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Let $M$ be an $n$-dimensional K\"ahler manifold with positive holomorphic bisectional curvature and let $\Omega\Subset M$ be a pseudoconvex domain of order $n-q$, $1\leq q\leq n$, with $C^{2}$ smooth boundary. Then, we study the (weighted) $\overline\partial$-equation with support conditions in $\Omega$ and the closed range property of $\overline\partial$ on $\Omega$. Applications to the $\overline\partial$-closed extensions from the boundary are given. In particular, for $q=1$, we prove that there exists a number $\ell_{0}>0$ such that the $\overline\partial$-Neumann problem and the Bergman projection are regular in the Sobolev space $W^{\ell}(\Omega)$ for $\ell<\ell_{0}$.
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참고문헌 (27)
참고문헌 신청
O. Abdelkader / 2007 / Solution to ∂-equations with exact support on pseudoconvex manifolds / Int. J. Geom. Methods Mod. Phys 4(3):339~348
A. Andreotti / 1965 / Carleman estimates for the Laplace-Beltrami equation on complex manifolds / dInst. Hautes Etudes Sci. Publ. Math (25):81~130
B. Berndtsson / 2000 / A Sobolev mapping property of the Bergman kernel / Math. Z 235(1):1~10
H. P. Boas / 1990 / Equivalence of regularity for the Bergman projection and the ∂-Neumann problem / Manuscripta Math 67(1):25~33
J. Cao / 2007 / The ∂-Cauchy problem and nonexistence of Lipschitz Levi-flat hypersurfaces in CPn with n ≥ 3 / Math. Z 256(1):175~192
A. Andreotti / 1965 / Carleman estimates for the Laplace-Beltrami equation on complex manifolds / dInst. Hautes Etudes Sci. Publ. Math (25):81~130
B. Berndtsson / 2000 / A Sobolev mapping property of the Bergman kernel / Math. Z 235(1):1~10
H. P. Boas / 1990 / Equivalence of regularity for the Bergman projection and the ∂-Neumann problem / Manuscripta Math 67(1):25~33
J. Cao / 2007 / The ∂-Cauchy problem and nonexistence of Lipschitz Levi-flat hypersurfaces in CPn with n ≥ 3 / Math. Z 256(1):175~192
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