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

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.

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.

@article{AFST_2006_6_15_3_399_0,
     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},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 15},
     number = {3},
     year = {2006},
     pages = {399-426},
     doi = {10.5802/afst.1126},
     zbl = {1153.32024},
     mrnumber = {2246411},
     language = {en},
     url = {http://www.numdam.org/item/AFST_2006_6_15_3_399_0}
}
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, Serie 6, Volume 15 (2006) no. 3, pp. 399-426. doi : 10.5802/afst.1126. http://www.numdam.org/item/AFST_2006_6_15_3_399_0/

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