Reverse estimates in growth spaces

Abstract

Let H (B d ) denote the space of holomorphic functions on the unit ball B d of C d . Given a radial doubling weight w, we construct functions f, g ∈ H (B 1 ) such that | f | + |g| is comparable to w. Also, we obtain similar results for B d , d ≥ 2, and for circular, strictly convex domains with smooth boundary. As an application, we study weighted composition operators and related integral operators on growth spaces of holomorphic functions.

Jointly maximal products in weighted growth spaces

“…To see that this result is equivalent to [6,Theorem 1.1], that was published essentially at the same time as [1], we may argue as follows. If ω is doubling, then ψ(x) = ω(1−1/x) is almost subnormal; see [6] for the definitions.…”

“…Such a function ω is said to be doubling, if there exists a constant B > 1 such that (1) ω(1 − r/2) ≤ B ω(1 − r), 0 < r ≤ 1.…”

Hadamard gap series in growth spaces

“…Let v be a positive increasing continuous function on [0, 1), assume that v(0) = 1 and lim r→1 v(r) = +∞. We study the growth spaces of harmonic functions in the unit disk: (1) h ∞ v = {u : D → R, ∆u = 0, |u(z)| ≤ Kv(|z|) for some K > 0}. In this article we mainly deal with weights which satisfy the doubling condition (2) v(1 − d) ≤ Dv(1 − 2d).…”

“…General growth spaces can be found in the works of L. Rubel and A. Shields, and A. Shields and D. Williams, see [11,13]. Multidimensional analogs were recently considered in [1,6]. Various results on coefficients of functions in growth spaces were obtained in [2].…”

The Bloch space and the dual space of a Luecking-type subspace of

“…We refer to [1,11,14] for more results related to these constructions. Furthermore, the space X 0 is not a subset of the Bloch space.…”