Remote Access Journal of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0894-0347-98-00247-1
MathSciNet review: 1458816
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. L.V. Ahlfors, Conformal Invariants; Topics in Geometric Function Theory, McGraw-Hill, 1973. MR 50:10211
  • 2. K.G. Andersson, Propagation of analyticity of solutions of partial differential equations with constant coefficients, Ark. Mat. 8 (1971), 277-302. MR 45:8986
  • 3. R.W. Braun, Hörmander's Phragmén-Lindelöf principle and irreducible singularities of codimension 1, Boll. Un. Mat. Ital. 6 -A (1992), 339-348. MR 94b:35012
  • 4. R.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), 223-249. MR 96e:35025
  • 5. R.W. Braun and R. Meise, Generalized Fourier expansions for zero-solutions of surjective convolution operators on ${\mathcal D}'_{\{w\}}(\mathbb{R})$, Arch. Math. 55 (1990), 55-63. MR 91i:46037
  • 6. R.W. Braun, R. Meise, and B.A. Taylor, Ultradifferentiable functions and Fourier analysis, Res. Math. 17 (1990), 207-237. MR 91h:46072
  • 7. R.W. Braun, R. Meise, and D. Vogt, Applications of the projective limit functor to convolution and partial differential equations, in Advances in the Theory of Fréchet Spaces, T. Terzio\u{g}lu (Ed.), NATO ASI Series C, Vol. 287 (Kluwer), 1989, 29-46. MR 92b:46119
  • 8. R.W. Braun, R. Meise, and D. Vogt, Characterisation of the linear partial differential operators with constant coefficients which are surjective on non-quasianalytic classes of Roumieu type on $\mathbb{R}^N$, Math. Nachr. 168 (1994), 19-54. MR 95g:35004
  • 9. E.M. Chirka, Complex Analytic Sets, Kluwer Academic Publishers, Dordrecht/Boston/London, 1989. MR 92b:32016
  • 10. D.K. Cohoon, Nonexistence of a continuous linear right inverse for parabolic differential operators, J. Diff. Equ. 6 (1969), 503-511. MR 40:3364
  • 11. D.K. Cohoon, Nonexistence of a continuous linear right inverse for partial differential operators with constant coefficients, Math. Scand. 29 (1971), 337-342. MR 47:5659
  • 12. J. Fornaess and R. Narasimhan, The Levi problem on complex spaces with singularities, Math. Ann. 248 (1980), 47-72. MR 81f:32020
  • 13. L. Hörmander, Linear Partial Differential Operators, Springer, 1963. MR 28:4221
  • 14. L. Hörmander, Complex Analysis in Several Variables, 2nd Ed. North Holland, 1973. MR 49:9246
  • 15. L. Hörmander, On the existence of real analytic solutions of partial differential equations with constant coefficients, Invent. Math. 21 (1973), 151-183. MR 49:817
  • 16. A. Kaneko, On Hartogs type continuation theorem for regular solutions of linear differential equations with constant coefficients, J. Fac. Sci. Tokyo Sect. IA 35 (1988), 1-26. MR 89m:35043
  • 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 generalized Fourier expansions, Indiana Univ. J. Math. 36 (1987), 729-750. MR 89c:46058
  • 19. R. Meise, B.A. Taylor, and D. Vogt, Charactérisation des opérateurs linéaires aux dérivées partielles avec coefficients constants sur ${\mathcal E}(\mathbb{R}^{n})$ admettant un inverse à droite qui est linéaire et continu, C. R. Acad. Paris 307 (1988), 239-242. MR 89m:35042
  • 20. R. Meise, B.A. Taylor, and D. Vogt, Partial differential operators with continuous linear right inverse, in Advances in the Theory of Fréchet Spaces, T. Terzio\u{g}lu (Ed.), NATO ASI Series C, Vol. 289 (Kluwer), 1989, pp. 47-72. MR 92b:46122
  • 21. R. Meise, B.A. Taylor, and D. Vogt, Characterization of the linear partial operators with constant coefficients that admit a continuous linear right inverse, Ann. Inst. Fourier (Grenoble), 40 (1990), 619-655. MR 92e:46083
  • 22. R. Meise, B.A. Taylor, and D. Vogt, Equivalence of analytic and plurisubharmonic Phragmén-Lindelöf principles on algebraic varieties, Proceedings of Symposia in Pure Mathematics, 52 (1991), 287-308. MR 93a:32023
  • 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, in Geometrical and Algebraical Aspects in Several Complex Variables, C.A. Berenstein and D.C. Struppa (Eds.), EditEl (1991), pp. 231-250. MR 94d:32018
  • 25. R. Meise, B.A. Taylor, and D. Vogt, Continuous linear right inverses for partial differential operators with constant coefficients and Phragmén-Lindelöf conditions, in Functional Analysis, K.D. Bierstedt, A. Pietsch, W.M. Ruess, and D. Vogt (Eds.), Lecture Notes in Pure and Applied Math., Vol. 150, Marcel Dekker, 1994, pp. 357-389. MR 94k:35064
  • 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 96j:32019
  • 27. R. Meise, B.A. Taylor, and D. Vogt, $\omega$-Hyperbolicity of linear partial differential operators with constant coefficients, in Complex Analysis, Harmonic Analysis and Applications (Bordeaux, 1995), Pitman Res. Notes Math. Ser., 347, Longman, Harlow, 1996, 157-182. MR 97h:35027
  • 28. S. Momm, 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
  • 29. R. Narasimhan, Introduction to the Theory of Analytic Spaces, Lecture Notes in Math. 25 (1966). MR 36:428
  • 30. V.P. Palamodov, 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
  • 31. D. Vogt, Some results on continuous linear maps between Fréchet spaces, in Functional Analysis: Surveys and Recent Results II, K.D. Bierstedt and B. Fuchssteiner (Eds), North-Holland Mathematics Studies, 90 (1984), pp. 349-381. MR 86i:46075
  • 32. G. Zampieri, 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

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 32F05, 46J99

Retrieve articles in all journals with MSC (1991): 32F05, 46J99


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: https://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

American Mathematical Society