Extremal bases, geometrically separated domains and applications

Ph. Charpentier ; Y. Dupain

Algebra i analiz, Tome 26 (2014) no. 1, pp. 196-269.

Résumé

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 $\mathbb C^n$, $n\geq3$. 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 Szegö projections. As an application, a result by C. Fefferman, J. J. Kohn, and M. Machedon for the local Hölder estimate of the Szegö projection is refined, by removing the arbitrarily small loss in the Hölder index and giving a stronger nonisotropic estimate.

Détail

Keywords: finite type, extremal basis, complex geometry, adapted plurisubharmonic function, Bergman and Szegö projections.

Bibliographie

[1] Ahn H., Cho S., “On the mapping properties of the Bergman projection on pseudoconvex domains with one degenerate eigenvalue”, Complex Variable Theory Appl., 39:4 (1999), 365–379 | DOI | MR | Zbl

[2] Bruna J., Charpentier Ph., Dupain Y., “Zero varieties for the Nevanlinna class in convex domains of finite type in $\mathbb C^n$”, Ann. of Math., 147:2 (1998), 391–415 | DOI | MR | Zbl

[3] Bloom T., “On the contact between complex manifolds and real hypersurfaces in $\mathbb C^3$”, Trans. Amer. Math. Soc., 263:2 (1981), 515–529 | MR | Zbl

[4] Catlin D., “Subelliptic estimates for $\bar\partial$-Neumann problem on pseudoconvex domains”, Ann. Math. (2), 126:1 (1987), 131–191 | DOI | MR | Zbl

[5] Catlin D., “Estimates of invariant metrics on pseudoconvex domains of dimension two”, Math. Z., 200:3 (1989), 429–466 | DOI | MR | Zbl

[6] Charpentier P., Dupain Y., “Une estimation des coefficients tangents d'un courant positif fermé dans un domaine de $\mathbb C^3$”, Publ. Mat., 36:1 (1992), 319–349 | DOI | MR | Zbl

[7] Charpentier P., Dupain Y., “Estimates for the Bergman and Szegö projections for pseudo-convex domains of finite type with locally diagonalizable Levi form”, Publ. Mat., 50:2 (2006), 413–446 | DOI | MR | Zbl

[8] Charpentier P., Dupain Y., “Geometry of pseudo-convex domains of finite type with locally diagonalizable Levi form and Bergman kernel”, J. Math. Pures Appl., 85:1 (2006), 71–118 | DOI | MR | Zbl

[9] Der-Chen Chang, Grellier S., “Estimates for the Szegö kernel on decoupled domains”, J. Math. Anal. Appl., 187:2 (1994), 628–649 | DOI | MR | Zbl

[10] Cho S., “Boundary behavior of the Bergman kernel function on some pseudoconvex domains in $\mathbb C^n$”, Trans. Amer. Math. Soc., 345:2 (1994), 803–817 | MR | Zbl

[11] Cho S., “Estimates of the Bergman kernel function on certain pseudoconvex domains in $\mathbb C^n$”, Math. Z., 222:2 (1996), 329–339 | DOI | MR | Zbl

[12] Cho S., “Estimates of invariants metrics on pseudoconvex domains with comparable Levi form”, J. Math. Kyoto Univ., 42:2 (2002), 337–349 | MR | Zbl

[13] Cho S., “Estimates of the Bergman kernel function on pseudoconvex domains with comparable Levi form”, J. Korean Math. Soc., 39:3 (2002), 425–437 | DOI | MR | Zbl

[14] Cho S., “Boundary behavior of the Bergman kernel function on pseudoconvex domains with comparable Levi form”, J. Math. Anal. Appl., 283:3 (2003), 386–397 | DOI | MR | Zbl

[15] Christ M., “Regularity peoperties of the $\bar\partial_b$ equation on weakly pseudoconvex CR manifold of dimension 3”, J. Amer. Math. Soc., 1:3 (1988), 587–646 | MR | Zbl

[16] Conrad M., Anisotrope optimale Pseudometriken für lineal konvex Gebeite von endlichem Typ (mit Anwendungen), PhD Thesis, Bergische Universität-GHS Wuppertal, 2002

[17] D'Angelo J. P., “Real hypersurfaces, orders of contact, and applications”, Ann. Math., 115:3 (1982), 615–637 | DOI | MR

[18] Derridj M., “Régularité hölderienne pour $\square_b$, sur des hypersurfaces de $\mathbb C^n$, à forme de Levi décomposable en blocs”, J. Geom. Anal., 9:4 (1999), 627–652 | DOI | MR | Zbl

[19] Diederich K., Fornaess J. E., “Support functions for convex domains of finite type”, Math. Z., 230:1 (1999), 145–164 | DOI | MR | Zbl

[20] Diederich K., Fornaess J. E., “Lineally convex domains of finite type: holomorphic support functions”, Manuscripta Math., 112:4 (2003), 403–431 | DOI | MR | Zbl

[21] Diederich K., Fischer B., “Hölder estimates on lineally convex domains of finite type”, Michigan Math. J., 54:2 (2006), 341–452 | DOI | MR

[22] Fefferman C. L., Kohn J. J., “Hölder estimates on domains of complex dimension two and on three dimensional CR manifold”, Adv. in Math., 69:2 (1988), 233–303 | DOI | MR

[23] Fefferman C. L., Kohn J. J., Machedon M., “Hölder estimates on CR manifold with a diagonalizable Levi form”, Adv. Math., 84:1 (1990), 1–90 | DOI | MR | Zbl

[24] Gay R., Sebbar A., “Division et extension dans l'algèbre $A^\infty(\Omega)$ d'un ouvert pseudoconvexe à bord lisse de $\mathbb C^n$”, Math. Z., 189:3 (1985), 421–447 | DOI | MR | Zbl

[25] Hefer T., “Extremal bases and Hölder estimates for $\bar\partial$ on convex domains of finite type”, Michigan Math. J., 52:3 (2004), 573–602 | DOI | MR | Zbl

[26] Herbort G., “Logarithmic growth of the Bergman kernel for weakly pseudoconvex domains in $\mathbb C^n$ of finite type”, Manuscripta Math., 45:1 (1983), 69–76 | DOI | MR | Zbl

[27] Kang H., “An approximation theorem for Szegö kernels and applications”, Michigan Math. J., 37:3 (1990), 447–458 | DOI | MR | Zbl

[28] Kerzman N., “The Bergman kernel function. Differentiability at the boundary”, Math. Ann., 195 (1972), 149–158 | DOI | MR

[29] Kiselman C. O., “A differential inequality characterizing weak lineal convexity”, Math. Annalen, 311:1 (1998), 1–10 | DOI | MR | Zbl

[30] Kohn J. J., Nirenberg L., “Non coercive boundary value problems”, Comm. Pure Appl. Math., 18 (1965), 443–492 | DOI | MR | Zbl

[31] Koenig K. D., “On maximal Sobolev and Hölder estimates for the tangential Cauchy–Riemann operator and boundary Laplacian”, Amer. J. Math., 124:2 (2002), 129–197 | DOI | MR | Zbl

[32] Koenig K. D., “Comparing the Bergman and Szegö projections on domains with subelliptic boundary Laplacian”, Math. Ann., 339:3 (2007), 667–693 | DOI | MR | Zbl

[33] Kohn J. J., “Estimates for $\bar\partial_b$ on pseudoconvex CR manifolds”, Proc. Sympos. Pure Math., 43, Amer. Math. Soc., Providence, RI, 1985, 207–217 | DOI | MR

[34] Machedon M., “Szegö kernels on pseudoconvex domains with one degenerate eigenvalue”, Ann. of Math. (2), 128:3 (1988), 619–640 | DOI | MR | Zbl

[35] McNeal J., “Boundary behavior of the Bergman kernel function in $\mathbb C^2$”, Duke Math. J., 58:2 (1989), 499–512 | DOI | MR | Zbl

[36] McNeal J., “Local geometry of decoupled pseudoconvex domains”, Complex analysis (Wuppertal, 1991), Aspekts Math., 1, Vieweg, Braunschweig, 1991, 223–230 | DOI | MR

[37] McNeal J., “Estimates on Bergman kernels of convex domains”, Adv. Math., 109:1 (1994), 108–139 | DOI | MR | Zbl

[38] McNeal J., “Uniform subelliptic estimates on scaled convex domains of finite type”, Proc. Amer. Math. Soc., 130:1 (2002), 39–47 | DOI | MR | Zbl

[39] McNeal J. D., Stein E. M., “Mapping properties of the Bergman projection on convex domains of finite type”, Duke Math. J., 73:1 (1994), 177–199 | DOI | MR | Zbl

[40] McNeal J. D., Stein E. M., “The Szegö projection on convex domains”, Math. Z., 224:4 (1997), 519–553 | DOI | MR | Zbl

[41] Nagel A., Rosay J. P., Stein E. M., Wainger S., “Estimates for the Bergman and Szegö kernels in $\mathbb C^2$”, Annals of Math. (2), 129:1 (1989), 113–147 | DOI | MR

[42] Nagel A., Stein E., Wainger S., “Balls and metrics defined by vectors fields: basic properties”, Acta Math., 155:1–2 (1985), 103–147 | DOI | MR | Zbl