A new characterization of the analytic surfaces in ℂ 3 that satisfy the local Phragmén-Lindelöf condition

Rüdiger W. Braun[1]; Reinhold Meise[2]; B. A. Taylor[3]

  • [1] Mathematisches Institut, Heinrich-Heine-Universität  Universitätsstraße 1, 40225 Düsseldorf, Germany
  • [2] Mathematisches Institut, Heinrich-Heine-Universität, Universitätsstraße 1, 40225 Düsseldorf, Germany
  • [3] Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, U.S.A.

Annales de la faculté des sciences de Toulouse Mathématiques (2011)

  • Volume: 20, Issue: S2, page 71-99
  • ISSN: 0240-2963

Abstract

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We prove that an analytic surface V in a neighborhood of the origin in ℂ 3 satisfies the local Phragmén-Lindelöf condition PL loc at the origin if and only if V satisfies the following two conditions: (1) V is nearly hyperbolic; (2) for each real simple curve γ in ℝ 3 and each d ≥ 1 , the (algebraic) limit variety T γ , d V satisfies the strong Phragmén-Lindelöf condition. These conditions are also necessary for any pure k -dimensional analytic variety V to satisify PL loc .

How to cite

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Braun, Rüdiger W., Meise, Reinhold, and Taylor, B. A.. "A new characterization of the analytic surfaces in $\mathbb{C}^3$ that satisfy the local Phragmén-Lindelöf condition." Annales de la faculté des sciences de Toulouse Mathématiques 20.S2 (2011): 71-99. <http://eudml.org/doc/219702>.

@article{Braun2011,
abstract = {We prove that an analytic surface $V$ in a neighborhood of the origin in $\mathbb\{C\}^3$ satisfies the local Phragmén-Lindelöf condition $\mathop \{\rm PL\}_\{\{\rm loc\}\}$ at the origin if and only if $V$ satisfies the following two conditions: (1) $V$ is nearly hyperbolic; (2) for each real simple curve $\gamma $ in $\mathbb\{R\}^3$ and each $d \ge 1$, the (algebraic) limit variety $T_\{\gamma ,d\}V$ satisfies the strong Phragmén-Lindelöf condition. These conditions are also necessary for any pure $k$-dimensional analytic variety $V$ to satisify $\mathop \{\rm PL\}_\{\{\rm loc\}\}$.},
affiliation = {Mathematisches Institut, Heinrich-Heine-Universität  Universitätsstraße 1, 40225 Düsseldorf, Germany; Mathematisches Institut, Heinrich-Heine-Universität, Universitätsstraße 1, 40225 Düsseldorf, Germany; Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, U.S.A.},
author = {Braun, Rüdiger W., Meise, Reinhold, Taylor, B. A.},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {analytic surfaces; Phragmén-Lindelöf condition; algebraic limit varieties; local hyperbolicity},
language = {eng},
month = {4},
number = {S2},
pages = {71-99},
publisher = {Université Paul Sabatier, Toulouse},
title = {A new characterization of the analytic surfaces in $\mathbb\{C\}^3$ that satisfy the local Phragmén-Lindelöf condition},
url = {http://eudml.org/doc/219702},
volume = {20},
year = {2011},
}

TY - JOUR
AU - Braun, Rüdiger W.
AU - Meise, Reinhold
AU - Taylor, B. A.
TI - A new characterization of the analytic surfaces in $\mathbb{C}^3$ that satisfy the local Phragmén-Lindelöf condition
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2011/4//
PB - Université Paul Sabatier, Toulouse
VL - 20
IS - S2
SP - 71
EP - 99
AB - We prove that an analytic surface $V$ in a neighborhood of the origin in $\mathbb{C}^3$ satisfies the local Phragmén-Lindelöf condition $\mathop {\rm PL}_{{\rm loc}}$ at the origin if and only if $V$ satisfies the following two conditions: (1) $V$ is nearly hyperbolic; (2) for each real simple curve $\gamma $ in $\mathbb{R}^3$ and each $d \ge 1$, the (algebraic) limit variety $T_{\gamma ,d}V$ satisfies the strong Phragmén-Lindelöf condition. These conditions are also necessary for any pure $k$-dimensional analytic variety $V$ to satisify $\mathop {\rm PL}_{{\rm loc}}$.
LA - eng
KW - analytic surfaces; Phragmén-Lindelöf condition; algebraic limit varieties; local hyperbolicity
UR - http://eudml.org/doc/219702
ER -

References

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  3. Braun (R. W.), Meise (R.), Taylor (B. A.).— Higher order tangents to analytic varieties along curves, Canad. J. Math., 55, p. 64-90 (2003). Zbl1030.32008MR1952326
  4. Braun (R. W.), Meise (R.), Taylor (B. A.).— Perturbation results for the local Phragmen-Lindeloef condition and stable homogeneous polynomials, Rev. R. Acad. Cien. Serie A. Mat. 97, p. 189-208 (2003).  Zbl1061.32026MR2068173
  5. Braun (R. W.), Meise (R.), Taylor (B. A.).— The geometry of analytic varieties satisfying the local Phragmén-Lindelöf condition and a geometric characterization of partial differential operators that are surjective on 𝒜 ( ℝ 4 ) , Trans. Amer. Math. Soc., 365, p. 1315-1383 (2004). Zbl1039.32010MR2034311
  6. Braun (R. W.), Meise (R.), Taylor (B. A.).— Nearly Hyperbolic Varieties and Phragmén-Lindelöf Conditions, p. 81-95, in “Harmonic Analysis, Signal Processing, and Complexity”, I. Sabadini, D. C. Struppa, D.F. Walnut (Eds.), Progress in Mathematics, 238 (2005). Zbl1089.32030MR2174311
  7. Braun (R. W.), Meise (R.), Taylor (B. A.).— The algebraic surfaces on which the classical Phragmén-Lindelöf theorem holds, Math. Z., 253, p. 387-417 (2006). Zbl1096.31003MR2218707
  8. Chirka (E. M.).— Complex Analytic sets. Kluver, Dordrecht (1989). Zbl0683.32002MR1111477
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  10. Heinrich (T.).— Eine geometrische Charakterisierung des lokalen Phragmén-Lindelöf Prinzips für algebraische Flächen in ℂ n . Dissertation, Düsseldorf, 2008. Electronic version http//deposit.ddb.de/cgi-bin/dokserv?idn=989795861. 
  11. Hörmander (L.).— On the existence of real analytic solutions of partial differential equations with constant coefficients, Invent. Math., 21, p. 151-183 (1973). Zbl0282.35015MR336041
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  13. 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, 40, p. 619-655 (1990). Zbl0703.46025MR1091835
  14. Meise (R.), Taylor (B. A.), Vogt (D.).— Extremal plurisubharmonic functions of linear growth on algebraic varieties, Math. Z., 219, p. 515-537 (1995). Zbl0835.32008MR1343660
  15. Meise (R.), Taylor (B. A.), Vogt (D.).— Phragmén-Lindelöf principles for algebraic varieties, J. of the Amer. Math. Soc., 11, p. 1-39 (1998). Zbl0896.32008MR1458816
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