In this note we construct -equations (inhomogeneous Cauchy-Riemann equations) without solutions. The construction involves Bochner-Martinelli type kernels and differentiation with respect to certain parameters in appropriate directions.
For some given logarithmically convex sequence M of positive numbers we construct a subspace of the space of rapidly decreasing infinitely differentiable functions on an unbounded closed convex set in ℝn. Due to the conditions on M each function of this space admits a holomorphic extension in ℂn. In the current article, the space of holomorphic extensions is considered and Paley-Wiener type theorems are established. To prove these theorems, some auxiliary results on extensions of holomorphic functions...
We show that a certain solution operator for ∂ in a space of forms square integrable against e-|z|2 is canonical, i.e., that it gives the minimal solution when applied to a ∂-closed form, and gives zero when applied to a form orthogonal to Ker ∂.As an application, we construct a canonical homotopy operator for i∂∂.
Convergence of special Green integrals for matrix factorization of the Laplace operator in is proved. Explicit formulae for solutions of -equation in strictly pseudo-convex domains in are obtained.
In this survey we shall tour the area of multidimensional complex analysis which centers around ∂̅-problems (i.e., the Cauchy-Riemann equations) on pseudoconvex domains. Along the way we shall highlight some of the classical milestones as well as more recent landmarks, and we shall discuss some of the major open problems and conjectures. For the sake of simplicity we will only consider domains in ; intriguing phenomena occur already in the simple setting of (Euclidean) convex domains. We will not...