BiBloch-type Maps: Existence and Beyond

Abstract

It is shown that for > 0 there are two constants c 1 , c 2 > 0 and a holomorphic map f from the unit disk D of C into C 2 such that c 1 ð1 À jzjÞ À j f 0 ðzÞj c 2 ð1 À jzjÞ À for all z 2 D. Moreover, this existence is effectively used in the study of invariance of the Bloch-type spaces under composition, but also in the discussion of embedding the Bloch-type spaces via derivation into the Lebesgue, mixed-norm and Coifman-Meyer-Stein tent spaces.

Reverse estimates in growth spaces

“…For example, see [13,29], where w(t) = 1/(1 − t) α , α > 0; see [14], where w(t) = log e 1−t . Multiplicative logarithmic perturbations of w(t) = 1/(1 − t) were considered in [12,18].…”

BiBloch mappings and composition operators from Bloch type spaces to BMOA

“…We consider and resolve an extended version of the problem on the settings of B(ψ). 3 In view of (1.1),…”

“…An extension of the Ramey-Ullrich theorem to the case where ψ(x) = x β , β > 0, was proved by Gauthier and Xiao [3] (see also Xiao [15]). In [2], in connection with a problem on composition operators, Galanopulos considered the case where ψ(x) = x(1 + log x), which was extended to ψ(x) = x γ (1 + log x), γ > 0, by Liu and Li [4].…”

Carleson–Sobolev measures for weighted Bloch spaces

“…Namely, the first part of Lemma 3.3 was proved by Ramey and Ullrich [14] for Ω = B 1 and α = 1; see [8,19] for the case Ω = B 1 and α > 0. The second part of Lemma 3.3 was proved in [9] for Ω = B 1 .…”