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Let $X$ be a complex manifold of dimension $n\geqslant 2$ and let $\Omega\Subset X$ be a weakly $q$-convex domain with smooth boundary. Assume that $E$ is a holomorphic line bundle over $X$ and $E^{\otimes m}$ is the $m$-times tensor product of $E$ for positive integer $m$. If there exists a strongly plurisubharmonic function on a neighborhood of $b\Omega$, then we solve the $\overline{\partial}$-problem with support condition in $\Omega$ for forms of type $(r,s)$, $s\geqslant q$ with values in $E^{\otimes m}$. Moreover, the solvability of the $\overline{\partial}_{b}$-problem on boundaries of weakly $q$-convex domains with smooth boundary in K\"{a}hler manifolds are given. Furthermore, we shall establish an extension theorem for the $\overline\partial_{b}$-closed forms.
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참고문헌 신청
A. Andreotti / 1961 / Sopra un teorema di Kodaira / Ann. Scuola Norm. Sup. Pisa 15(3):283
J. Cao / 2004 / Estimates for the $\partial$ -Neumann problem and nonexistence of $C^2$ Levi-flat hypersurfaces in $P^n$ / Math. Z. 248(1):183~221
M. Derridj / 1978 / Regularitepour $\partial$ dans quelques domaines faiblement pseudo-convexes / J. Differential Geom. 13(4):559~576
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P. A. Griffths / 1966 / The extension problem in complex analysis. II. Embeddings with positive normal bundle / Amer. J. Math. 88:366~446
J. Cao / 2004 / Estimates for the
M. Derridj / 1978 / Regularitepour
G. B. Folland / 1972 / The Neumann Problem for the Cauchy-Riemann Complex / Princeton University Press
P. A. Griffths / 1966 / The extension problem in complex analysis. II. Embeddings with positive normal bundle / Amer. J. Math. 88:366~446
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