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Optimal Hölder and $ L\sp p$ estimates for $ \overline\partial\sb b$ on the boundaries of real ellipsoids in $ {\bf C}\sp n$


Author: Mei-Chi Shaw
Journal: Trans. Amer. Math. Soc. 324 (1991), 213-234
MSC: Primary 32F20; Secondary 32A25, 32F15, 35N15
DOI: https://doi.org/10.1090/S0002-9947-1991-1005084-7
MathSciNet review: 1005084
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Abstract: Let $ D$ be a real ellipsoid in $ {{\mathbf{C}}^n},n \geq 3$, with defining function $ \rho (z) = \sum\nolimits_{k = 1}^n {(x_k^{2{n_k}} + y_k^{2{m_k}})} - 1$, $ {z_k} = {x_k} + i{y_k}$, where $ {n_k},{m_k} \in N$. In this paper we study the sharp Hàlder and $ {L^p}$ estimates for the solutions of the tangential Cauchy-Riemann equations $ {\overline \partial _b}$ on the boundary $ \partial D$ of $ D$ using the integral kernel method. In particular, we proved that if $ \alpha \in L_{(0,1)}^\infty (\partial D)$ such that $ {\overline \partial _b}\alpha = 0$ on $ \partial D$ in the distribution sense, then there exists a $ u \in {\Lambda _{1/2m}}(\partial D)$ satisfying $ {\overline \partial _b}u = \alpha $ and $ {\left\Vert u \right\Vert _{{\Lambda _{1/2m}}(\partial D)}} \leq c{\left\Vert \alpha \right\Vert _{{L^\infty }(\partial D)}}$ for some constant $ c > 0$ independent of $ \alpha$, where $ {\Lambda _{1/2m}}(\partial D)$ is the Lipschitz space with exponent $ \frac{1} {{2m}}$ and $ 2m = {\max _{1 \leq k \leq n}}\min (2{n_k},2{m_k})$ is the type of the domain $ D$.


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DOI: https://doi.org/10.1090/S0002-9947-1991-1005084-7
Article copyright: © Copyright 1991 American Mathematical Society

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