Sobolev estimates for the local extension of boundary holomorphic forms on real hypersurfaces in $\mathbb C^n$

Sanghyun Cho

doi:10.4134/JKMS.2013.50.3.479

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Korean Mathematical Society JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2013; 50(3): 479-491

Printed May 1, 2013

https://doi.org/10.4134/JKMS.2013.50.3.479

Sobolev estimates for the local extension of boundary holomorphic forms on real hypersurfaces in $\mathbb C^n$

Sanghyun Cho

Sogang University

Abstract

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Abstract

Let $M$ be a smooth real hypersurface in complex space of dimension $n$, $n\ge 3$, and assume that the Levi-form at $z_0$ on $M$ has at least ($q+1$)-positive eigenvalues, $1\leq q\leq n-2$. We estimate solutions of the local $\dibar$-closed extension problem near $z_0$ for $(p,q)$-forms in Sobolev spaces. Using this result, we estimate the local solution of tangential Cauchy-Riemann equation near $z_0$ in Sobolev spaces.

Keywords: tangential Cauchy-Riemann equation, boundary holomorphic forms

MSC numbers: Primary 32W05; Secondary 32W10