November 19, 2012. E. S. Dubtsov. Sums of moduli for holomorphic functions.

Informally, we give an answer to the following question: How do the sums |f| + |g| look like when f and g are functions holomorphic in the unit disk?

To formalize the above question, assume that w is a weight function, that is, w is a positive, non-decreasing, continuous, unbounded function on the interval [0, 1). The corresponding radial weight is defined by the identity w(z) = w(|z|) for z in the unit disk. Consider the following approximation problem:

Given a radial weight w, construct holomorphic functions f and g such that the sum |f| + |g| is equivalent to w, that is,

The main result of the talk gives an explicit description of those radial weights for which the problem is solvable. Also, we have an answer when two functions are replaced by a finite set of holomorphic functions. Similar results hold in several complex variables for circular, strictly convex domains with smooth boundary.

About the proofs.

1. The restrictions on the admissible weight functions w follow from the classical Hadamard's theorem. Also, we use basic properties of thelogarithmically convex functions.

2. Constructive part: the desired holomorphic functions are obtained as appropriate lacunary series. Namely, given an admissible weight function w, we consider an auxiliary convex function v. Applying the convexity of v, we use a geometric argument to construct by induction the frequencies and the coefficients of the required lacunary series.

Applications.

The test functions obtained are useful in the studies of Carleson measures, weighted composition operators, extended Cesaro operators and other concrete linear operators.

The talk is based on a joint work with E.Abakumov.

List of talks at previous sessions of the seminar.