Let ψ be a holomorphic mapping on the upper half-plane Π+={z∈ℂ:𝔍z>0} and φ be a holomorphic self-map of Π+. We characterize bounded weighted composition operators acting from the weighted Bergman space to the weighted-type space on the upper half-plane. Under a mild condition on ψ, we also characterize the compactness of these operators.

1. Introduction and Preliminaries

Let Π+={z∈ℂ:ℑz>0} be the upper half-plane, Ω a domain in ℂ or ℂn, and H(Ω) the space of all holomorphic functions on Ω. Let ψ∈H(Ω), and letφ be a holomorphic self-map of Ω. Then by Wφ,ψ(f)(z)=ψ(f∘φ)(z),z∈Ω,
is defined a linear operator on H(Ω) which is called weighted composition operator. If ψ(z)=1, then Wφ,ψ becomes composition operator and is denoted by Cφ, and if φ(z)=z, then Wφ,ψ becomes multiplication operator and is denoted by Mψ.

During the past few decades, composition operators and weighted composition operators have been studied extensively on spaces of holomorphic functions on various domains in ℂ or ℂn (see, e.g., [1–22] and the references therein). There are many reasons for this interest, for example, it is well known that the surjective isometries of Hardy and Bergman spaces are certain weighted composition operators (see [23, 24]). For some other operators related to weighted composition operators, see [25–30] and the references therein.

While there is a vast literature on composition and weighted composition operators between spaces of holomorphic functions on the unit disk 𝔻, there are few papers on these and related operators on spaces of functions holomorphic in the upper half-plane (see, e.g., [2, 3, 5, 7–9, 11, 12, 16–18, 31] and the references therein). For related results in the setting of the complex plane see also papers [19–21].

The behaviour of composition operators on spaces of functions holomorphic in the upper half-plane is considerably different from the behaviour of composition operators on spaces of functions holomorphic in the unit disk 𝔻. For example, there are holomorphic self-maps of Π+ which do not induce composition operators on Hardy and Bergman spaces on the upper half-plane, whereas it is a well-known consequence of the Littlewood subordination principle that every holomorphic self-map φ of 𝔻 induces a bounded composition operator on the Hardy and weighted Bergman spaces on 𝔻. Also, Hardy and Bergman spaces on the upper half-plane do not support compact composition operators (see [3, 5]).

For 0<p<∞ and α∈(-1,∞), let 𝔏p(Π+,dAα) denote the collection of all Lebesgue p-integrable functions f:Π+→ℂ such that ∫Π+|f(z)|pdAα(z)<∞,
where dAα(z)=1π(α+1)(2Iz)αdA(z),dA(z)=dxdy, and z=x+iy.

Let 𝒜αp(Π+)=𝔏p(Π+,dAα)∩H(Π+). For 1≤p<∞,𝒜αp(Π+) is a Banach space with the norm defined by ‖f‖Aαp(Π+)=(∫Π+|f(z)|pdAα(z))1/p<∞.
With this norm 𝒜αp(Π+) becomes 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‖Aαp(Π+)p,f,g∈Aαp(Π+).

Recall that for every f∈𝒜αp(Π+) the following estimate holds: |f(x+iy)|p≤C∥f∥Aαp(Π+)pyα+2,
where C is a positive constant independent of f.

Let β>0. The weighted-type space (or growth space) on the upper half-plane 𝒜β∞(Π+) consists of all f∈H(Π+) such that ‖f‖Aβ∞(Π+)=supz∈Π+(Iz)β|f(z)|<∞.
It is easy to check that 𝒜β∞(Π+) is a Banach space with the norm defined above. For weighted-type spaces on the unit disk, polydisk, or the unit ball see, for example, papers [10, 32, 33] and the references therein.

Given two Banach spaces Y and Z, we recall that a linear map T:Y→Z is bounded if T(E)⊂Z is bounded for every bounded subset E of Y. In addition, we say that T is compact if T(E)⊂Z is relatively compact for every bounded set E⊂Y.

In this paper, we consider the boundedness and compactness of weighted composition operators acting from 𝒜αp(Π+) to the weighted-type space 𝒜β∞(Π+). Related results on the unit disk and the unit ball can be found, for example, in [6, 13, 15].

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. Main Results

The boundedness and compactness of the weighted composition operator Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) are characterized in this section.

Theorem 2.1.

Let 1≤p<∞,α>-1,β>0,ψ∈H(Π+), and let φ be a holomorphic self-map of Π+. Then Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) is bounded if and only if
M:=supz∈Π+(Iz)β(Iφ(z))(α+2)/p|ψ(z)|<∞.
Moreover, if the operator Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) is bounded then the following asymptotic relationship holds:
‖Wφ,ψ‖Aαp(Π+)→Aβ∞(Π+)≍M.

Proof.

First suppose that (2.1) holds. Then for any z∈Π+ and f∈𝒜αp(Π+), by (1.6) we have
(Iz)β|(Wφ,ψf)(z)|=(Iz)β|ψ(z)||f(φ(z))|⪯(Iz)β(Iφ(z))(α+2)/p|ψ(z)|‖f‖Aαp(Π+),
and so by (2.1), Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) is bounded and moreover
‖Wφ,ψ‖Aαp(Π+)→Aβ∞(Π+)⪯M.
Conversely suppose Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) is bounded. Consider the function
fw(z)=(Iw)(α+2)/p(z-w¯)(2α+4)/p,w∈Π+.
Then fw∈𝒜αp(Π+) and moreover supw∈Π+∥fw∥𝒜αp(Π+)⪯1 (see, e.g., Lemma 1 in [18]).

Thus the boundedness of Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) implies that
(Iz)β|ψ(z)||fw(φ(z))|≤‖Wφ,ψfw‖Aβ∞(Π+)≼‖Wφ,ψ‖Aαp(Π+)→Aβ∞(Π+),
for every z,w∈Π+. In particular, if z∈Π+ is fixed then for w=φ(z), we get
(Iz)β(Iφ(z))(α+2)/p|ψ(z)|⪯‖Wφ,ψ‖Aαp(Π+)→Aβ∞(Π+).
Since z∈Π+ is arbitrary, (2.1) follows and moreover
M⪯‖Wφ,ψ‖Aαp(Π+)→Aβ∞(Π+).
If Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) is bounded then from (2.4) and (2.8) asymptotic relationship (2.2) follows.

Corollary 2.2.

Let 1≤p<∞,α>-1, and β>0 be such that βp≥α+2 and ψ∈H(Π+). Then Mψ:𝒜αp(Π+)→𝒜β∞(Π+) is bounded if and only if ψ∈X, where
X={Aβ-((α+2)/p)∞(Π+)ifα+2<βp,H∞(Π+)ifα+2=βp.

Example 2.3.

Let 1≤p<∞,α>-1 and β>0 be such that βp≥α+2 and w∈Π+. Let ψw be a holomorphic map of Π+ defined as
ψw(z)={1(z-w¯)β-((α+2)/p)ifα+2<βp,Iwz-w¯ifα+2=βp.
For z=x+iy and w=u+iv in Π+, we have
supz∈Π+(Iz)β-(α+2)/p|ψw(z)|=supz=x+iy∈Π+yβ-(α+2)/p((x-u)2+(y+v)2)(βp-(α+2))/2p≤supz=x+iy∈Π+yβ-(α+2)/p(y+v)β-(α+2)/p≤1.
Thus ψw∈𝒜β-(α+2)/p∞(Π+) if α+2<βp. Similarly ψw∈H∞(Π+) if α+2=βp. By Corollary 2.2, it follows that Mψw:𝒜αp(Π+)→𝒜β∞(Π+) is bounded.

Corollary 2.4.

Let 1≤p<∞,α>-1,β>0, and letφ be a holomorphic self-map of Π+. Then Cφ:𝒜αp(Π+)→𝒜β∞(Π+) is bounded if and only if
supz∈Π+(Iz)β(Iφ(z))(α+2)/p<∞.

Corollary 2.5.

Let φ be the linear fractional map
φ(z)=az+bcz+d,a,b,c,d∈R,ad-bc>0.
Then necessary and sufficient condition that Cφ:𝒜αp(Π+)→𝒜β∞(Π+) is bounded is that c=0 and α+2=βp.

Proof.

Assume that Cφ:𝒜αp(Π+)→𝒜β∞(Π+) is bounded. Then
supz∈Π+(Iz)β(Iφ(z))(α+2)/p=supz=x+iy∈Π+((cx+d)2+c2y2)(α+2)/pyβ(ad-bc)(α+2)/py(α+2)/p,
which is finite only if c=0 and α+2=βp.

Conversely, if c=0 and α+2=βp, then from (2.13) we get a≠0, and by some calculation
supz∈Π+(Iz)β(Iφ(z))(α+2)/p=(da)β<∞.
Hence Cφ:𝒜αp(Π+)→𝒜β∞(Π+) is bounded.

Corollary 2.6.

Let 1≤p<∞,α>-1, and β>0 be such that βp=α+2. Let φ be a holomorphic self-map of Π+ and ψ=(φ′)β. Then the weighted composition operator Wφ,ψ acts boundedly from 𝒜αp(Π+) to 𝒜β∞(Π+).

Proof.

By Theorem 2.1, Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) is bounded if and only if
supz∈Π+(Iz)β(Iφ(z))β|φ′(z)|β<∞.
By the Schwarz-Pick theorem on the upper half-plane we have that for every holomorphic self-map φ of Π+ and all z∈Π+|φ′(z)|Iφ(z)≤1Iz,
where the equality holds when φ is a Möbius transformation given by (2.13). From (2.17), condition (2.16) follows and consequently the boundedness of the operator Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+).

Corollary 2.6 enables us to show that there exist 1≤p<∞,α>-1,β>0, and holomorphic maps φ and ψ of the upper half-plane Π+ such that neither Cφ:𝒜αp(Π+)→𝒜β∞(Π+) nor Mψ:𝒜αp(Π+)→𝒜β∞(Π+) is bounded, but Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) is bounded.

Example 2.7.

Let 1≤p<∞,α>-1, and β>0 be such that βp=α+2. Let φ(z)=(az+b)/(cz+d),a,b,c,d∈ℝ,ad-bc>0, and c≠0. Then by Corollary 2.5, Cφ:𝒜αp(Π+)→𝒜β∞(Π+) is not bounded. On the other hand, if
ψ(z)=(φ′(z))β=(ad-bc(cz+d)2)β,
then ψ∉H∞(Π+) and so by Corollary 2.2, Mψ:𝒜αp(Π+)→𝒜β∞(Π+) is not bounded. However, by Corollary 2.6, we have that Wφ,(φ′)β:𝒜αp(Π+)→𝒜β∞(Π+) is bounded.

The next Schwartz-type lemma characterizes compact weighted composition operators Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) and it follows from standard arguments ([4]).

Lemma 2.8.

Let 1≤p<∞,α>-1,β>0,ψ∈H(Π+), and letφ be a holomorphic self-map of Π+. Then Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) is compact if and only if, for any bounded sequence (fn)n∈ℕ⊂𝒜αp(Π+) converging to zero on compacts of Π+, one has
limn→∞‖Wφ,ψfn‖Aβ∞(Π+)=0.

Theorem 2.9.

Let 1≤p<∞,α>-1,β>0,ψ∈H(Π+) and φ be a holomorphic self-map of Π+. If Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) is compact, then
limr→0supIφ(z)<r(Iz)β(Iφ(z))(α+2)/p|ψ(z)|=0.

Proof.

Suppose Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) is compact and (2.20) does not hold. Then there is a δ>0 and a sequence (zn)n∈ℕ⊂Π+ such that ℑφ(zn)→0 and
(Izn)β(Iφ(zn))(α+2)/p|ψ(zn)|>δ
for all n∈ℕ. Let wn=φ(zn), n∈ℕ, and
fn(z)=(Iwn)(α+2)/p(z-w¯n)(2α+4)/p,n∈N.
Then fn is a norm bounded sequence and fn→0 on compacts of Π+ as ℑφ(zn)→0. By Lemma 2.8 it follows that
limn→∞‖Wφ,ψfn‖Aβ∞(Π+)=0.
On the other hand,
‖Wφ,ψfn‖Aβ∞(Π+)≥(Izn)β|(Wφ,ψfn)(zn)|=(Izn)β|ψ(zn)||fn(φ(zn))|=(Izn)β2(2α+4)/p(Iφ(zn))(α+2)/p|ψ(zn)|>δ2(2α+4)/p,
which is a contradiction. Hence (2.20) must hold, as claimed.

Before we formulate and prove a converse of Theorem 2.9, we define, for every a,b∈(0,∞) such that a<b, the following subset of Π+: Γa,b={z∈Π+:a≤Iz≤b}.

Theorem 2.10.

Let 1≤p<∞,α>-1,β>0,ψ∈H(Π+), and letφ be a holomorphic self-map of Π+ and Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) be bounded. Suppose that ψ∈𝒜β∞(Π+) and (ℑz)β|ψ(z)|→0 as |ℜφ(z)|→∞ within Γa,b for all a and b,0<a<b<∞. Then Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) is compact if condition (2.20) holds.

Proof.

Assume (2.20) holds. Then for each ɛ>0, there is an M1>0 such that
(Iz)β(Iφ(z))(α+2)/p|ψ(z)|<ɛ,wheneverIφ(z)<M1.
Let (fn)n∈ℕ be a sequence in 𝒜αp(Π+) such that supn∈ℕ∥fn∥𝒜αp(Π+)≤M and fn→0 uniformly on compact subsets of Π+ as n→∞. Thus for z∈Π+ such that ℑφ(z)<M1 and each n∈ℕ, we have
(Iz)β|ψ(z)||fn(φ(z))|⪯(Iz)β(Iφ(z))(α+2)/p|ψ(z)|‖fn‖Aαp(Π+)<ɛM.

From estimate (1.6) we have
|fn(z)|⪯‖fn‖Aαp(Π+)(Iz)(α+2)/p⪯M(Iz)(α+2)/p.
Thus there is an M2>M1 such that
|fn(φ(z))|<ɛ,
whenever ℑφ(z)>M2. Hence for z∈Π+ such that ℑφ(z)>M2 and each n∈ℕ we have
(Iz)β|ψ(z)||fn(φ(z))|<ɛ‖ψ‖Aβ∞(Π+).
If M1≤ℑφ(z)≤M2, then by the assumption there is an M3>0 such that (ℑz)β|ψ(z)|<ɛ, whenever |ℜφ(z)|>M3. Therefore, for each n∈ℕ we have
(Iz)β|ψ(z)||fn(φ(z))|⪯ɛ‖fn‖Aαp(Π+)(Iφ(z))(α+2)/p≤ɛMM1(α+2)/p,
whenever M1≤ℑφ(z)≤M2 and |ℜφ(z)|>M3.

If M1≤ℑφ(z)≤M2 and |ℜφ(z)|≤M3, then there exists some n0∈ℕ such that |fn(φ(z))|<ɛ for all n≥n0, and so
(Iz)β|ψ(z)||fn(φ(z))|<ɛ‖ψ‖Aβ∞(Π+).
Combining (2.27)–(2.32), we have that
‖Wφ,ψfn‖Aβ∞(Π+)<ɛC,
for n≥n0 and some C>0 independent of n. Since ɛ is an arbitrary positive number, by Lemma 2.8, it follows that Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) is compact.

Example 2.11.

Let 1≤p<∞,α>-1, and β>0 be such that α+2=βp. Let φ(z)=z+i and ψ(z)=1/(z+i)β, then ℜφ(z)=x and ℑφ(z)=y+1. It is easy to see that ψ∈𝒜β∞(Π+). Beside this, for z∈Γa,b, we have
(Iz)β|ψ(z)|=yβ(x2+(y+1)2)β/2≤bβ(x2+a2)β/2⟶0asRφ(z)=x⟶∞.

Also
supz∈Π+(Iz)β(Iφ(z))β|ψ(z)|=supz=x+iy∈Π+yβ(y+1)β1(x2+(y+1)2)β/2≤1<∞,
and the set {z:ℑφ(z)<1} is empty. Thus φ and ψ satisfy all the assumptions of Theorem 2.10, and so Wφ,ψ:𝒜αp(Π+)→𝒜β∞(Π+) is compact.

Acknowledgments

The work is partially supported by the Serbian Ministry of Science, projects III 41025 and III 44006. The work of the second author is a part of the research project sponsored by National Board of Higher Mathematics (NBHM)/DAE, India (Grant no. 48/4/2009/R&D-II/426).

CowenC. C.MacCluerB. D.MatacheV.Composition operators on H^{p} of the upper half-planeMatacheV.Composition operators on Hardy spaces of a half-planeSchwartzH. J.ShapiroJ. H.SmithW.Hardy spaces that support no compact composition operatorsSharmaA. K.SharmaS. D.Weighted composition operators between Bergman-type spacesSharmaS. D.SharmaA. K.AhmedS.Carleson measures in a vector-valued Bergman spaceSharmaS. D.SharmaA. K.AhmedS.Composition operators between Hardy and Bloch-type spaces of the upper half-planeSharmaS. D.SharmaA. K.AbbasZ.Weighted composition operators on weighted vector-valued Bergman spacesShieldsA. L.WilliamsD. L.Bonded projections, duality, and multipliers in spaces of analytic functionsSinghR. K.SharmaS. D.Composition operators on a functional Hilbert spaceSinghR. K.SharmaS. D.Noncompact composition operatorsStevićS.Weighted composition operators between mixed norm spaces and Hα∞ spaces in the unit ballStevićS.Norm of weighted composition operators from Bloch space to Hμ∞ on the unit ballStevićS.Weighted composition operators from weighted Bergman spaces to weighted-type spaces on the unit ballStevićS.Composition operators from the Hardy space to Zygmund-type spaces on the upper half-plane and the unit discStevićS.Composition operators from the Hardy space to the n-th weighted-type space on the unit disk and the half-planeStevićS.Composition operators from the weighted Bergman space to the n-th weighted-type space on the upper half-planeUekiS.-I.Hilbert-Schmidt weighted composition operator on the Fock spaceUekiS.-I.Weighted composition operators on the Bargmann-Fock spaceUekiS.-I.Weighted composition operators on some function spaces of entire functionsZhuX.Weighted composition operators from area Nevanlinna spaces into Bloch spacesForelliF.The isometries of H^{p}spacesKolaskiC. J.Isometries of weighted Bergman spacesLiS.StevićS.Generalized composition operators on Zygmund spaces and Bloch type spacesStevićS.On a new operator from H∞ to the Bloch-type space on the unit ballStevićS.On an integral operator from the Zygmund space to the Bloch-type space on the unit ballStevićS.On a new integral-type operator from the Bloch space to Bloch-type spaces on the unit ballStevićS.On an integral operator between Bloch-type spaces on the unit ballZhuX.Generalized weighted composition operators from Bloch type spaces to weighted Bergman spacesAvetisyanK. L.Integral representations in general weighted Bergman spacesAvetisyanK. L.Hardy-Bloch type spaces and lacunary series on the polydiskBierstedtK. D.SummersW. H.Biduals of weighted Banach spaces of analytic functions