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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Optimal $ L\sp p$ and Hölder estimates for the Kohn solution of the $ \overline\partial$-equation on strongly pseudoconvex domains


Author: Der-Chen E. Chang
Journal: Trans. Amer. Math. Soc. 315 (1989), 273-304
MSC: Primary 32F20; Secondary 35N15
DOI: https://doi.org/10.1090/S0002-9947-1989-0937241-0
MathSciNet review: 937241
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Abstract: Let $ \Omega $ be an open, relatively compact subset in $ {{\mathbf{C}}^{n + 1}}$, and assume the boundary of $ \Omega $, $ \partial \Omega $, is smooth and strongly pseudoconvex. Let $ \operatorname{Op}(K)$ be an integral operator with mixed type homogeneities defined on $ \overline \Omega $: i.e., $ K$ has the form as follows:

$\displaystyle \sum\limits_{k,l \geq 0} {{E_k}{H_l},} $

where $ {E_k}$ is a homogeneous kernel of degree $ - k$ in the Euclidean sense and $ {H_l}$ is homogeneous of degree $ - l$ in the Heisenberg sense. In this paper, we study the optimal $ {L^p}$ and Hölder estimates for the kernel $ K$. We also use Lieb-Range's method to construct the integral kernel for the Kohn solution $ \overline {{\partial^\ast}} {\mathbf{N}}$ of the Cauchy-Riemann equation on the Siegel upper-half space and then apply our results to $ \overline {{\partial^\ast}} {\mathbf{N}}$. On the other hand, we prove Lieb-Range's kernel gains $ 1$ in "good" directions (hence gains $ 1/2$ in all directions) via Phong-Stein's theory. We also discuss the transferred kernel from the Siegel upper-half space to $ \Omega $.

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DOI: https://doi.org/10.1090/S0002-9947-1989-0937241-0
Article copyright: © Copyright 1989 American Mathematical Society