Given a smoothly bounded pseudoconvex domain, being able to construct a holomorphic peak function associated to each of its boundary points has a number of important consequences. Such constructions are, in general, very difficult on weakly pseudoconvex domains. Typically, they involve two steps:
1) The construction of a local peak function, which involves a delicate analysis of the geometry of the boundary around the point in question
2) The extension of the local peak function to the whole of the given domain. The second step involves the solution of an inhomogeneous Cauchy--Riemann equation.
In talk, we shall discuss a principle for globally extending local holomorphic peak functions for a family of domains that includes most classes of pseudoconvex domains for which peak-function constructions are of interest. Time permitting, we shall also take a look at some problems whose solutions rely on the ability to construct holomorphic peak functions.