$C^k$-estimates for the $\overline{\partial }$-equation on concave domains of finite type

. 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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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 15.3 (2006): 399-426. <http://eudml.org/doc/10006>.

@article{Alexandre2006,
abstract = {$C^k$ estimates for convex domains of finite type in $\mathbb\{C\}^n$ 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 $\zeta $ 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.},
affiliation = {LMPA, Université du Littoral - Côte d’Opale, maison de la recherche Blaise Pascal, B.P. 699, 62 228 Calais Cedex (France).},
author = {Alexandre, William},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {pseudo-concave domains; regularity of solutions; Cauchy-Riemann equations},
language = {eng},
number = {3},
pages = {399-426},
publisher = {Université Paul Sabatier, Toulouse},
title = {$C^k$-estimates for the $\overline\{\partial \}$-equation on concave domains of finite type},
url = {http://eudml.org/doc/10006},
volume = {15},
year = {2006},
}

TY - JOUR
AU - Alexandre, William
TI - $C^k$-estimates for the $\overline{\partial }$-equation on concave domains of finite type
JO - Annales de la faculté des sciences de Toulouse Mathématiques
PY - 2006
PB - Université Paul Sabatier, Toulouse
VL - 15
IS - 3
SP - 399
EP - 426
AB - $C^k$ estimates for convex domains of finite type in $\mathbb{C}^n$ 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 $\zeta $ 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.
LA - eng
KW - pseudo-concave domains; regularity of solutions; Cauchy-Riemann equations
UR - http://eudml.org/doc/10006
ER -

References

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