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Sharp estimates for\(\bar \partial \) on convex domains of finite typeon convex domains of finite type

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Arkiv för Matematik

Abstract

Let Ω be a bounded convex domain in C n , with smooth boundary of finite typem.

The equation\(\bar \partial u = f\) is solved in Ω with sharp estimates: iff has bounded coefficients, the coefficients of our solutionu are in the Lipschitz space Λ. Optimal estimates are also given when data have coefficients belonging toL p(Ω),p≥1.

We solve the\(\bar \partial \)-equation by means of integral operators whose kernels are not based on the choice of a “good” support function. Weighted kernels are used; in order to reflect the geometry ofbΩ, we introduce a weight expressed in terms of the Bergman kernel of Ω.

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  1. Laboratoire Emile Picard, Université Paul Sabatier, FR-31062, Toulouse Cedex, France

    Anne Cumenge

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  1. Anne Cumenge
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Cumenge, A. Sharp estimates for\(\bar \partial \) on convex domains of finite typeon convex domains of finite type. Ark. Mat. 39, 1–25 (2001). https://doi.org/10.1007/BF02388789

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  • DOI: https://doi.org/10.1007/BF02388789

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