Extremal bases, geometrically separated domains and applications

Charpentier, Ph.; Dupain, Y.

doi:10.1090/S1061-0022-2014-01335-0

Extremal bases, geometrically separated domains and applications
HTML articles powered by AMS MathViewer

by Ph. Charpentier and Y. Dupain

St. Petersburg Math. J. 26 (2015), 139-191

DOI: https://doi.org/10.1090/S1061-0022-2014-01335-0

Published electronically: November 21, 2014

PDF | Request permission

Abstract:

The notion of an extremal basis of tangent vector fields is introduced for a boundary point of finite type of a pseudo-convex domain in $\mathbb {C}^n$, $n\geq 3$. By using this notion, the class of geometrically separated domains at a boundary point is defined and a description of their complex geometry is presented. Examples of such domains are given, for instance, by locally lineally convex domains, domains with locally diagonalizable Levi form at a point, or by domains for which the Levi form has comparable eigenvalues near a point. Moreover, it is shown that geometrically separated domains can be localized. An example of a not geometrically separated domain is presented. Next, the so-called “adapted plurisubharmonic functions” are defined and sufficient conditions, related to extremal bases, for their existence are given. Then, for these domains, when such functions exist, global and local sharp estimates are proved for the Bergman and Szegő projections. As an application, a result by C. Fefferman, J. J. Kohn, and M. Machedon for the local Hölder estimate of the Szegő projection is refined, by removing the arbitrarily small loss in the Hölder index and giving a stronger nonisotropic estimate.

References

H. Ahn and S. Cho, On the mapping properties of the Bergman projection on pseudoconvex domains with one degenerate eigenvalue, Complex Variables Theory Appl. 39 (1999), no. 4, 365–379. MR 1727631, DOI 10.1080/17476939908815203
Joaquim Bruna, Philippe Charpentier, and Yves Dupain, Zero varieties for the Nevanlinna class in convex domains of finite type in $\mathbf C^n$, Ann. of Math. (2) 147 (1998), no. 2, 391–415. MR 1626753, DOI 10.2307/121013
Thomas Bloom, On the contact between complex manifolds and real hypersurfaces in $\textbf {C}^{3}$, Trans. Amer. Math. Soc. 263 (1981), no. 2, 515–529. MR 594423, DOI 10.1090/S0002-9947-1981-0594423-0
David Catlin, Subelliptic estimates for the $\overline \partial$-Neumann problem on pseudoconvex domains, Ann. of Math. (2) 126 (1987), no. 1, 131–191. MR 898054, DOI 10.2307/1971347
David W. Catlin, Estimates of invariant metrics on pseudoconvex domains of dimension two, Math. Z. 200 (1989), no. 3, 429–466. MR 978601, DOI 10.1007/BF01215657
Ph. Charpentier and Y. Dupain, Une estimation des coefficients tangents d’un courant postif fermé dans un domaine de $\textbf {C}^3$, Publ. Mat. 36 (1992), no. 1, 319–349 (French, with English summary). MR 1179619, DOI 10.5565/PUBLMAT_{3}6192_{2}2
Philippe Charpentier and Yves Dupain, Estimates for the Bergman and Szegö projections for pseudoconvex domains of finite type with locally diagonalizable Levi form, Publ. Mat. 50 (2006), no. 2, 413–446. MR 2273668, DOI 10.5565/PUBLMAT_{5}0206_{0}8
Philippe Charpentier and Yves Dupain, Geometry of pseudo-convex domains of finite type with locally diagonalizable Levi form and Bergman kernel, J. Math. Pures Appl. (9) 85 (2006), no. 1, 71–118 (English, with English and French summaries). MR 2200592, DOI 10.1016/j.matpur.2005.10.001
Der-Chen Chang and Sandrine Grellier, Estimates for the Szegő kernel on decoupled domains, J. Math. Anal. Appl. 187 (1994), no. 2, 628–649. MR 1297047, DOI 10.1006/jmaa.1994.1379
Sanghyun Cho, Boundary behavior of the Bergman kernel function on some pseudoconvex domains in $\textbf {C}^n$, Trans. Amer. Math. Soc. 345 (1994), no. 2, 803–817. MR 1254189, DOI 10.1090/S0002-9947-1994-1254189-7
Sanghyun Cho, Estimates of the Bergman kernel function on certain pseudoconvex domains in $\textbf {C}^n$, Math. Z. 222 (1996), no. 2, 329–339. MR 1429340, DOI 10.1007/PL00004537
Sanghyun Cho, Estimates of invariant metrics on pseudoconvex domains with comparable Levi form, J. Math. Kyoto Univ. 42 (2002), no. 2, 337–349. MR 1966842, DOI 10.1215/kjm/1250283875
Sanghyun Cho, Estimates of the Bergman kernel function on pseudoconvex domains with comparable Levi form, J. Korean Math. Soc. 39 (2002), no. 3, 425–437. MR 1895763, DOI 10.4134/JKMS.2002.39.3.425
Sanghyun Cho, Boundary behavior of the Bergman kernel function on pseudoconvex domains with comparable Levi form, J. Math. Anal. Appl. 283 (2003), no. 2, 386–397. MR 1991815, DOI 10.1016/S0022-247X(03)00160-4
Michael Christ, Regularity properties of the $\overline \partial _b$ equation on weakly pseudoconvex CR manifolds of dimension $3$, J. Amer. Math. Soc. 1 (1988), no. 3, 587–646. MR 928903, DOI 10.1090/S0894-0347-1988-0928903-2
M. Conrad, Anisotrope optimale Pseudometriken für lineal konvex Gebeite von endlichem Typ (mit Anwendungen), PhD Thesis, Bergische Univ. Wuppertal, 2002.
John P. D’Angelo, Real hypersurfaces, orders of contact, and applications, Ann. of Math. (2) 115 (1982), no. 3, 615–637. MR 657241, DOI 10.2307/2007015
M. Derridj, Régularité höldérienne pour $\square _b$, sur des hypersurfaces de $\textbf {C}^n$, à forme de Levi décomposable en blocs, J. Geom. Anal. 9 (1999), no. 4, 627–652 (French). MR 1757582, DOI 10.1007/BF02921976
Klas Diederich and John Erik Fornæss, Support functions for convex domains of finite type, Math. Z. 230 (1999), no. 1, 145–164. MR 1671870, DOI 10.1007/PL00004683
Klas Diederich and John Erik Fornæss, Lineally convex domains of finite type: holomorphic support functions, Manuscripta Math. 112 (2003), no. 4, 403–431. MR 2064651, DOI 10.1007/s00229-003-0418-9
Klas Diederich and Bert Fischer, Hölder estimates on lineally convex domains of finite type, Michigan Math. J. 54 (2006), no. 2, 341–352. MR 2252763, DOI 10.1307/mmj/1156345598
Charles L. Fefferman and Joseph J. Kohn, Hölder estimates on domains of complex dimension two and on three-dimensional CR manifolds, Adv. in Math. 69 (1988), no. 2, 223–303. MR 946264, DOI 10.1016/0001-8708(88)90002-3
C. L. Fefferman, J. J. Kohn, and M. Machedon, Hölder estimates on CR manifolds with a diagonalizable Levi form, Adv. Math. 84 (1990), no. 1, 1–90. MR 1075233, DOI 10.1016/0001-8708(90)90036-M
Roger Gay and Ahmed Sebbar, Division et extension dans l’algèbre $A^\infty (\Omega )$ d’un ouvert pseudo-convexe à bord lisse de $\textbf {C}^n$, Math. Z. 189 (1985), no. 3, 421–447 (French). MR 783566, DOI 10.1007/BF01164163
Torsten Hefer, Extremal bases and Hölder estimates for $\overline \partial$ on convex domains of finite type, Michigan Math. J. 52 (2004), no. 3, 573–602. MR 2097399, DOI 10.1307/mmj/1100623414
Gregor Herbort, Logarithmic growth of the Bergman kernel for weakly pseudoconvex domains in $\textbf {C}^{3}$ of finite type, Manuscripta Math. 45 (1983), no. 1, 69–76. MR 722923, DOI 10.1007/BF01168581
Hyeonbae Kang, An approximation theorem for Szegő kernels and applications, Michigan Math. J. 37 (1990), no. 3, 447–458. MR 1077328, DOI 10.1307/mmj/1029004202
Norberto Kerzman, The Bergman kernel function. Differentiability at the boundary, Math. Ann. 195 (1972), 149–158. MR 294694, DOI 10.1007/BF01419622
Christer O. Kiselman, A differential inequality characterizing weak lineal convexity, Math. Ann. 311 (1998), no. 1, 1–10. MR 1624326, DOI 10.1007/s002080050172
J. J. Kohn and L. Nirenberg, Non-coercive boundary value problems, Comm. Pure Appl. Math. 18 (1965), 443–492. MR 181815, DOI 10.1002/cpa.3160180305
Kenneth D. Koenig, On maximal Sobolev and Hölder estimates for the tangential Cauchy-Riemann operator and boundary Laplacian, Amer. J. Math. 124 (2002), no. 1, 129–197. MR 1879002
Kenneth D. Koenig, Comparing the Bergman and Szegö projections on domains with subelliptic boundary Laplacian, Math. Ann. 339 (2007), no. 3, 667–693. MR 2336063, DOI 10.1007/s00208-007-0128-9
J. J. Kohn, Estimates for $\bar \partial _b$ on pseudoconvex CR manifolds, Pseudodifferential operators and applications (Notre Dame, Ind., 1984) Proc. Sympos. Pure Math., vol. 43, Amer. Math. Soc., Providence, RI, 1985, pp. 207–217. MR 812292, DOI 10.1090/pspum/043/812292
Matei Machedon, Szegő kernels on pseudoconvex domains with one degenerate eigenvalue, Ann. of Math. (2) 128 (1988), no. 3, 619–640. MR 970613, DOI 10.2307/1971438
Jeffery D. McNeal, Boundary behavior of the Bergman kernel function in $\textbf {C}^2$, Duke Math. J. 58 (1989), no. 2, 499–512. MR 1016431, DOI 10.1215/S0012-7094-89-05822-5
Jeffery D. McNeal, Local geometry of decoupled pseudoconvex domains, Complex analysis (Wuppertal, 1991) Aspects Math., E17, Friedr. Vieweg, Braunschweig, 1991, pp. 223–230. MR 1122183
Jeffery D. McNeal, Estimates on the Bergman kernels of convex domains, Adv. Math. 109 (1994), no. 1, 108–139. MR 1302759, DOI 10.1006/aima.1994.1082
Jeffery D. McNeal, Uniform subelliptic estimates on scaled convex domains of finite type, Proc. Amer. Math. Soc. 130 (2002), no. 1, 39–47. MR 1855617, DOI 10.1090/S0002-9939-01-06373-0
J. D. McNeal and E. M. Stein, Mapping properties of the Bergman projection on convex domains of finite type, Duke Math. J. 73 (1994), no. 1, 177–199. MR 1257282, DOI 10.1215/S0012-7094-94-07307-9
J. D. McNeal and E. M. Stein, The Szegő projection on convex domains, Math. Z. 224 (1997), no. 4, 519–553. MR 1452048, DOI 10.1007/PL00004593
A. Nagel, J.-P. Rosay, E. M. Stein, and S. Wainger, Estimates for the Bergman and Szegő kernels in $\textbf {C}^2$, Ann. of Math. (2) 129 (1989), no. 1, 113–149. MR 979602, DOI 10.2307/1971487
Alexander Nagel, Elias M. Stein, and Stephen Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math. 155 (1985), no. 1-2, 103–147. MR 793239, DOI 10.1007/BF02392539