C k -estimates for the ¯-equation on concave domains of finite type
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 15 (2006) no. 3, pp. 399-426.

Les estimées C k pour les domaines convexes de type fini ont été établies dans [7] pour k=0 et dans [2] pour k>0. Nous voulons ici étudier le cas des domaines concaves de type fini. Comme pour le cas strictement pseudoconvexe, nous adaptons les outils utilisés par K. Diederich, B. Fisher et J.E. Fornæss et W. Alexandre en échangeant le rôle des variables dans les noyaux intégraux de leurs opérateurs. Cependant le comportement au bord des nouveaux noyaux n’est plus le même et il faut modifier la fonction de support de K. Diederich et J.E. Fornæss. Elle perdra son holomorphie et générera un terme résiduel dans la formule d’homotopie dont il faudra tenir compte.

C k estimates for convex domains of finite type in n are known from [7] for k=0 and from [2] for k>0. We want to show the same result for concave domains of finite type. As in the case of strictly pseudoconvex domain, we fit the method used in the convex case to the concave one by switching z and ζ in the integral kernel of the operator used in the convex case. However the kernel will not have the same behavior on the boundary as in the Diederich-Fischer-Fornæss-Alexandre work. To overcome this problem we have to alter the Diederich-Fornæss support function. Also we have to take care of the so generated residual term in the homotopy formula.

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     author = {Alexandre, William},
     title = {$C^k$-estimates for the $\overline{\partial }$-equation on concave domains of finite type},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {399--426},
     publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 15},
     number = {3},
     year = {2006},
     doi = {10.5802/afst.1126},
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     url = {http://www.numdam.org/articles/10.5802/afst.1126/}
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Alexandre, William. $C^k$-estimates for the $\overline{\partial }$-equation on concave domains of finite type. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 15 (2006) no. 3, pp. 399-426. doi : 10.5802/afst.1126. http://www.numdam.org/articles/10.5802/afst.1126/

[1] Alexandre, W. Construction d’une fonction de support à la Diederich-Fornæss, PUB. IRMA, Lille, Volume 54 (2001) no. III

[2] Alexandre, W. C k estimates for ¯ on convex domain of finite type (to appear in Mat. Z.)

[3] Bruna, J.; Charpentier, P.; Dupain, Y. Zero varieties for the Nevanlinna class in convex domains of finite type in n , Ann. Math., Volume 147 (1998), pp. 391-415 | MR 1626753 | Zbl 0912.32001

[4] Cumenge, A. Estimées Lipscitz optimales dans les convexes de type fini, C. R. Acad. Sci. Paris, Volume 325 (1997), pp. 1077-1080 | MR 1614008 | Zbl 0904.32013

[5] Cumenge, A. Sharp estimates for ¯ in convex domains of finite type (1998) (Prépublication du lab. de math. E. Picard, Univ. Paul Sabatier, Toulouse, p. 1-22)

[6] Diederich, K.; Fornæss, J.E. Support functions for convex domains of finite type, Math. Z., Volume 230 (1999), pp. 145-164 | MR 1671870 | Zbl 1045.32016

[7] Diederich, K.; Fischer, B.; Fornæss, J. E. Hölder estimates on convex domains of finite type, Math. Z., Volume 232 (1999), pp. 43-61 | MR 1714279 | Zbl 0932.32008

[8] Diederich, K.; Mazzilli, E. Zero varieties for the Nevanlinna class on all convex domains of finite type, Nagoya Math. J., Volume 163 (2001), pp. 215-227 | MR 1855196 | Zbl 0994.32003

[9] Fischer, B. L p estimates on convex domains of finite type, Math. Z., Volume 236 (2001), pp. 401-418 | MR 1815835 | Zbl 0984.32022

[10] Hefer, T. Hölder and L p estimates for ¯ on convex domains of finite type depending on Catlin’s multitype, Math. Z., Volume 242 (2002), pp. 367-398 | MR 1980628 | Zbl 1048.32024

[11] Lieb, I.; Range, R. M. Lösungsoperatoren für den Cauchy-Riemann-Komplex mit C k -Abschätzungen, Math. Ann., Volume 253 (1980), pp. 145-164 | MR 597825 | Zbl 0441.32007

[12] McNeal, J. D. Convex domains of finite type, J. Functional Anal., Volume 108 (1992), pp. 361-373 | MR 1176680 | Zbl 0777.31007

[13] McNeal, J. D. Estimates on the Bergman kernels of convex domains, Adv. in Math., Volume 109 (1994) no. 1, pp. 108-139 | MR 1302759 | Zbl 0816.32018

[14] Michel, J. ¯-Problem für stückweise streng pseudokonvexe Gebiete in n , Math. Ann., Volume 280 (1988), pp. 46-68 | MR 928297 | Zbl 0617.32032

[15] Range, R. M. Holomorphic Functions and Integral Representations in Several Complex Variables, Springer-Verlag, New York, 1986 | MR 847923 | Zbl 0591.32002

[16] Seeley, R. T. Extension of C -functions defined in a half space, Proc. Amer. Soc., Volume 15 (1964), pp. 625-626 | MR 165392 | Zbl 0127.28403

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