Here we introduce the nth
weighted space on the upper half-plane Π+={z∈ℂ:Imz>0} in the complex plane ℂ. For the case n=2, we
call it the Zygmund-type space, and denote it by 𝒵(Π+). The main result of the
paper gives some necessary and sufficient conditions for the boundedness of
the composition operator Cφf(z)=f(φ(z)) from the Hardy space Hp(Π+) on the upper half-plane, to the Zygmund-type space, where φ is an analytic
self-map of the upper half-plane.

1. Introduction

Let Π+ be the upper half-plane, that is, the set {z∈ℂ:Imz>0} and H(Π+) the space of
all analytic functions on Π+. The Hardy space Hp(Π+)=Hp, p>0, consists of all f∈H(Π+) such that∥f∥Hpp=supy>0∫−∞∞f|(x+iy)|pdx<∞. With this norm Hp(Π+) is a Banach
space when p≥1, while for p∈(0,1) it is a Fréchet
space with the translation invariant metric d(f,g)=∥f−g∥Hpp, f,g∈Hp(Π+), [1].

We introduce here the nth weighted space on the upper
half-plane. The nth weighted space consists of all f∈H(Π+) such thatsupz∈Π+Imz|f(n)(z)|<∞,where n∈ℕ0. For n=0 the space is
called the growth space and
is denoted by 𝒜∞(Π+)=𝒜∞ and for n=1 it is called
the Bloch spaceℬ∞(Π+)=ℬ∞ (for Bloch-type
spaces on the unit disk, polydisk, or the unit ball and some operators on them,
see, e.g., [2–14] and the references therein).

When n=2, we call the
space the Zygmund-type space on the upper half-plane (or simply the Zygmund
space) and denote it by 𝒵(Π+)=𝒵. Recall that the space consists of all f∈H(Π+) such thatb𝒵(f)=supz∈Π+Imz|f′′(z)|<∞. The quantity is a seminorm on the Zygmund space or a norm on 𝒵/ℙ1, where ℙ1 is the set of
all linear polynomials. A natural norm on the Zygmund space can be introduced
as follows:∥f∥𝒵=|f(i)|+|f′(i)|+b𝒵(f).
With this norm the Zygmund space becomes a Banach space.

To clarify the notation we have just introduced, we
have to say that the main reason for this name is found in the fact that for
the case of the unit disk 𝔻={z:|z|<1} in the complex
palne ℂ, Zygmund (see, e.g., [1, Theorem 5.3]) proved that a
holomorphic function on 𝔻 continuous on
the closed unit disk 𝔻¯ satisfies the
following condition:suph>0,θ∈[0,2π]|f(ei(θ+h))+f(ei(θ−h))−2f(eiθ)|h<∞ if and only ifsupz∈𝔻(1−|z|2)|f′′(z)|<∞.

The family of all analytic functions on 𝔻 satisfying
condition (1.6) is called the Zygmund class on the unit disk.

With the norm∥f∥=|f(0)|+|f′(0)|+supz∈𝔻(1−|z|2)|f′′(z)|,
the Zygmund class becomes a Banach space. Zygmund class with this norm is
called the Zygmund space and is denoted by 𝒵(𝔻). For some other information on this space and some
operators on it, see, for example, [15–19].

Now note that 1−|z| is the distance
from the point z∈𝔻 to the boundary
of the unit disc, that is, ∂𝔻, and that Imz is the distance
from the point z∈Π+ to the real
axis in ℂ which is the
boundary of Π+.

In two main theorems in [20], the authors proved the
following results, which we now incorporate in the next theorem.

Theorem A.

Assume p≥1 and φ is a
holomorphic self-map of Π+. Then the following statements true hold.

The operator Cφ:Hp(Π+)→𝒜∞(Π+) is bounded if
and only ifsupz∈Π+Imz(Imφ(z))1/p<∞.

The operator Cφ:Hp(Π+)→ℬ∞(Π+) is bounded if
and only ifsupz∈Π+Imz(Imφ(z))1+1/p|φ′(z)|<∞.

Motivated by Theorem A, here we investigate the
boundedness of the operator Cφ:Hp(Π+)→𝒵(Π+). Some recent
results on composition and weighted composition operators can be found, for example, in
[4, 6, 7, 10, 12, 18, 21–27].

Throughout this paper, constants are denoted by C, they are positive and may differ from one occurrence
to the other. The notation a⪯b means that
there is a positive constant C such that a≤Cb. Moreover, if both a⪯b and b⪯a hold, then one
says that a≍b.

2. An Auxiliary Result

In this section we prove an auxiliary result which
will be used in the proof of the main result of the paper.

Lemma 2.1.

Assume that p≥1, n∈ℕ, and w∈Π+.
Then the
function fw,n(z)=(Imw)n−1/p(z−w¯)n,
belongs to Hp(Π+). Moreoversupw∈Π+∥fw,n∥Hp≤π1/p.

Proof.

Let z=x+iy and w=u+iυ. Then, we have∥fw,n∥Hpp=supy>0∫−∞∞|fw,n(x+iy)|pdx=(Imw)np−1supy>0∫−∞∞dx|z−w¯|np−2|z−w¯|2≤vnp−1supy>0∫−∞∞dx((y+v)2)(np−2)/2((x−u)2+(y+v)2)≤vnp−1supy>01(y+v)np−1∫−∞∞y+v(x−u)2+(y+v)2dx=supy>0vnp−1(y+v)np−1∫−∞∞dtt2+1=π,
where we have used the change of variables x=u+t(y+v).

3. Main Result

Here we formulate and prove the main result of the
paper.

Theorem 3.1.

Assume p≥1 and φ is a
holomorphic self-map of Π+. Then Cφ:Hp(Π+)→𝒵(Π+) is bounded if
and only if supz∈Π+Imz(Imφ(z))2+1/p|φ′(z)|2<∞,supz∈Π+Imz(Imφ(z))1+1/p|φ′′(z)|<∞.

Moreover, if the operator Cφ:Hp(Π+)→𝒵/ℙ1(Π+) is bounded,
then∥Cφ∥Hp(Π+)→𝒵/ℙ1(Π+)≍supz∈Π+Imz(Imφ(z))2+1/p|φ′(z)|2+supz∈Π+Imz(Imφ(z))1+1/p|φ′′(z)|.

Proof.

First assume that the operator Cφ:Hp(Π+)→𝒵(Π+) is bounded.

For w∈Π+, setfw(z)=(Imw)2−1/pπ1/p(z−w¯)2.

By Lemma 2.1 (case n=2) we
know that fw∈Hp(Π+) for every w∈Π+. Moreover, we
have thatsupw∈Π+∥fw∥Hp(Π+)≤1.

From (3.5) and since the operator Cφ:Hp(Π+)→𝒵(Π+) is bounded, for
every w∈Π+, we obtainsupz∈Π+Imz|fw′′(φ(z))(φ′(z))2+fw′(φ(z))φ′′(z)|=∥Cφ(fw)∥𝒵(Π+)≤∥Cφ∥Hp(Π+)→𝒵(Π+).

We also have thatfw′(z)=−2(Imw)2−1/pπ1/p(z−w¯)3 ,fw′′(z)=6(Imw)2−1/pπ1/p(z−w¯)4.

Replacing (3.7) in (3.6) and taking w=φ(z), we obtainImz|38(φ′(z))2(Imφ(z))2+1/p−i4φ′′(z)(Imφ(z))1+1/p|≤π1/p∥Cφ∥Hp(Π+)→𝒵(Π+), and consequently14Imz(Imφ(z))1+1/p|φ′′(z)|≤π1/p∥Cφ∥Hp(Π+)→𝒵(Π+)+38Imz(Imφ(z))2+1/p|φ′(z)|2.

Hence if we show that (3.1) holds then from the last
inequality, condition (3.2) will follow.

For w∈Π+, setgw(z)=3i(Imw)2−1/pπ1/p(z−w¯)2−4(Imw)3−1/pπ1/p(z−w¯)3.

Then it is easy to see thatgw′(w)=0 ,gw′′(w)=Cw2+1/p, and by Lemma 2.1 (cases n=2 and n=3) it is
easy to see thatsupw∈Π+∥gw∥Hp<∞.

From this, since Cφ:Hp(Π+)→𝒵(Π+) is bounded and
by taking w=φ(z), it follows thatCImz(Imφ(z))2+1/p|φ′(z)|2≤∥Cφ(gw)∥𝒵(Π+)≤C∥Cφ∥Hp(Π+)→𝒵(Π+),
from which (3.1) follows, as desired.

Moreover, from (3.9) and (3.13) it follows thatsupz∈Π+Imz(Imφ(z))2+1/p|φ′(z)|2+supz∈Π+Imz(Imφ(z))1+1/p|φ′′(z)|≤C∥Cφ∥Hp(Π+)→𝒵(Π+).

Now assume that conditions (3.1) and (3.2) hold. By the
Cauchy integral formula in Π+ for Hp(Π+) functions (note
that p≥1), we
have

f(z)=12πi∫−∞∞f(t)t−zdt,z∈Π+.

By differentiating formula (3.15), we obtainf(n)(z)=n!2πi∫−∞∞f(t)(t−z)n+1dt,z∈Π+, for each n∈ℕ, from which it follows that

By using the change t−x=sy, we have that∫−∞∞yn[(t−x)2+y2](n+1)/2dt=∫−∞∞ds(s2+1)(n+1)/2=:cn<∞,n∈ℕ.

From this, applying Jensen's inequality on (3.17) and an
elementary inequality, we obtain|f(n)(z)|p≤dn∫−∞∞|f(t)|pynpyn[(t−x)2+y2](n+1)/2dt≤dn∫−∞∞|f(t)|pynp+1dt≤dn∥f∥Hp(Π+)pynp+1,wheredn=(cnn!2π)p, from which it follows that|f(n)(z)|≤C∥f∥Hp(Π+)yn+1/p.

Assume that f∈Hp(Π+). By applying (3.21) and Lemma 1 in [1, page 188], we have ∥Cφf∥𝒵(Π+)=|f(φ(i))|+|(f∘φ)′(i)|+supz∈Π+Imz|(Cφf)′′(z)|=|f(φ(i))|+|f′(φ(i))||φ′(i)|+supz∈Π+Imz|f′′(φ(z))(φ′(z))2+f′(φ(z))φ′′(z)|≤C∥f∥Hp(Π+)(1+supz∈Π+Imz(Imφ(z))2+1/p|φ′(z)|2+supz∈Π+Imz(Imφ(z))1+1/p|φ′′(z)|).

From this and by conditions (3.1) and (3.2), it follows
that the operator Cφ:Hp(Π+)→𝒵(Π+) is bounded.
Moreover, if we consider the space 𝒵/ℙ1(Π+), we have that∥Cφ∥Hp(Π+)→𝒵/ℙ1(Π+)≤C(supz∈Π+Imz(Imφ(z))2+1/p|φ′(z)|2+supz∈Π+Imz(Imφ(z))1+1/p|φ′′(z)|).

From (3.14) and (3.23), we obtain the asymptotic relation
(3.3).

DurenP.AvetisyanK. L.Hardy-Bloch type spaces and lacunary series on the polydiskAvetisyanK. L.Weighted integrals and Bloch spaces of n-harmonic functions on the polydiscClahaneD. D.StevićS.Norm equivalence and composition operators between Bloch/Lipschitz spaces of the ballLiS.Fractional derivatives of Bloch type functionsLiS.StevićS.Weighted composition operators from α-Bloch space to H∞ on the polydiscLiS.StevićS.Weighted composition operators from H∞ to the Bloch space on the polydiscLiS.StevićS.Weighted composition operators between H∞ and α-Bloch spaces in the unit ballLiS.WulanH.Characterizations of α-Bloch spaces on the unit ballStevićS.Composition operators between H∞ and α-Bloch spaces on the polydiscStevićS.On Bloch-type functions with Hadamard gapsStevićS.Norm of weighted composition operators from Bloch space to Hμ∞ on the unit ballYamashitaS.Gap series and α-Bloch functionsZhuX.Generalized weighted composition operators from Bloch type spaces to weighted Bergman spacesLiS.StevićS.Volterra-type operators on Zygmund spacesLiS.StevićS.Generalized composition operators on Zygmund spaces and Bloch type spacesLiS.StevićS.Products of Volterra type operator and composition operator from H∞ and Bloch spaces to Zygmund spacesLiS.StevićS.Weighted composition operators from Zygmund spaces into Bloch spacesStevićS.On an integral operator from the Zygmund space to the Bloch-type space on the unit ballto appear in Glasgow Mathematical JournalSharmaS. D.SharmaA. K.AhmedS.Composition operators between Hardy and Bloch-type spaces of the upper half-planeCowenC. C.MacCluerB. D.FuX.ZhuX.Weighted composition operators on some weighted spaces in the unit ballLuoL.UekiS. I.Weighted composition operators between weighted Bergman spaces and Hardy spaces on the unit ball of ℂnStevićS.Weighted composition operators between mixed norm spaces and Hα∞ spaces in the unit ballStevićS.Essential norms of weighted composition operators from the α-Bloch space to a weighted-type space on the unit ballUekiS. I.LuoL.Compact weighted composition operators and multiplication operaors
between Hardy spacesYeS.Weighted composition operator between the little α-Bloch spaces and the logarithmic Bloch