logo AFST

A new characterization of the analytic surfaces in 3 that satisfy the local Phragmén-Lindelöf condition
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 20 (2011) no. S2, pp. 71-99.

On démontre qu’une surface analytique V dans un voisinage de l’origine dans 3 satisfait à la condition Phragmén-Lindelöf locale PL loc à l’origine si et seulement si V satisfait aux deux conditions suivantes : (1) V is presque hyperbolique ; (2) pour chaque courbe réelle simple γ dans 3 et chaque d1, la variété (algebrique) limite T γ,d V satisfait à la condition de Phragmén-Lindelöf forte. Ces conditions sont aussi nécessaires que pour toute variété analytique V de dimension pure k vérifie la condition PL loc .

We prove that an analytic surface V in a neighborhood of the origin in 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 3 and each d1, 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 .

@article{AFST_2011_6_20_S2_71_0,
     author = {R\"udiger W. Braun and Reinhold Meise and B. A. Taylor},
     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, Toulouse},
     volume = {Ser. 6, 20},
     number = {S2},
     year = {2011},
     doi = {10.5802/afst.1306},
     zbl = {1228.32011},
     mrnumber = {2858168},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1306/}
}
Rüdiger W. Braun; Reinhold Meise; B. A. Taylor. 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, Série 6, Tome 20 (2011) no. S2, pp. 71-99. doi : 10.5802/afst.1306. https://afst.centre-mersenne.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 1714282 | Zbl 0933.32047

[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 1974640 | Zbl 1023.32018

[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 1952326 | Zbl 1030.32008

[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 2068173 | Zbl 1061.32026

[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( 4 ), Trans. Amer. Math. Soc., 365, p. 1315-1383 (2004). | MR 2034311 | Zbl 1039.32010

[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 2174311 | Zbl 1089.32030

[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 2218707 | Zbl 1096.31003

[8] Chirka (E. M.).— Complex Analytic sets. Kluver, Dordrecht (1989). | MR 1111477 | Zbl 0683.32002

[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 2181757 | Zbl 1088.31004

[10] Heinrich (T.).— Eine geometrische Charakterisierung des lokalen Phragmén-Lindelöf Prinzips für algebraische Flächen in n . Dissertation, Düsseldorf, 2008. Electronic version .

[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 336041 | Zbl 0282.35015

[12] Meise (R.), Taylor (B. A.).— Phragmén-Lindelöf conditions for graph varieties, Result. Math., 36, p. 121-148 (1999). | MR 1706532 | Zbl 0941.32032

[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 1091835 | Zbl 0703.46025

[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 1343660 | Zbl 0835.32008

[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 1458816 | Zbl 0896.32008

[16] Vogt (D.).— Extension operators for real analytic functions on compact subvarieties of d , J. Reine Angew. Math., 606, p. 217-233 (2007). | MR 2337649 | Zbl 1133.46014

[17] Whitney (W.).— ‘Complex analytic varieties’, Addison-Wesley Pub. Co. (1972). | MR 387634 | Zbl 0265.32008