Math. Z. 236, 401–418 (2001) Digital Object Identifier (DOI) 10.1007/s002090000184
Lp estimates on convex domains of finite type Bert Fischer Mathematik, Bergische Universit¨at-GHS Wuppertal, Gaußstr. 20, 42097 Wuppertal (e-mail: fi
[email protected]) Received May 31, 1999; in final form May 31, 2000 / c Springer-Verlag 2000 Published online December 8, 2000 -
1 Introduction
This paper continues the investigation of convex finite type domains by means of explicite integral formulas which started with [DiFo] and [DiFiFo]. In [DiFo] Diederich and Fornæss constructed smooth support functions for convex domains of finite type and proved that these support functions satisfy some nice estimates on the given domain. In [DiFiFo] the authors used these ¯ support functions to construct some ∂-solving Cauchy-Fantappi´e kernels. After proving some additional estimates for the support functions and their Leray decomposition they could prove that the solutions given by these kernels satisfy the best possible H¨older estimates. These results have also been used in [DiMa] to improve some theorem of [BrChDu] about the zero sets of functions of the Nevanlinna class in convex domains of finite type. ¯ In this paper we construct some ∂-solving integral operators that satisfy the best possible estimates with respect to Lp norms. More precisely we prove the following theorem. Theorem 1.1 Let D ⊂⊂ Cn be a linearly convex domain with C ∞ -smooth boundary of finite type m. We denote by Lp(0,r) (D) the Banach space of (0, r)-forms whose coefficients belong to Lp (D) by Λα(0,r) (D) the Banach space of (0, r)-forms whose coefficients are uniformly H¨older continuous of order α on D and by BM O(0,r) (D) the space of (0, r)-forms with BMOcoefficients.
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