PhragménLindelöf principles on algebraic varieties
Authors:
R. Meise, B. A. Taylor and D. Vogt
Journal:
J. Amer. Math. Soc. 11 (1998), 139
MSC (1991):
Primary 32F05; Secondary 46J99
MathSciNet review:
1458816
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Estimates of PhragménLindelöf (PL) type for plurisubharmonic functions on algebraic varieties in 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 PLcondition is the only method to check whether a given operator has the property in question. In the present paper the property which characterizes the existence of continuous linear right inverses is investigated. It is also the one closest in spirit to the classical PhragménLindelöf Theorem as various equivalent formulations for homogeneous varieties show. These also clarify the relation between and the PLcondition used by Hörmander to characterize the surjectivity of differential operators on realanalytic functions. We prove the property for an algebraic variety implies that , the tangent cone of at infinity, also has this property. The converse implication fails in general. However, if is a manifold outside the origin, then satisfies if and only if the real points in have maximal dimension and if the distance of to is bounded by as tends to infinity. In the general case, no geometric characterization of the algebraic varieties which satisfy is known, nor any of the other PLconditions 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.
 1.
Lars
V. Ahlfors, Conformal invariants: topics in geometric function
theory, McGrawHill Book Co., New York, 1973. McGrawHill Series in
Higher Mathematics. MR 0357743
(50 #10211)
 2.
Karl
Gustav Andersson, Propagation of analyticity of solutions of
partial differential equations with constant coefficients, Ark. Mat.
8 (1971), 277–302. MR 0299938
(45 #8986)
 3.
Rüdiger
W. Braun, Hörmander’s PhragménLindelöf
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
(94b:35012)
 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
(96e:35025)
 5.
Rüdiger
W. Braun and Reinhold
Meise, Generalized Fourier expansions for zerosolutions of
surjective convolution operators on
𝒟_{{𝜔}}(ℛ)’, Arch. Math. (Basel)
55 (1990), no. 1, 55–63. MR 1059516
(91i:46037), http://dx.doi.org/10.1007/BF01199116
 6.
R.
W. Braun, R.
Meise, and B.
A. Taylor, Ultradifferentiable functions and Fourier analysis,
Results Math. 17 (1990), no. 34, 206–237. MR 1052587
(91h:46072), http://dx.doi.org/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
(92b:46119)
 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
(95g:35004), http://dx.doi.org/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 (92b:32016)
 10.
David
K. Cohoon, Nonexistence of a continuous right inverse for parabolic
differential operators, J. Differential Equations 6
(1969), 503–511. MR 0250124
(40 #3364)
 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
(47 #5659)
 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 (81f:32020), http://dx.doi.org/10.1007/BF01349254
 13.
Lars
Hörmander, Linear partial differential operators, Die
Grundlehren der mathematischen Wissenschaften, Bd. 116, Academic Press
Inc., Publishers, New York, 1963. MR 0161012
(28 #4221)
 14.
Lars
Hörmander, An introduction to complex analysis in several
variables, Second revised edition, NorthHolland Publishing Co.,
Amsterdam, 1973. NorthHolland Mathematical Library, Vol. 7. MR 0344507
(49 #9246)
 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
(49 #817)
 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
(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, 6372. 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
(89c:46058), http://dx.doi.org/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
(89m:35042)
 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
(92b:46122)
 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
(92e:46083)
 22.
Reinhold
Meise, B.
A. Taylor, and Dietmar
Vogt, Equivalence of analytic and plurisubharmonic
PhragménLindelöf 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
(93a:32023)
 23.
R. Meise, B.A. Taylor, and D. Vogt, Continuous linear right inverses for partial differential operators on nonquasianalytic classes and on ultradistributions, Math. Nachr. 180 (1996), 213242. CMP 96:15
 24.
R.
Meise, B.
A. Taylor, and D.
Vogt, Indicators of plurisubharmonic functions on algebraic
varieties and Kaneko’s PhragménLindelöf condition,
(Cetraro, 1989) Sem. Conf., vol. 8, EditEl, Rende, 1991,
pp. 231–250. MR 1222217
(94d:32018)
 25.
R.
Meise, B.
A. Taylor, and D.
Vogt, Continuous linear right inverses for partial differential
operators with constant coefficients and PhragménLindelöf
conditions, Functional analysis (Essen, 1991) Lecture Notes in Pure
and Appl. Math., vol. 150, Dekker, New York, 1994,
pp. 357–389. MR 1241689
(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 1343660
(96j:32019), http://dx.doi.org/10.1007/BF02572379
 27.
R.
Meise, B.
A. Taylor, and D.
Vogt, 𝜔hyperbolicity 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
(97h:35027)
 28.
Siegfried
Momm, On the dependence of analytic
solutions of partial differential equations on the righthand
side, Trans. Amer. Math. Soc.
345 (1994), no. 2,
729–752. MR 1254192
(95a:46036), http://dx.doi.org/10.1090/S00029947199412541927
 29.
Raghavan
Narasimhan, Introduction to the theory of analytic spaces,
Lecture Notes in Mathematics, No. 25, SpringerVerlag, Berlin, 1966. MR 0217337
(36 #428)
 30.
V.
P. Palamodov, A criterion for splitness of differential complexes
with constant coefficients, (Cetraro, 1989) Sem. Conf.,
vol. 8, EditEl, Rende, 1991, pp. 265–291. MR 1222219
(94d:58137)
 31.
Dietmar
Vogt, Some results on continuous linear maps between Fréchet
spaces, Functional analysis: surveys and recent results, III
(Paderborn, 1983), NorthHolland Math. Stud., vol. 90, NorthHolland,
Amsterdam, 1984, pp. 349–381. MR 761391
(86i:46075), http://dx.doi.org/10.1016/S03040208(08)714853
 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
(88a:35044)
 1.
 L.V. Ahlfors, Conformal Invariants; Topics in Geometric Function Theory, McGrawHill, 1973. MR 50:10211
 2.
 K.G. Andersson, Propagation of analyticity of solutions of partial differential equations with constant coefficients, Ark. Mat. 8 (1971), 277302. MR 45:8986
 3.
 R.W. Braun, Hörmander's PhragménLindelöf principle and irreducible singularities of codimension 1, Boll. Un. Mat. Ital. 6 A (1992), 339348. 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), 223249. MR 96e:35025
 5.
 R.W. Braun and R. Meise, Generalized Fourier expansions for zerosolutions of surjective convolution operators on , Arch. Math. 55 (1990), 5563. MR 91i:46037
 6.
 R.W. Braun, R. Meise, and B.A. Taylor, Ultradifferentiable functions and Fourier analysis, Res. Math. 17 (1990), 207237. 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, 2946. 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 nonquasianalytic classes of Roumieu type on , Math. Nachr. 168 (1994), 1954. 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), 503511. 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), 337342. MR 47:5659
 12.
 J. Fornaess and R. Narasimhan, The Levi problem on complex spaces with singularities, Math. Ann. 248 (1980), 4772. 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), 151183. 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), 126. 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, 6372. 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), 729750. MR 89c:46058
 19.
 R. Meise, B.A. Taylor, and D. Vogt, Charactérisation des opérateurs linéaires aux dérivées partielles avec coefficients constants sur admettant un inverse à droite qui est linéaire et continu, C. R. Acad. Paris 307 (1988), 239242. 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. 4772. 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), 619655. MR 92e:46083
 22.
 R. Meise, B.A. Taylor, and D. Vogt, Equivalence of analytic and plurisubharmonic PhragménLindelöf principles on algebraic varieties, Proceedings of Symposia in Pure Mathematics, 52 (1991), 287308. MR 93a:32023
 23.
 R. Meise, B.A. Taylor, and D. Vogt, Continuous linear right inverses for partial differential operators on nonquasianalytic classes and on ultradistributions, Math. Nachr. 180 (1996), 213242. CMP 96:15
 24.
 R. Meise, B.A. Taylor, and D. Vogt, Indicators of plurisubharmonic functions on algebraic varieties and Kaneko's PhragménLindelöf condition, in Geometrical and Algebraical Aspects in Several Complex Variables, C.A. Berenstein and D.C. Struppa (Eds.), EditEl (1991), pp. 231250. MR 94d:32018
 25.
 R. Meise, B.A. Taylor, and D. Vogt, Continuous linear right inverses for partial differential operators with constant coefficients and PhragménLindelöf 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. 357389. 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, 515537. MR 96j:32019
 27.
 R. Meise, B.A. Taylor, and D. Vogt, 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, 157182. MR 97h:35027
 28.
 S. Momm, On the dependence of analytic solutions of partial differential equations on the righthand side, Trans. Amer. Math. Soc. 345 (1994), 729752. 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. 265290. 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), NorthHolland Mathematics Studies, 90 (1984), pp. 349381. 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), 361392. 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, HeinrichHeineUniversität, Universitätsstraße 1, 40225 Düsseldorf, Germany
Email:
meise@cs.uniduesseldorf.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.uniwuppertal.de
DOI:
http://dx.doi.org/10.1090/S0894034798002471
PII:
S 08940347(98)002471
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 HeinrichHeineUniversität Düsseldorf”.
Article copyright:
© Copyright 1998 American Mathematical Society
