esearch Article
Composition Operators on Some Banach Spaces of
Harmonic Mappings
Munirah Aljuaid
1
and Flavia Colonna
2
1
Department of Mathematics, Northern Borders University, Arar 73222, Saudi Arabia
2
Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA
Correspondence should be addressed to Flavia Colonna; fcolonna@gmu.edu
Received 26 November 2019; Accepted 17 December 2019; Published 19 February 2020
Academic Editor: Stanislav Hencl
Copyright©2020MunirahAljuaidandFlaviaColonna.isisanopenaccessarticledistributedundertheCreativeCommonsAttribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study the composition operators on Banach spaces of harmonic mappings that extend several well-known Banach spaces of
analytic functions on the open unit disk in the complex plane, including the α-Bloch spaces, the growth spaces, the Zygmund
space, the analytic Besov spaces, and the space BMOA.
1.Introduction
Givenasimplyconnectedregion Ω inthecomplexplane C,
a harmonic mapping with domain Ω is a complex-valued
function h defined on Ω satisfying the Laplace equation:
Δh ≔ 4h
z z
≡ 0, on Ω, (1)
where h
z z
is the mixed complex second partial derivative of h.
It is well known that a harmonic mapping h admits a
representation of the form f + g, where f and g are analytic
functions.isrepresentationisuniqueif,forafixedabasepoint
z
0
in the domain, the function g is chosen so that g(z
0
)� 0.
In this paper, we shall assume all the functions under
consideration are defined on D � z ∈ C: |z| < 1 { }. Let us
denote H(D) asthesetofharmonicmappingson D, H(D)
as the set of analytic functions on D, S(D) as the set of
analytic self-maps of D, and Aut(D) as the group of
(conformal) disk automorphisms of D.
Given an analytic self-map φ of D, the composition
operator induced by φ is defined as the operator
C
φ
f � f ∘ φ, (2)
for all f belonging to a selected class. It is immediate to see
that such an operator preserves harmonic mappings.
Since analytic functions are clearly harmonic, an inter-
esting question is how to extend to harmonic mappings
Banachspacestructuresofknownspacesofanalyticfunctions
X insuchawaythatthenormonthelargerspaceagreeswith
the norm of X when restricting to the elements of X.
Anexampleofaspaceofharmonicmappingson D that
extends a Banach space of analytic functions is BMOH,
defined as the space of harmonic mappings on D which are
Poissonintegralsoffunctionsontheunitcircle zD belonging
to BMO, which was thoroughly studied by Girela [1]. In that
work, it was shown that H(D) ∩ BMOH is the space BMOA
of analytic functions of bounded mean oscillation.
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),theZygmundspace Z,andtheanalyticBesovspaces
B
p
for p > 1.Inparticular,thelinearstructureandproperties
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
werestudiedin[4,5].eharmonicBesovspaces B
p
H
for
p > 1 were introduced in [2].
Inthiswork,aftergivinginSection2somepreliminaries
on the spaces of harmonic mappings mentioned above, we
introduce the harmonic Besov space B
1
H
and an alternative
extension of BMOA to harmonic mappings denoted by
BMOA
H
.Wethenanalyzethecompositionoperatorsacting
on all such spaces. Specifically, we characterize the com-
position operators that are bounded, compact, or bounded
Hindawi
Journal of Function Spaces
Volume 2020, Article ID 9034387, 11 pages
https://doi.org/10.1155/2020/9034387