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Abstract

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 $ℂ^n$; intriguing phenomena occur already in the simple setting of (Euclidean) convex domains. We will not discuss at all the closely related theory of the induced Cauchy-Riemann equations on boundaries of domains or on submanifolds of higher codimension. The reader interested in such $∂̅_b$-problems may consult the recent monograph of Boggess [Bo].