\mathbb R}^n,\omega)$ if and only if the real points in $V_h$ have maximal dimension and if the distance of $z\in V$ to $V_h$ is bounded by $C\omega(|z|)$ as $z$ tends to infinity. In the general case, no geometric characterization of the algebraic varieties which satisfy $\textup{PL}({\mathbb R}^n,\omega)$ is known, nor any of the other PL-conditions alluded to above.

Besides these main results the paper contains several auxiliary necessary conditions and sufficient conditions which make it possible to treat interesting examples completely. Since it was submitted they have been applied by several authors to achieve further progress on questions left open here.


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  • 1. Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR 0357743
  • 2. Karl Gustav Andersson, Propagation of analyticity of solutions of partial differential equations with constant coefficients, Ark. Mat. 8 (1971), 277–302. MR 0299938
  • 3. Rüdiger W. Braun, Hörmander’s Phragmén-Lindelöf principle and irreducible singularities of codimension 1, Boll. Un. Mat. Ital. A (7) 6 (1992), no. 3, 339–348 (English, with Italian summary). MR 1196128
  • 4. Rüdiger W. Braun, 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), no. 1, 223–249 (English, with English and French summaries). MR 1324131
  • 5. Rüdiger W. Braun and Reinhold Meise, Generalized Fourier expansions for zero-solutions of surjective convolution operators on 𝒟_{{𝜔}}(ℛ)’, Arch. Math. (Basel) 55 (1990), no. 1, 55–63. MR 1059516, 10.1007/BF01199116
  • 6. R. W. Braun, R. Meise, and B. A. Taylor, Ultradifferentiable functions and Fourier analysis, Results Math. 17 (1990), no. 3-4, 206–237. MR 1052587, 10.1007/BF03322459
  • 7. Rüdiger W. Braun, Reinhold Meise, and Dietmar Vogt, Applications of the projective limit functor to convolution and partial differential equations, Advances in the theory of Fréchet spaces (Istanbul, 1988) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 287, Kluwer Acad. Publ., Dordrecht, 1989, pp. 29–46. MR 1083556
  • 8. R. W. Braun, R. Meise, and D. Vogt, Characterization of the linear partial differential operators with constant coefficients which are surjective on nonquasianalytic classes of Roumieu type on 𝑅^{𝑁}, Math. Nachr. 168 (1994), 19–54. MR 1282630, 10.1002/mana.19941680103
  • 9. 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
  • 10. David K. Cohoon, Nonexistence of a continuous right inverse for parabolic differential operators, J. Differential Equations 6 (1969), 503–511. MR 0250124
  • 11. D. K. Cohoon, Nonexistence of a continuous right inverse for linear partial differential operators with constant coefficients, Math. Scand. 29 (1971), 337–342 (1972). MR 0317111
  • 12. John Erik Fornæss and Raghavan Narasimhan, The Levi problem on complex spaces with singularities, Math. Ann. 248 (1980), no. 1, 47–72. MR 569410, 10.1007/BF01349254
  • 13. Lars Hörmander, Linear partial differential operators, Die Grundlehren der mathematischen Wissenschaften, Bd. 116, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0161012
  • 14. Lars Hörmander, An introduction to complex analysis in several variables, Second revised edition, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. North-Holland Mathematical Library, Vol. 7. MR 0344507
  • 15. Lars Hörmander, On the existence of real analytic solutions of partial differential equations with constant coefficients, Invent. Math. 21 (1973), 151–182. MR 0336041
  • 16. Akira Kaneko, On Hartogs type continuation theorem for regular solutions of linear partial differential equations with constant coefficients, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 35 (1988), no. 1, 1–26. MR 931441
  • 17. A. Kaneko, 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. Terzio\u{g}lu (Ed.), NATO ASI Series C, Vol. 289 (Kluwer), 1989, 63-72. CMP 91:05
  • 18. R. Meise, B. A. Taylor, and D. Vogt, Equivalence of slowly decreasing conditions and local Fourier expansions, Indiana Univ. Math. J. 36 (1987), no. 4, 729–756. MR 916742, 10.1512/iumj.1987.36.36042
  • 19. Reinhold Meise, B. Alan Taylor, and Dietmar Vogt, Caractérisation des opérateurs linéaires aux dérivées partielles avec coefficients constants sur ℰ(ℛ^{𝒩}) admettant un inverse à droite qui est linéaire et continu, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 6, 239–242 (French, with English summary). MR 956814
  • 20. R. Meise, B. A. Taylor, and D. Vogt, Partial differential operators with continuous linear right inverse, Advances in the theory of Fréchet spaces (Istanbul, 1988) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 287, Kluwer Acad. Publ., Dordrecht, 1989, pp. 47–62. MR 1083557
  • 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), no. 3, 619–655 (English, with French summary). MR 1091835
  • 22. Reinhold Meise, B. A. Taylor, and Dietmar Vogt, Equivalence of analytic and plurisubharmonic Phragmén-Lindelöf conditions, Several complex variables and complex geometry, Part 3 (Santa Cruz, CA, 1989), Proc. Sympos. Pure Math., vol. 52, Amer. Math. Soc., Providence, RI, 1991, pp. 287–308. MR 1128602
  • 23. R. Meise, B.A. Taylor, and D. Vogt, Continuous linear right inverses for partial differential operators on non-quasianalytic classes and on ultradistributions, Math. Nachr. 180 (1996), 213-242. CMP 96:15
  • 24. R. Meise, B. A. Taylor, and D. Vogt, Indicators of plurisubharmonic functions on algebraic varieties and Kaneko’s Phragmén-Lindelöf condition, Geometrical and algebraical aspects in several complex variables (Cetraro, 1989) Sem. Conf., vol. 8, EditEl, Rende, 1991, pp. 231–250. MR 1222217
  • 25. R. Meise, B. A. Taylor, and D. Vogt, Continuous linear right inverses for partial differential operators with constant coefficients and Phragmén-Lindelöf conditions, Functional analysis (Essen, 1991) Lecture Notes in Pure and Appl. Math., vol. 150, Dekker, New York, 1994, pp. 357–389. MR 1241689
  • 26. R. Meise, B. A. Taylor, and D. Vogt, Extremal plurisubharmonic functions of linear growth on algebraic varieties, Math. Z. 219 (1995), no. 4, 515–537. MR 1343660, 10.1007/BF02572379
  • 27. R. Meise, B. A. Taylor, and D. Vogt, 𝜔-hyperbolicity of linear partial differential operators with constant coefficients, Complex analysis, harmonic analysis and applications (Bordeaux, 1995), Pitman Res. Notes Math. Ser., vol. 347, Longman, Harlow, 1996, pp. 157–182. MR 1402027
  • 28. Siegfried Momm, On the dependence of analytic solutions of partial differential equations on the right-hand side, Trans. Amer. Math. Soc. 345 (1994), no. 2, 729–752. MR 1254192, 10.1090/S0002-9947-1994-1254192-7
  • 29. Raghavan Narasimhan, Introduction to the theory of analytic spaces, Lecture Notes in Mathematics, No. 25, Springer-Verlag, Berlin-New York, 1966. MR 0217337
  • 30. V. P. Palamodov, A criterion for splitness of differential complexes with constant coefficients, Geometrical and algebraical aspects in several complex variables (Cetraro, 1989) Sem. Conf., vol. 8, EditEl, Rende, 1991, pp. 265–291. MR 1222219
  • 31. Dietmar Vogt, Some results on continuous linear maps between Fréchet spaces, Functional analysis: surveys and recent results, III (Paderborn, 1983), North-Holland Math. Stud., vol. 90, North-Holland, Amsterdam, 1984, pp. 349–381. MR 761391, 10.1016/S0304-0208(08)71485-3
  • 32. Giuseppe Zampieri, An application of the fundamental principle of Ehrenpreis to the existence of global Gevrey solutions of linear differential equations, Boll. Un. Mat. Ital. B (6) 5 (1986), no. 2, 361–392 (English, with Italian summary). MR 860634

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Additional Information

R. 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@math.lsa.umich.edu

D. Vogt
Affiliation: Fachbereich Mathematik, Bergische Universität, Gaußstraße 20, 42097 Wuppertal, Germany
Email: vogt@math.uni-wuppertal.de

DOI: http://dx.doi.org/10.1090/S0894-0347-98-00247-1
Received by editor(s): June 28, 1994
Received by editor(s) in revised form: July 13, 1994
Additional Notes: The second named author gratefully acknowledges the support of his research by the “Gesellschaft von Freunden und Förderern der Heinrich-Heine-Universität Düsseldorf”.
Article copyright: © Copyright 1998 American Mathematical Society