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1. CMB 2010 (vol 53 pp. 311)
Remark on Zero Sets of Holomorphic Functions in Convex Domains of Finite Type We prove that if the $(1,1)$-current of integration on an analytic subvariety $V\subset D$ satisfies the uniform Blaschke condition, then $V$ is the zero set of a holomorphic function $f$ such that $\log |f|$ is a function of bounded mean oscillation in $bD$. The domain $D$ is assumed to be smoothly bounded and of finite d'Angelo type. The proof amounts to non-isotropic estimates for a solution to the $\overline{\partial}$-equation for Carleson measures.
Categories:32A60, 32A35, 32F18 |
2. CMB 2008 (vol 51 pp. 618)
Vanishing Theorems in Colombeau Algebras of Generalized Functions Using a canonical linear embedding of the algebra
${\mathcal G}^{\infty}(\Omega)$ of Colombeau generalized functions in the space of
$\overline{\C}$-valued $\C$-linear maps on the space
${\mathcal D}(\Omega)$ of smooth functions with compact support, we give vanishing
conditions for functions and linear integral operators of class
${\mathcal G}^\infty$. These results are then applied to the zeros of holomorphic
generalized functions in dimension greater than one.
Keywords:Colombeau generalized functions, linear integral operators, generalized holomorphic functions Categories:32A60, 45P05, 46F30 |