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Let $\Omega$ be a bounded, uniformly totally pseudoconvex domain in $\mathbb{C}^2$ with the smooth boundary $b\Omega$. Assuming that $\Omega$ satisfies the negative $\bar\partial$ property. Let $M$ be a positive, finite area divisor of $\Omega$. In this paper, we will prove that: if $\Omega$ admits a maximal type $F$ and the $\rm \check{C}$eck cohomology class of the second order vanishes in $\Omega$, there is a bounded holomorphic function in $\Omega$ such that its zero set is $M$. The proof is based on the method given by Shaw \cite{Sha89}.
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참고문헌 (28)
참고문헌 신청
A. Bonami / 1982 / Solutions de l’equation ?∂ et z´eros de la class de Nevanlinna dans certains domaines faiblement pseudoconvexes / Ann. Inst. Fourier (Grenoble) 32 : 53
A. V. Romanov / 1976 / A formula and estimates for solutions of the tangential Cauchy-Riemann equation / Math. Sb 99 : 58 ~ 83
B. Berndtsson / 1980 / Integral formulas for the ∂ ?∂-equation and zeros of bounded holomorphic functions in the unit ball / Math. Ann 249 (2) : 163 ~ 176
G. M. Henkin / 1969 / Integral representations of functions holomorphic in strictly-pseudoconvex domains and some applications / Math. USSR Sbornik 7 (4) : 597 ~ 616
G. M. Henkin / 1977 / H. Lewy’s equation and analysis on a pseudoconvex manifold II / Math. USSR Sbornik 1 : 63 ~ 94
A. V. Romanov / 1976 / A formula and estimates for solutions of the tangential Cauchy-Riemann equation / Math. Sb 99 : 58 ~ 83
B. Berndtsson / 1980 / Integral formulas for the ∂ ?∂-equation and zeros of bounded holomorphic functions in the unit ball / Math. Ann 249 (2) : 163 ~ 176
G. M. Henkin / 1969 / Integral representations of functions holomorphic in strictly-pseudoconvex domains and some applications / Math. USSR Sbornik 7 (4) : 597 ~ 616
G. M. Henkin / 1977 / H. Lewy’s equation and analysis on a pseudoconvex manifold II / Math. USSR Sbornik 1 : 63 ~ 94
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