Moduli for pointed convex domains.
Multipliers and weighted ∂ operator estimates.
We study estimates for the solution of the equation du=f in one variable. The new ingredient is the use of holomorphic functions with precise growth restrictions in the construction of explicit solution to the equation.
Non-solvability of the tangential -systems
We prove that for a real analytic generic submanifold of whose Levi-form has constant rank, the tangential -system is non-solvable in degrees equal to the numbers of positive and negative Levi-eigenvalues. This was already proved in [1] in case the Levi-form is non-degenerate (with non-necessarily real analytic). We refer to our forthcoming paper [7] for more extensive proofs.
On a class of -equations without solutions
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.
On a space of smooth functions on a convex unbounded set in ℝn admitting holomorphic extension in ℂn
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...
On canonical homotopy operators for ∂ in Fock type spaces in Cn.
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∂∂.
On construction of exact complexes connected with the Dolbeault complex.
On duality and interpolation for spaces of polyharmonic functions
On Integral Representations and a priori Lipschitz Estimates for the Canonical Solution of the ...-Equation.
On iterations of Green type integrals for matrix factorizations of the Laplace operator
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.
On ∂̅-problems on (pseudo)-convex domains
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...
On the absence of Poincaré lemma in tangential Cauchy-Riemann complexes
On the compactness of the -Neumann operator
On the extension of L2 holomorphic functions III: negligible weights.
On the Hartogs-Bochner extension phenomenon for differential forms.
On the Kuranishi family of deformations of complex structures over a tubular neighborhood of a strongly pseudo convex boundary with complex dimension 3.
On the parameter dependence of solutions to the ...-equation.
On the theorem of Frobenius for complex vector fields
On the -equation in a Banach space
Optimal Lipschitz estimates for the equation on a class of convex domains