We prove that an analytic surface in a neighborhood of the origin in satisfies the local Phragmén-Lindelöf condition at the origin if and only if satisfies the following two conditions: (1) is nearly hyperbolic; (2) for each real simple curve in and each , the (algebraic) limit variety satisfies the strong Phragmén-Lindelöf condition. These conditions are also necessary for any pure -dimensional analytic variety to satisify .
On démontre qu’une surface analytique dans un voisinage de l’origine dans satisfait à la condition Phragmén-Lindelöf locale à l’origine si et seulement si satisfait aux deux conditions suivantes : (1) is presque hyperbolique ; (2) pour chaque courbe réelle simple dans et chaque , la variété (algebrique) limite satisfait à la condition de Phragmén-Lindelöf forte. Ces conditions sont aussi nécessaires que pour toute variété analytique de dimension pure vérifie la condition .
@article{AFST_2011_6_20_S2_71_0, author = {Braun, R\"udiger W. and Meise, Reinhold and Taylor, B. A.}, title = {A new characterization of the analytic surfaces in $\mathbb{C}^3$ that satisfy the local {Phragm\'en-Lindel\"of} condition}, journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques}, pages = {71--99}, publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques}, address = {Toulouse}, volume = {Ser. 6, 20}, number = {S2}, year = {2011}, doi = {10.5802/afst.1306}, zbl = {1228.32011}, mrnumber = {2858168}, language = {en}, url = {http://www.numdam.org/articles/10.5802/afst.1306/} }
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 PY - 2011 SP - 71 EP - 99 VL - 20 IS - S2 PB - Université Paul Sabatier, Institut de mathématiques PP - Toulouse UR - http://www.numdam.org/articles/10.5802/afst.1306/ DO - 10.5802/afst.1306 LA - en ID - AFST_2011_6_20_S2_71_0 ER -
%0 Journal Article %A Braun, Rüdiger W. %A Meise, Reinhold %A Taylor, B. A. %T A new characterization of the analytic surfaces in $\mathbb{C}^3$ that satisfy the local Phragmén-Lindelöf condition %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2011 %P 71-99 %V 20 %N S2 %I Université Paul Sabatier, Institut de mathématiques %C Toulouse %U http://www.numdam.org/articles/10.5802/afst.1306/ %R 10.5802/afst.1306 %G en %F AFST_2011_6_20_S2_71_0
Braun, Rüdiger W.; Meise, Reinhold; 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, Serie 6, Numéro Spécial : Actes du colloque Analyse Complexe et Applications en l’honneur de Nguyen Than Van, Volume 20 (2011) no. S2, pp. 71-99. doi : 10.5802/afst.1306. http://www.numdam.org/articles/10.5802/afst.1306/
[1] Braun (R. W.), Meise (R.), Taylor (B. A.).— The algebraic varieties on which the classical Phragmén-Lindelöf estimate holds for plurisubharmonic functions, Math. Z., 232, p. 103-135 (1999). | MR | Zbl
[2] Braun (R. W.), Meise (R.), Taylor (B. A.).— Local radial Phragmén-Lindelöf estimates for plurisubharmonic functions on analytic varieties, Proc. Amer. Math. Soc., 131, p. 2423-2433 (2002). | MR | Zbl
[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). | MR | Zbl
[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). | MR | Zbl
[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 , Trans. Amer. Math. Soc., 365, p. 1315-1383 (2004). | MR | Zbl
[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). | MR | Zbl
[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). | MR | Zbl
[8] Chirka (E. M.).— Complex Analytic sets. Kluver, Dordrecht (1989). | MR | Zbl
[9] Heinrich (T.).— A new necessary condition for analytic varieties satisfying the local Phragmén-Lindelöf condition, Ann. Polon. Math., 85, p. 283-290 (2005). | MR | Zbl
[10] Heinrich (T.).— Eine geometrische Charakterisierung des lokalen Phragmén-Lindelöf Prinzips für algebraische Flächen in . 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). | MR | Zbl
[12] Meise (R.), Taylor (B. A.).— Phragmén-Lindelöf conditions for graph varieties, Result. Math., 36, p. 121-148 (1999). | MR | Zbl
[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). | Numdam | MR | Zbl
[14] Meise (R.), Taylor (B. A.), Vogt (D.).— Extremal plurisubharmonic functions of linear growth on algebraic varieties, Math. Z., 219, p. 515-537 (1995). | MR | Zbl
[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). | MR | Zbl
[16] Vogt (D.).— Extension operators for real analytic functions on compact subvarieties of , J. Reine Angew. Math., 606, p. 217-233 (2007). | MR | Zbl
[17] Whitney (W.).— ‘Complex analytic varieties’, Addison-Wesley Pub. Co. (1972). | MR | Zbl
Cited by Sources: