Characterizations of Bloch-Type Spaces of Harmonic Mappings

Abstract

We study the Banach space BHα (α>0) of the harmonic mappings h on the open unit disk D satisfying the condition supz∈D⁡(1-z2)α(hzz+hz¯z)<∞, where hz and hz¯ denote the first complex partial derivatives of h. We show that several properties that are valid for the space of analytic functions known as the α-Bloch space extend to BHα. In particular, we prove that for α>0 the mappings in BHα can be characterized in terms of a Lipschitz condition relative to the metric defined by dH,α(z,w)=sup⁡{hz-hw:h∈BHα,hBHα≤1}. When α>1, the harmonic α-Bloch space can be viewed as the harmonic growth space of order α-1, while for 0<α<1, BHα is the space of harmonic mappings that are Lipschitz of order 1-α.

Composition Operators on Some Banach Spaces of Harmonic Mappings

“…e mapping h⟼h A − α H defines a Banach space structure on A − α H , which again extends that of the (analytic) growth [2,4] for details; for more information on the analytic growth space, see [7]).…”

“…Consequently, a harmonic mapping h belongs to B α H if and only if the unique functions f and g analytic on D such that h � f + g with g(0) � 0 are in B α (for more information on the spaces B α H , see [4]). For α � 1, the space B α is the classical Bloch space B, and the corresponding harmonic extension will be denoted by B H .…”

“…In [2], the first author pursued this study by extending several classes of Banach spaces, including the Bloch space B and its generalizations B α known as α-Bloch spaces introduced by Zhu in [3], the growth spaces A − α (where α > 0), the Zygmund space Z, and the analytic Besov spaces B p for p > 1. In particular, the linear structure and properties of the harmonic α-Bloch spaces B α H , the harmonic growth spaces A − α H (for α > 0), and the harmonic Zygmund space Z H were studied in [4,5]. e harmonic Besov spaces B p H for p > 1 were introduced in [2].…”

“…where m is the Lebesgue measure. Since by [3] and its extension to the corresponding harmonic spaces provided in [4], for α > 0, the spaces A − α and A − α H are equivalent to B α+1 and B α+1 H , respectively, and eorem 1 can be applied to the growth spaces as well. In particular, with the appropriate modification of the parameter, Corollary 1 yields a characterization of compactness on the harmonic growth spaces.…”

On the Harmonic Zygmund Spaces

“…Since several well-known Banach spaces of analytic functions have alternative characterisations (for example, the α-Bloch spaces [13], the Zygmund space [7] and the space BMOA of analytic functions of bounded mean oscillation [10]), it is of interest to establish whether such characterisations also hold for the associated space of harmonic mappings. In [1], we carried out this study on the harmonic α-Bloch spaces and showed that several known results on the analytic α-Bloch spaces introduced by Zhu in [13] do indeed extend to the harmonic setting. Let H(D) denote the class of analytic functions on D. A space that has been thoroughly studied in complex function theory is the classical Zygmund space Z, defined as the collection of analytic functions f on D whose extension to the unit circle obtained by means of the radial limits satisfies the smoothness condition…”