We consider the (characteristic and non-characteristic) Cauchy problem for a system of constant coefficients partial differential equations with initial data on an affine subspace of arbitrary codimension. We show that evolution is equivalent to the validity of a principle on the complex characteristic variety and we study the relationship of this condition with the one introduced by Hörmander in the case of scalar operators and initial data on a hypersurface.

We show that the projections of the pluripolar hull of the graph of an analytic function in a subdomain of the complex plane are open in the fine topology.

We consider the solution operator $S{ℱ}_{\mu ,(p,q)}\to L^2{\left(\mu \right)}_{(p,q)}$ to the $\overline{\partial }$-operator restricted to forms with coefficients in ${ℱ}_{\mu }=\left\{ff\text{is}\text{entire}\text{and}{\int }_{{ℂ}^n}{\left|f\left(z\right)\right|}^2\mathrm{d}\mu \left(z\right)<\infty \right\}$. Here ${ℱ}_{\mu ,(p,q)}$ denotes $(p,q)$-forms with coefficients in ${ℱ}_{\mu }$, $L^2\left(\mu \right)$ is the corresponding $L^2$-space and $\mu$ is a suitable rotation-invariant absolutely continuous finite measure. We will develop a general solution formula $S$ to $\overline{\partial }$. This solution operator will have the property $Sv\perp {ℱ}_{(p,q)}\forall v\in {ℱ}_{(p,q+1)}$. As an application of the solution formula we will be able to characterize compactness of the solution operator in terms of compactness of commutators...

The Serre problem is solved for fiber bundles whose fibers are two-dimensional pseudoconvex hyperbolic Reinhardt domains.

Generalizations of the theorem of Forelli to holomorphic mappings into complex spaces are given.

General versions of Glicksberg's theorem concerning zeros of holomorphic maps and of Hurwitz's theorem on sequences of analytic functions is extended to infinite dimensional Banach spaces.

This is a summary of recent work where we introduced a class of D-modules adapted to study ideals generated by exponential polynomials.

We prove that compactness of the canonical solution operator to $\overline{\partial }$ restricted to $(0,1)$-forms with holomorphic coefficients is equivalent to compactness of the commutator $[𝒫,\overline{M}]$ defined on the whole $L_{(0,1)}^2\left(\Omega \right),$ where $\overline{M}$ is the multiplication by $\overline{z}$ and $𝒫$ is the orthogonal projection of $L_{(0,1)}^2\left(\Omega \right)$ to the subspace of $(0,1)$ forms with holomorphic coefficients. Further we derive a formula for the $\overline{\partial }$-Neumann operator restricted to $(0,1)$ forms with holomorphic coefficients expressed by commutators of the Bergman projection and the multiplications...