Abstract:
nformally, 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,
$$c w(z) < |f| + |g| < C w(z)$$
for all $z$ in the unit disk and for some constants $C > c > 0$.
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 the
logarithmically 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.
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