# The Cauchy-Riemann equations in several variables

June 9 to June 13, 2014

at the

American Institute of Mathematics, San Jose, California

organized by

John P. D'Angelo, Bernhard Lamel, and Dror Varolin

## Original Announcement

This workshop will focus on the many interesting questions that remain about the interaction between estimates for solutions of the Cauchy-Riemann equations and the behavior of the Bergman kernel associated to the given norm. Does knowledge of the Bergman kernel provide Hörmander type estimates for the solution of ${\overline \partial}$ under weaker pseudoconvexity assumptions? Another question concerns the link between the off-diagonal decay of the Bergman kernel and the ability to solve ${\overline \partial}$ with better hypotheses than those given by Hörmander's theorem. The workshop will consider many test scenarios in which we can formulate and explore conjectural answers.

Most results about ${\overline \partial}$ require stringent curvature or potential-theoretic conditions. It is of great geometrical interest to find solutions for sections of bundles appearing naturally in the CR setting, such as the infinitesimal CR automorphisms. The workshop will also discuss how results about ${\overline \partial}$ can be used in Hermitian analogues of Hilbert's $17$-th problem and related ideas.

## Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

A list of open problems.

Papers arising from the workshop:

Sobolev inequalities and the d-bar-Neumann operator
by  Friedrich Haslinger,  J. Geom. Anal. 26 (2016), no. 1, 287-293  MR3441514
$L^p$ Mapping Properties of the Bergman Projection on the Hartogs Triangle
by  Debra Chakrabarti, Zeytuncu, Yunus E.,  Proc. Amer. Math. Soc. 144 (2016), no. 4, 1643-1653  MR3451240