On a product type operator between Hardy and α Bloch spaces of the upper half plane

(1)

R E S E A R C H

Open Access

On a product-type operator between

Hardy and

α-Bloch spaces of the upper

half-plane

Stevo Stevi´c

1,2*

_{and Ajay K. Sharma}

3

*_{Correspondence:}_{sstevic@ptt.rs}

1_{Mathematical Institute of the}

Serbian Academy of Sciences, Beograd, Serbia

2_{Department of Medical Research,}

China Medical University Hospital, China Medical University, Taichung, Taiwan, Republic of China Full list of author information is available at the end of the article

Abstract

Recently we have introduced a product-type operator and studied it on some spaces of analytic functions on the unit disc. Here we start investigating the operator on the space of analytic functions on the upper half-plane. We characterize the boundedness and compactness of the operator between Hardy and

α

-Bloch spaces on the domain.

MSC: Primary 47B38; 46E10; secondary 30D45

Keywords: Product-type operator; Hardy space; Bloch space; Boundedness; Compactness; Upper half-plane

1 Introduction

LetDbe the unit disc in the complex planeC,+₌_{_z_∈_C_:_z_{> 0}}_{the upper half-plane}

inC, and+₌+_{∪ {∞}}_{. Let}_{be a domain in}_C_{. We denote by}_H₍_{) the space of all}

analytic functions onand byS() the class of all analytic self-maps of.

For 0 <p<∞, theHardy spaceof+, denoted byHp₍+_{), consists of all}_f _∈_H₍+_{) such}

that

fp_Hp₍+₎=sup

y>0

∞

–∞

_f₍_x₊_iy₎p

dx<∞.

For 1≤p<∞,Hp₍+_{) is a Banach space.}

Letα> 0. Theα-BlochorBloch-typespace on+, denoted byBα₍+_{), consists of all}

f∈H(+_{) such that}

fBα₍+₎:=f(i)+sup

z∈+(z)

α

f(z)<∞. (1)

With the norm (1), theα-Bloch space is a Banach space. For Bloch-type spaces on various domains and operators on them, see, for example, [1–13] and the references therein. For a natural extension of Bloch-type spaces and for Zygmud-type spaces, see [14–16].

Forϕ∈S(), thecomposition operator Cϕis the linear operator deﬁned by

Cϕ(f)(z) = (f◦ϕ)(z) forf ∈H(). (2)

(2)

Forψ∈H(), themultiplication operator Mψis deﬁned onH() by

Mψf(z) =ψ(z)f(z) forf ∈H(). (3)

The product of these two operators

Wϕ,ψ=Mψ◦Cϕ (4)

is the so-calledweighted composition operator. ByDwe denote thediﬀerentiation operator, that is,

Df =f, f∈H(). (5)

These concrete operators, along with some integral-type ones, are among those con-siderably studied, during the last ﬁve decades, on spaces of analytic functions on various domains inCor domains in the complex-vector spaceCn_{. Majority of the papers on the} operators are devoted to investigating them on spaces of analytic functions onD. Much less papers consider the operators on spaces of analytic functions on other domains, in-cluding the upper half-plane. Even for some popular operators, such as the composition ones, up to the end of the previous century, there are no many papers on popular spaces such asHp₍+_{) (see, e.g., [}₁₇_–₂₀_{] and the references therein). Hence, any new result on}

the spaces and operators on+is of some interest. Regarding some operators on the men-tioned spaces, let us mention that some basic results on the boundedness of composition operators fromHp₍+_{) to the classical Bloch space}_B₍+_{) can be found in note [}₂₁_{]. For}

related investigations of composition or weighted composition operators on some other spaces, see [14,15,22–25]. Let us mention that the behavior of composition operators on spaces of analytic functions in+is considerably diﬀerent from the behavior of composi-tion operators on spaces of analytic funccomposi-tions inD. For example, every analytic self-map ofDinduces a bounded composition operator on the corresponding Hardy space, which is not always the case on the spaceHp₍+_{) [}₁₇_,₁₈_].

From 1968 to 2005, experts more or less studied theoretic properties of only opera-tors (2)–(5) and integral-type ones on spaces of analytic functions in terms of their sym-bols. The only product-type operator among (2)–(5) is the weighted composition operator. Since 2005, some experts started studying some other product-type operators. The first product-type operators different from weighted composition ones that attracted some at-tention were the products of composition and differentiation operators (see, e.g., [7,11,13,

26–29] and the references therein). Some generalizations of the products of composition and differentiation operators, containing iterated differentiation, have appeared soon af-ter them (see [12,30–33]). Around 2008, Li and Stević have initiated studying products of integral and composition operators, including in some cases the differentiation operator, on spaces of analytic functions onD(see, e.g., [34]), which have been considerably studied recently (see, e.g., [3,35–40]). For some other results and related product-type operators, see, for example, [4–6,35,41–46].

(3)

[5,42] on the operator on some other spaces of functions deﬁned by

Tψ1,ψ2,ϕf(z) =ψ1(z)f

ϕ(z)+ψ2(z)f

ϕ(z), f ∈H(D), (6)

whereψ1,ψ2∈H(D) andϕ∈S(D).

It is clear that operator (6) includes the composition, multiplication, diﬀerentiation, weighted composition, weighted diﬀerentiation composition, and many other concrete operators, including those in [6,41,43], which are obtained by some concrete choices of the symbolsψ1,ψ2, andϕ. This is one of the reasons why the operator is of some

impor-tance for investigation.

So far the operator has not been considered between spaces of analytic functions on the upper half-plane. Here we start investigating the operator on such spaces by characteriz-ing the boundedness and compactness of the operator between the Hardy andα-Bloch spaces on the domain, that is, of Tψ1,ψ2,ϕ:Hp(+)→Bα(+). We provide a complete

characterization of the compactness. The paper can be regarded as a continuation of our investigations in [14,15,22–25].

Throughout this paper, constants are denoted byC; they are positive and not necessarily the same at each occurrence. The notation ABmeans thatBAB, whereAB

means that there is a positive constantCsuch thatA≤CB.

2 Auxiliary results

First, we quote a point evaluation lemma, which is a folklore result (see, e.g., [15, Lemma 3]).

Lemma 1 Let p∈(0,∞)and n∈N0.Then

f(n)(z)≤CfHp(+) yn+1p

for some positive constant C=C(p,n)independent of f.

The following lemma can be found in [14, Lemma 2.1].

Lemma 2 Let p∈[1,∞),k∈N0,ζ∈+,and

fζ,k(z) =

(ζ)k–1p

(z–ζ)k.

Then

sup

ζ∈+fζ,kH

p₍+₎≤π 1 p_.

To deal with the compactness of the operatorTψ1,ψ2,ϕ:H

p₍+₎_→_Bα₍+_{), in the lemma}

(4)

composition operator. It is interesting that the proof of our present lemma is more com-plicated than those of the corresponding lemmas that we have had so for (for example, the lemma in [24]). Hence we will present a detailed proof of the lemma.

We say that a sequence (fn)n∈NinHp(+) converges weakly to zero if it is norm bounded

inHp₍+_{) and converges to zero on compacts of}+_.

Lemma 3 Let p≥ 1, α ∈ (0, +∞), ψ1,ψ2 ∈ H(+), and ϕ ∈ S(+). Then Tψ1,ψ2,ϕ : Hp(+)→Bα₍+₎_{is compact if and only if for any sequence}₍_f

n)n∈N weakly convergent

to zero,we have

lim

n→∞Tψ1,ψ2,ϕfnBα(

+₎= 0. (7)

Proof Let the operator be compact. Suppose that (fn)n∈Nweakly converges to zero. Then

there are a subsequence (fnk)k∈Nandg∈B

α₍+_{) such that}

lim

k→∞Tψ1,ψ2,ϕfnk–gBα(+)= 0. (8)

LetK⊂+_{be compact. Then}

dK:=d

K,∂+= inf

z∈K,x∈R|z–x|> 0.

From this and from (1) we have

Tψ1,ψ2,ϕfnk(i) –g(i)≤ Tψ1,ψ2,ϕfnk–gBα(+) (9)

and

sup z∈K

Tψ1,ψ2,ϕfnk(z) –g(z)

≤ 1

dK

Tψ1,ψ2,ϕfnk–gBα(+). (10)

LetA⊂+_and_A

ε={z∈+:d(z,K)≤ε}, whereε> 0. Note that ifAis compact, then

for eachε<dK, the setAεis also a compact set as bounded and closed.

On the other hand, we have

f(z) =f(i) +

z

i

f(ζ)dζ (11)

for allf ∈H(+) andz∈+. LetA⊂+and

A(i) =w∈+:∃z∈A,w∈[i,z].

Note that ifAis compact, thenA(i) is also compact. Hence from (11) we easily get

f(z)≤f(i)+diamK(i) sup w∈K(i)

f(w) (12)

(5)

From (9), (10), and (12) it follows that

sup z∈K

Tψ1,ψ2,ϕfnk(z) –g(z)≤(Tψ1,ψ2,ϕfnk(i) –g(i)

+diamK(i) sup z∈K(i)

Tψ1,ψ2,ϕfnk(z) –h(z)

≤ 1 +diam(K(i))

dK(i)

Tψ1,ψ2,ϕfnk–gBα(+). (13)

From (8) and (13) it follows that

Tψ1,ψ2,ϕfnk(z) –g(z)⇒0, k→ ∞, (14)

on each compactK∈+. Further, note that

Mj:=sup z∈K

ψj(z)<∞, j= 1, 2,

for each compactK. The compactness of the setϕ(K) also implies that maxfnk

ϕ(z),f_n_kϕ(z)⇒0, k→ ∞,

on each compactK∈+. From this, since

Tψ1,ψ2,ϕfnk

ϕ(z)≤M1 sup w∈ϕ(K)

fnk(w)+M2 sup

w∈ϕ(K) f_n_k(w),

it follows that

Tψ1,ψ2,ϕfnk(z)⇒0, k→ ∞, (15)

on each compactK∈+_.

From (14) and (15) we obtaing(z) = 0 for everyz∈+_{, since each}_z_∈+_{lies in a}

com-pact subset of+.

Using the fact in (8), it follows that

lim

k→∞Tψ1,ψ2,ϕfnkB

α₍+₎= 0. (16)

Such a procedure can be applied to any subsequence of (fn)n∈N, from which it follows that (7) holds.

Now assume that, for any sequence (fn)n∈N weakly convergent to zero, we have limn→∞Tψ1,ψ2,ϕfnBα(+) = 0. Let (fn)n∈N be a sequence of functions such that M:=

sup_n_∈_NfnHp₍+₎ < +∞. By Lemma1 the sequence is uniformly bounded on compacts

of+_{, and consequently normal. Hence there are a subsequence (}_f

nk)k∈N andf ∈H(+)

such that

(6)

on each compactK∈+_{. The Fatou lemma along with (}₁₇_{) implies}_f

Hp₍+₎≤M. Hence,

the sequence (fnk–f)k∈Nweakly converges to zero, and consequently

lim

k→∞Tψ1,ψ2,ϕ

fnk–Tψ1,ψ2,ϕfBα₍+₎= 0,

from which the compactness of the operatorTψ1,ψ2,ϕ:H

p₍+₎_→_Bα₍+_{) follows.}

3 Boundedness and compactness ofTψ1,ψ2,ϕ:H

p₍

+₎_→_Bα₍

+₎

In this section, we characterize the boundedness and compactness of the operator

Tψ1,ψ2,ϕ:Hp(+)→B

α₍+_{). We also give upper and lower bounds for the norm of}_T

ψ1,ψ2,ϕ

acting between these spaces.

Theorem 4 Let 1 ≤p< ∞, α > 0, ψ1,ψ2 ∈ H(+) and ϕ ∈S(+). Then Tψ1,ψ2,ϕ : Hp₍+₎_→_Bα₍+₎_{is bounded if and only if the following conditions are satisﬁed}_:

(i) M1=sup z∈+

(z)α

(ϕ(z))1/pψ

1(z)<∞;

(ii) M2=sup z∈+

(z)α

(ϕ(z))1+1/pψ1(z)ϕ

₍_z_{) +}_ψ

2(z)<∞;

(iii) M3=sup z∈+

(z)α

(ϕ(z))2+1/pψ2(z)ϕ

₍_z₎_<_∞.

Moreover,

M1+M2+M3Tψ1,ψ2,ϕHp(+₎_→_Bα₍+₎

|ψ1(i)|

(ϕ(i))1/p +

|ψ2(i)|

(ϕ(i))1+1/p +M1+M2+M3. (18)

Proof First, suppose that conditions (i)–(iii) hold. Then by Lemma1we have

(z)α₍_T

ψ1,ψ2,ϕf)(z)

= (z)αψ₁(z)fϕ(z)+ψ1(z)ϕ(z) +ψ2(z)

fϕ(z)+ψ2(z)ϕ(z)f

ϕ(z) ≤CfHp₍+₎ (z)

α

(ϕ(z))1/pψ

1(z)+

(z)α

(ϕ(z))1+1/pψ1(z)ϕ

₍_z_{) +}_ψ

2(z)

+ (z)

α

(ϕ(z))2+1/pψ2(z)ϕ

₍_z₎

≤C(M1+M2+M3)fHp₍+₎. (19)

We also have

Tψ1,ψ2,ϕf(i)=ψ1(i)f

ϕ(i)+ψ2(i)f

ϕ(i) ≤CfHp₍+₎ |ψ1(i)|

(ϕ(i))1/p +

|ψ2(i)|

(ϕ(i))1+1/p

(7)

Combining (19) and (20), we have

Tψ1,ψ2,ϕfBα(+)

|ψ1(i)|

(ϕ(i))1/p +

|ψ2(i)|

(ϕ(i))1+1/p +M1+M2+M3

fHp₍+₎,

from which it follows that

Tψ1,ψ2,ϕHp₍+₎_→_Bα₍+₎ |ψ1(i)|

(ϕ(i))1/p +

|ψ2(i)|

(ϕ(i))1+1/p+M1+M2+M3. (21) Conversely, suppose thatTψ1,ψ2,ϕ:H

p₍+₎_→_Bα₍+_{) is bounded. Consider the family}

of functions

fw(z) =

(w)2–1/p

π1/p₍_z_–_w₎2 – 4i

(w)3–1/p

π1/p₍_z_–_w₎3 – 4

(w)4–1/p

π1/p₍_z_–_w₎4,

wherew∈+.

Since the functions fware linear combinations of the functions in Lemma 2(fork= 2, 3, 4), from this by using the lemma it follows that

L1:= sup w∈+fwH

p₍+₎<∞. (22)

We also have

f_w(z) =–2(w)

2–1/p

π1/p₍_z_–_w₎3 + 12i

(w)3–1/p

π1/p₍_z_–_w₎4+ 16

(w)4–1/p

π1/p₍_z_–_w₎5,

f_w(z) = 6(w)

2–1/p

π1/p₍_z_–_w₎4 – 48i

(w)3–1/p

π1/p₍_z_–_w₎5 – 80

(w)4–1/p

π1/p₍_z_–_w₎6,

from which with some simple calculation we obtain

fw(w) = 0, fw(w) = 0 and fw(w) =

1

8π1/p(_w₎2+1/p. (23)

SinceTψ1,ψ2,ϕ:Hp(+)→Bα(+) is bounded, we have

Tψ1,ψ2,ϕfwBα(+)≤ Tψ1,ψ2,ϕHp(+)→Bα(+)fwHp(+)

≤L1Tψ1,ψ2,ϕHp₍+₎_→_Bα₍+₎

for everyw∈+_.

Thus for eachζ∈+, we have

L1Tψ1,ψ2,ϕHp(+₎_→_Bα₍+₎≥(ζ)α(T_ψ₁_,_ψ₂_,_ϕf_ϕ₍_ζ₎)(ζ)

= (ζ)α_ψ

1(ζ)fϕ(ζ)

ϕ(ζ)+ψ1(ζ)ϕ(ζ) +ψ2(ζ)

f_ϕ₍_ζ₎ϕ(ζ) +ψ2(ζ)ϕ(ζ)fϕ(ζ)

ϕ(ζ) =(ζ)

α_|_ψ

2(ζ)ϕ(ζ)|

(8)

Sinceζ∈+_{is arbitrary, we have that}

M3= sup

ζ∈+

(ζ)α_|_ψ

2(ζ)ϕ(ζ)|

(ϕ(ζ))2+1/p ≤8π 1/p_L

1Tψ1,ψ2,ϕHp₍+₎_→_Bα₍+₎. (24)

Now, consider the family of functions

gw(z) = 12

(w)2–1/p

π1/p₍_z_–_w₎2 – 32i

(w)3–1/p

π1/p₍_z_–_w₎3– 24

(w)4–1/p

π1/p₍_z_–_w₎4.

Since the functionsgware also linear combinations of the functions in Lemma2, we have

L2:=supw∈+gwHp₍+₎≤1.

We also have

g_w(z) = –24 (w)

2–1/p

π1/p₍_z_–_w₎3+ 96i

(w)3–1/p

π1/p₍_z_–_w₎4 + 96

(w)4–1/p

π1/p₍_z_–_w₎5,

g_w(z) = 72 (w)

2–1/p

π1/p₍_z_–_w₎4 – 384i

(w)3–1/p

π1/p₍_z_–_w₎5 – 480

(w)4–1/p

π1/p₍_z_–_w₎6,

from which it follows that

g_w(w) = 0, g_w(w) = 0, and gw(w) = – 1 2π1/p₍_w₎1/p.

SinceTψ1,ψ2,ϕ:Hp(+)→Bα(+) is bounded, for eachζ∈+, we have

L2Tψ1,ψ2,ϕHp(+)→Bα(+)≥ Tψ1,ψ2,ϕgϕ(ζ)Bα(+)

= (ζ)αψ₁(ζ)gϕ(ζ)

ϕ(ζ)+ψ1(ζ)ϕ(ζ) +ψ2(ζ)

gϕ(ζ)

ϕ(ζ) +ψ2(ζ)ϕ(ζ)gϕ(ζ)

ϕ(ζ) = (ζ)

α_|_ψ

1(ζ)|

2π1/p₍_ϕ₍_ζ₎₎1/p. Sinceζ∈+is arbitrary, we have that

M1= sup

ζ∈+

(ζ)α_|_ψ

1(ζ)|

(ϕ(ζ))1/p ≤2π 1/p_L

2Tψ1,ψ2,ϕHp(+)→Bα(+). (25)

Now, consider the family of functions

hw(z) = 8 (

w)2–1/p

π1/p₍_z_–_w₎2 – 28i

(w)3–1/p

π1/p₍_z_–_w₎3 – 24

(w)4–1/p

π1/p₍_z_–_w₎4.

Once again proceeding as before, we can show thatsup_w_∈+h_w_Hp₍+₎1.

We have

h_w(z) = –16 (w)

2–1/p

π1/p₍_z_–_w₎3+ 84i

(w)3–1/p

π1/p₍_z_–_w₎4 + 96

(w)4–1/p

π1/p₍_z_–_w₎5;

h_w(z) = 48 (w)

2–1/p

π1/p₍_z_–_w₎4– 336i

(w)3–1/p

π1/p₍_z_–_w₎5– 480

(w)4–1/p

(9)

Thus

hw(w) = 0, hw(w) = 0 and hw(w) =

i

4π1/p(_w₎1+1/p.

Therefore by the boundedness ofTψ1,ψ2,ϕ:Hp(+)→Bα(+), for eachζ ∈+, we have

L3Tψ1,ψ2,ϕHp(+)→Bα(+)≥(ζ)

α_ψ

1(ζ)hϕ(ζ)

ϕ(ζ)+ψ1(ζ)ϕ(ζ) +ψ2(ζ)

hϕ(ζ)

ϕ(ζ) +ψ2(ζ)ϕ(ζ)hϕ(ζ)

ϕ(ζ) =(ζ)

α_|_ψ

1(ζ)ϕ(ζ) +ψ2(ζ)|

4π1/p(_ϕ₍_ζ₎₎1+1/p , and consequently

M2= sup

ζ∈+

(ζ)α_|_ψ

1(ζ)ϕ(ζ) +ψ2(ζ)|

(ϕ(ζ))1+1/p ≤4π 1/p_L

3Tψ1,ψ2,ϕHp(+)→Bα(+). (26)

Combining (24), (25), and (26), we have

M1+M2+M3Tψ1,ψ2,ϕHp₍+₎_→_Bα₍+₎,

ﬁnishing the proof of the theorem.

Corollary 5 Let1≤p<∞,α> 0andϕ∈S(+).Then Cϕ:Hp(+)→Bα(+)is bounded

if and only if

M4=sup z∈+

(z)α

(ϕ(z))1+1/pϕ

₍_z₎_<_∞.

Moreover,

M4CϕHp₍+₎_→_Bα₍+₎ 1

(ϕ(i))1/p+M4. (27)

Corollary 6 Let1≤p<∞,α> 0,ψ∈H(+)andϕ∈S(+).Then MψCϕ:Hp(+)→

Bα₍+₎_{is bounded if and only if}

(i) M5=sup z∈+

(z)α

(ϕ(z))1/pψ

₍_z₎_<_∞,

(ii) M6=sup z∈+

(z)α

(ϕ(z))1+1/pψ(z)ϕ

₍_z₎_<_∞.

Moreover,

M5+M6MψCϕHp₍+₎_→_Bα₍+₎ |ψ(i)|

(ϕ(i))1/p +M5+M6. (28)

Corollary 7 Let1≤p<∞,α> 0,ψ∈H(+)andϕ∈S(+).Then CϕMψ:Hp(+)→

(i) M7=sup z∈+

(z)α

(ϕ(z))1/pψ

_ϕ₍_z₎_ϕ₍_z₎_<_∞,

(ii) M8=sup z∈+

(z)α

(ϕ(z))1+1/pψ

(10)

Moreover,

M7+M8CϕMψHp₍+₎_→_Bα₍+₎ |ψ(ϕ(i))|

(ϕ(i))1/p +M7+M8. (29)

Corollary 8 Let 1≤p<∞, α > 0 and ϕ∈S(+). Then CϕD:Hp(+)→Bα(+) is

bounded if and only if

M9=sup z∈+

(z)α

(ϕ(z))2+1/pϕ

₍_z₎_<_∞.

Moreover,

M9CϕDHp₍+₎_→_Bα₍+₎ 1

(ϕ(i))1+1/p +M9. (30)

Corollary 9 Let1 ≤p<∞, α > 0 andϕ ∈S(+). Then DCϕ :Hp(+)→Bα(+) is

bounded if and only if

(i) M10=sup z∈+

(z)α

(ϕ(z))1+1/pϕ

₍_z₎_<_∞,

(ii) M11=sup z∈+

(z)α

(ϕ(z))2+1/pϕ

₍_z₎2

<∞.

Moreover,

M10+M11DCϕHp₍+₎_→_Bα₍+₎ |ϕ ₍_i₎_|

(ϕ(i))1+1/p +M10+M11. (31) Corollary 10 Let1≤p<∞,α> 0,ψ∈H(+)andϕ∈S(+).Then MψCϕD:Hp(+)→

(i) M12=sup z∈+

(z)α

(ϕ(z))1+1/pψ

₍_z₎_<_∞_,

(ii) M13=sup z∈+

(z)α

(ϕ(z))2+1/pψ(z)ϕ

₍_z₎_<_∞_.

Moreover,

M12+M13MψCϕDHp₍+₎_→_Bα₍+₎ |ψ(i)|

(ϕ(i))1+1/p +M12+M13. (32) Corollary 11 Let1≤p<∞,α> 0,ψ∈H(+)andϕ∈S(+).Then CϕMψD:Hp(+)→

(i) M14=sup z∈+

(z)α

(ϕ(z))1+1/pψ

_ϕ₍_z₎_ϕ₍_z₎_<_∞,

(ii) M15=sup z∈+

(z)α

(ϕ(z))2+1/pψ

(11)

Moreover,

M14+M15CϕMψDHp(+₎_→_Bα₍+₎ |

ψ(ϕ(i))|

(ϕ(i))1+1/p +M14+M15. (33) Corollary 12 Let1≤p<∞,α> 0,ψ∈H(+₎_and_ϕ_∈_S₍+_)._{Then M}

ψDCϕ:Hp(+)→

(i) M16=sup z∈+

(z)α

(ϕ(z))1+1/pψ ϕ

(z)<∞,

(ii) M17=sup z∈+

(z)α

(ϕ(z))2+1/pψ(z)

ϕ(z)2<∞.

Moreover,

M16+M17MψDCϕHp₍+₎_→_Bα₍+₎ |ψ(i)ϕ ₍_i_)|

(ϕ(i))1+1/p +M16+M17. (34) Corollary 13 Let1≤p<∞,α> 0,ψ∈H(+)andϕ∈S(+).Then DMψCϕ:Hp(+)→

(i) M18=sup z∈+

(z)α

(ϕ(z))1/pψ

₍_z₎_<_∞,

(ii) M19=sup z∈+

(z)α

(ϕ(z))1+1/p2ψ

₍_z₎_ϕ₍_z_{) +}_ψ₍_z₎_ϕ₍_z₎_<_∞,

(iii) M20=sup z∈+

(z)α

(ϕ(z))2+1/pψ(z)

ϕ(z)2<∞.

Moreover,

M18+M19+M20DMψCϕHp₍+₎_→_Bα₍+₎

₍|ψ(i)|

ϕ(i))1/p +

|ψ(i)ϕ(i)|

(ϕ(i))1+1/p +M18+M19+M20. Corollary 14 Let1≤p<∞,α> 0,ψ∈H(+₎_and_ϕ_∈_S₍+_)._{Then C}

ϕDMψ:Hp(+)→

(i) M21=sup z∈+

(z)α

(ϕ(z))1/pψ

_ϕ₍_z₎_ϕ₍_z₎_<_∞,

(ii) M22=sup z∈+

(z)α

(ϕ(z))1+1/pψ

_ϕ₍_z₎_ϕ₍_z₎_<_∞,

(iii) M23=sup z∈+

(z)α

(ϕ(z))2+1/pψ

ϕ(z)ϕ(z)<∞.

Moreover,

M21+M22+M23CϕDMψHp₍+₎_→_Bα₍+₎

|₍ψ(ϕ(i))|

ϕ(i))1/p +

|ψ(ϕ(i))|

(12)

Corollary 15 Let1≤p<∞,α> 0,ψ∈H(+₎_and_ϕ_∈_S₍+_)._{Then DC}

ϕMψ:Hp(+)→

(i) M24=sup z∈+

(z)α

(ϕ(z))1/pψ

_ϕ₍_z₎_ϕ₍_z_{) +}_ψ_ϕ₍_z₎_ϕ₍_z₎2_<_∞_,

(ii) M25=sup z∈+

(z)α

(ϕ(z))1+1/p2ψ

_ϕ₍_z₎_ϕ₍_z₎2₊_ψ_ϕ₍_z₎_ϕ₍_z₎_<_∞,

(iii) M26=sup z∈+

(z)α

(ϕ(z))2+1/pψ

ϕ(z)ϕ(z)2<∞.

Moreover,

M24+M25+M26DCϕMψHp₍+₎_→_Bα₍+₎

|ψ(ϕ(i))ϕ(i)| (ϕ(i))1/p +

|ψ(ϕ(i))ϕ(i)|

(ϕ(i))1+1/p +M24+M25+M26.

Recall that for a functionf deﬁned in+,lim_z_→_∂+f(z) = 0 if and only if for everyε> 0,

there is a compact setK⊂+such that|f(z)|<εforz∈+\K.

The following result characterizes the compactness of the operatorTψ1,ψ2,ϕ:Hp(+)→

Bα₍+_).

Theorem 16 Let 1 ≤p<∞, α > 0, ψ1,ψ2 ∈H(+), and ϕ ∈S(+). Then Tψ1,ψ2,ϕ :

Hp₍+₎_→_Bα₍+₎_{is compact if and only if it is bounded and the following conditions}

are satisﬁed:

(i) lim

ϕ(z)→∂+

(z)α

(ϕ(z))1/pψ

1(z)= 0,

(ii) lim

ϕ(z)→∂+

(z)α

(ϕ(z))1+1/pψ1(z)ϕ

₍_z_{) +}_ψ

2(z)= 0,

(iii) lim

ϕ(z)→∂+

(z)α

(ϕ(z))2+1/pψ2(z)ϕ

₍_z₎_{= 0.}

Proof Assume that the operator is bounded and conditions (i)–(iii) hold. Then by Theo-rem4the quantitiesMj,j= 1, 3, are ﬁnite, whereas from conditions (i)–(iii) we have that, for eachε> 0, there exists a compact setKin+such that

(z)α

(ϕ(z))1/pψ

1(z)<ε, (35)

(z)α

(ϕ(z))1+1/pψ1(z)ϕ

₍_z_{) +}_ψ

2(z)<ε, (36)

(z)α

(ϕ(z))2+1/pψ2(z)ϕ

₍_z₎_<_ε_, ₍₃₇₎

provided thatϕ(z)∈+\K.

(13)

(z)α(Tψ1,ψ2,ϕfn) ₍_z₎

≤(z)αψ₁(z)fn

ϕ(z)+ψ1(z)ϕ(z) +ψ2(z)

f_nϕ(z)+ψ2(z)ϕ(z)fn

ϕ(z)

≤C (z)

α

(ϕ(z))1/pψ

1(z)+

(z)α

(ϕ(z))1+1/pψ1(z)ϕ

₍_z_{) +}_ψ

2(z)

+ (z)

α

(ϕ(z))2+1/pψ2(z)ϕ

₍_z₎_f

nHp₍+₎

<Cε.

SinceKis compact,ϕ(K) is also compact. Hence

M27:= sup w∈ϕ(z)

w∈(0,∞).

Choosen0∈Nsuch that

max z∈K

fn(z)<

ε

M1M1/27p

, max

z∈K

f_n(z)< ε

M2M1/27p+1

,

max z∈Kf

n(z)<

ε

M3M1/27p+2

,

(38)

and

maxfn(i),fn(i)<ε (39)

for alln≥n0.

Then, by the preceding and (38), for everyz∈+_{such that}_ϕ₍_z₎_∈_K_{, we have}

(z)αψ₁(z)fn

ϕ(z)+ψ1(z)ϕ(z) +ψ2(z)

f_nϕ(z)+ψ2(z)ϕ(z)fn

ϕ(z) ≤(z)αψ₁(z)max

w∈K

fn(w)+ (z)αψ1(z)ϕ(z) +ψ2(z)max w∈K

f_n(w)

+ (z)αψ2(z)ϕ(z)max w∈Kf

n(w)

≤M1

ϕ(z)1/pmax w∈K

fn(w)+M2

ϕ(z)1+1/pmax w∈K

f_n(w)

+M3

ϕ(z)2+1/pmax w∈K

f_n(w)

≤M1max w∈K(w)

1/p_max w∈K

fn(w)+M2max w∈K(w)

1+1/p_max w∈K

f_n(w)

+M3(w)2+1/pmax w∈K

f_n(w)

< 3ε. (40)

Further, using (39), we have

Tψ1,ψ2,ϕfn(i)=ψ1(i)fn

ϕ(i)+ψ2(i)fn

ϕ(i)≤ψ1(i)+ψ2(i)ε (41)

(14)

From (40) and (41) it follows that

Tψ1,ψ2,ϕfnBα(+)<

3 +ψ1(i)+ψ2(i)ε

forn≥n0.

Sinceε> 0 is arbitrary, by Lemma3it follows thatTψ1,ψ2,ϕ:H

p₍+₎_→_Bα₍+_{) is}

com-pact.

Conversely, suppose that Tψ1,ψ2,ϕ:H

p₍+₎_→_Bα₍+_{) is compact. Then, clearly, it is}

bounded. Let (zn)n∈N be a sequence in+such thatϕ(zn)→∂+asn→ ∞. If such a sequence does not exist, then conditions (i)–(iii) vacuously hold.

Letwn=ϕ(zn),n∈N, and let the family of functions (fn)n∈Nbe deﬁned by

fn(z) =

(wn)2–1/p

π1/p₍_z_–_w n)2

– 4i (wn)

3–1/p

π1/p₍_z_–_w n)3

– 4 (wn)

4–1/p

π1/p₍_z_–_w n)4

.

Then by using (22) and some simple estimates it is easy to see that (fn)n∈Nweakly converges to zero asn→ ∞(including the case whenwn→ ∞asn→ ∞). Hence

lim

n→∞Tψ1,ψ2,ϕfnB

α₍+₎= 0. (42)

The boundedness ofTψ1,ψ2,ϕ:Hp(+)→Bα(+), together with (23), implies

Tψ1,ψ2,ϕfnBα(+)≥(zn)

α_ψ

1(zn)fn

ϕ(zn)

+ψ1(zn)ϕ(zn) +ψ2(zn)

f_nϕ(zn)

+ψ2(zn)ϕ(zn)fn

ϕ(zn)

= (zn)

α

8π1/p₍_ϕ₍_z n))2+1/p

ψ2(zn)ϕ(zn). (43)

From (42) and (43) we obtain

lim

ϕ(zn)→∂+

(zn)α (ϕ(zn))2+1/p

ψ2(zn)ϕ(zn)= 0, (44)

from which it follows that condition (i) holds. By considering families of functions

gn(z) = 12 (

wn)2–1/p

π1/p₍_z_–_w n)2

– 32i (wn)

3–1/p

π1/p₍_z_–_w n)3

– 24 (wn)

4–1/p

π1/p₍_z_–_w n)4

and

hn(z) = 8

(wn)2–1/p

π1/p₍_z_–_w n)2

– 28i (wn)

3–1/p

π1/p₍_z_–_w n)3

– 24 (wn)

4–1/p

π1/p₍_z_–_w n)4

and proceeding as before, we can similarly show that

lim

ϕ(zn)→∂+

(zn)α (ϕ(zn))1+1/p

(15)

and

lim

ϕ(zn)→∂+

(zn)α (ϕ(zn))1/p

ψ₁(zn)= 0, (46)

from which it follows that conditions (ii) and (iii) hold, respectively.

Corollary 17 Let1≤p<∞,α> 0andϕ∈S(+).Then Cϕ:Hp(+)→Bα(+)is

com-pact if and only if it is bounded and

lim

ϕ(z)→∂+

(z)α

(ϕ(z))1+1/pϕ

₍_z₎_{= 0.}

Corollary 18 Let1≤p<∞,α> 0,ψ∈H(+)andϕ∈S(+).Then MψCϕ:Hp(+)→

Bα₍+₎_{is compact if and only if it is bounded and}

(i) lim

ϕ(z)→∂+

(z)α

(ϕ(z))1/pψ

₍_z₎_{= 0,}

(ii) lim

ϕ(z)→∂+

(z)α

(ϕ(z))1+1/pψ(z)ϕ

₍_z₎_{= 0.}

Corollary 19 Let1≤p<∞,α> 0,ψ∈H(+)andϕ∈S(+).Then CϕMψ:Hp(+)→

(i) lim

ϕ(z)→∂+

(z)α

(ϕ(z))1/pψ

_ϕ₍_z₎_ϕ₍_z₎_{= 0,}

(ii) lim

ϕ(z)→∂+

(z)α

(ϕ(z))1+1/pψ

ϕ(z)ϕ(z)= 0.

Corollary 20 Let1≤p<∞,α> 0andϕ∈S(+_)._{Then C}

ϕD:Hp(+)→Bα(+)is

lim

ϕ(z)→∂+

(z)α

(ϕ(z))2+1/pϕ

₍_z₎_{= 0.}

Corollary 21 Let1≤p<∞,α> 0andϕ∈S(+).Then DCϕ:Hp(+)→Bα(+)is

(i) lim

ϕ(z)→∂+

(z)α

(ϕ(z))1+1/pϕ

₍_z₎_{= 0,}

(ii) lim

ϕ(z)→∂+

(z)α

(ϕ(z))2+1/pϕ

₍_z₎2

= 0.

Corollary 22 Let1≤p<∞,α> 0,ψ∈H(+₎_and_ϕ_∈_S₍+_)._{Then M}

ψCϕD:Hp(+)→

(i) lim

ϕ(z)→∂+

(z)α

(ϕ(z))1+1/pψ

₍_z₎_{= 0,}

(ii) lim

ϕ(z)→∂+

(z)α

(ϕ(z))2+1/pψ(z)ϕ

(16)

Corollary 23 Let1≤p<∞,α> 0,ψ∈H(+₎_and_ϕ_∈_S₍+_)._{Then C}

ϕMψD:Hp(+)→

(i) lim

ϕ(z)→∂+

(z)α

(ϕ(z))1+1/pψ

_ϕ₍_z₎_ϕ₍_z₎_{= 0,}

(ii) lim

ϕ(z)→∂+

(z)α

(ϕ(z))2+1/pψ

ϕ(z)ϕ(z)= 0.

Corollary 24 Let1≤p<∞,α> 0,ψ∈H(+₎_and_ϕ_∈_S₍+_)._{Then M}

ψDCϕ:Hp(+)→

(i) lim

ϕ(z)→∂+

(z)α

(ϕ(z))1+1/pψ ϕ

(z)= 0,

(ii) lim

ϕ(z)→∂+

(z)α

(ϕ(z))2+1/pψ(z)

ϕ(z)2= 0.

Corollary 25 Let1≤p<∞,α> 0,ψ∈H(+)andϕ∈S(+).Then DMψCϕ:Hp(+)→

(i) lim

ϕ(z)→∂+

(z)α

(ϕ(z))1/pψ

₍_z₎_{= 0,}

(ii) lim

ϕ(z)→∂+

(z)α

(ϕ(z))1+1/p2ψ

₍_z₎_ϕ₍_z_{) +}_ψ₍_z₎_ϕ₍_z₎_{= 0,}

(iii) lim

ϕ(z)→∂+

(z)α

(ϕ(z))2+1/pψ(z)

ϕ(z)2= 0.

Corollary 26 Let1≤p<∞,α> 0,ψ∈H(+₎_and_ϕ_∈_S₍+_)._{Then C}

ϕDMψ:Hp(+)→

(i) lim

ϕ(z)→∂+

(z)α

(ϕ(z))1/pψ

_ϕ₍_z₎_ϕ₍_z₎_{= 0,}

(ii) lim

ϕ(z)→∂+

(z)α

(ϕ(z))1+1/pψ

_ϕ₍_z₎_ϕ₍_z₎_{= 0,}

(iii) lim

ϕ(z)→∂+

(z)α

(ϕ(z))2+1/pψ

ϕ(z)ϕ(z)= 0.

Corollary 27 Let1≤p<∞,α> 0,ψ∈H(+)andϕ∈S(+).Then DCϕMψ:Hp(+)→

(i) lim

ϕ(z)→∂+

(z)α

(ϕ(z))1/pψ

_ϕ₍_z₎_ϕ₍_z_{) +}_ψ_ϕ₍_z₎_ϕ₍_z₎2_{= 0,}

(ii) lim

ϕ(z)→∂+

(z)α

(ϕ(z))1+1/p2ψ

_ϕ₍_z₎_ϕ₍_z₎2₊_ψ_ϕ₍_z₎_ϕ₍_z₎_{= 0,}

(iii) lim

ϕ(z)→∂+

(z)α

(ϕ(z))2+1/pψ

(17)

Acknowledgements

The study in the paper is a part of the investigation at the projects III 41025 and III 44006 by the Serbian Ministry of Education and Science.

Funding

Not applicable.

Availability of data and materials

Not applicable.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors have contributed equally to the writing of this paper. They read and approved the manuscript.

Author details

1_{Mathematical Institute of the Serbian Academy of Sciences, Beograd, Serbia.}2_{Department of Medical Research, China}

Medical University Hospital, China Medical University, Taichung, Taiwan, Republic of China.3_{Department of Mathematics,} Central University of Jammu, Jammu, India.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 10 August 2018 Accepted: 30 September 2018 References

1. Avetisyan, K.L.: Hardy-Bloch type spaces and lacunary series on the polydisk. Glasg. Math. J.49(2), 345–356 (2007) 2. Avetisyan, K.L.: Weighted integrals and Bloch spaces ofn-harmonic functions on the polydisc. Potential Anal.29(1),

49–63 (2008)

3. Benke, G., Chang, D.C.: A note on weighted Bergman spaces and the Cesáro operator. Nagoya Math. J.159, 25–43 (2000)

4. Jiang, Z.J.: Generalized product-type operators from weighted Bergman–Orlicz spaces to Bloch–Orlicz spaces. Appl. Math. Comput.268, 966–977 (2015)

5. Jiang, Z.J.: On Stevi´c–Sharma operator from the Zygmund space to the Bloch–Orlicz space. Adv. Diﬀer. Equ.2015, Article ID 228 (2015)

6. Li, H., Guo, Z.: On a product-type operator from Zygmund-type spaces to Bloch–Orlicz spaces. J. Inequal. Appl.2015, Article ID 132 (2015)

7. Li, S., Stevi´c, S.: Composition followed by diﬀerentiation from mixed-norm spaces toα-Bloch spaces. Sb. Math.

199(12), 1847–1857 (2008)

8. Li, S., Stevi´c, S.: Products of composition and diﬀerentiation operators from Zygmund spaces to Bloch spaces and Bers spaces. Appl. Math. Comput.217, 3144–3154 (2010)

9. Li, S., Stevi´c, S.: Weighted diﬀerentiation composition operators from the logarithmic Bloch space to the weighted-type space. An. ¸Stiin¸t. Univ. ‘Ovidius’ Constan¸ta, Ser. Mat.24(3), 223–240 (2016)

10. Sehba, B., Stevi´c, S.: On some product-type operators from Hardy–Orlicz and Bergman–Orlicz spaces to weighted-type spaces. Appl. Math. Comput.233C, 565–581 (2014)

11. Stevi´c, S.: Characterizations of composition followed by diﬀerentiation between Bloch-type spaces. Appl. Math. Comput.218, 4312–4316 (2011)

12. Zhu, X.: Generalized weighted composition operators from Bloch-type spaces to weighted Bergman spaces. Indian J. Math.49(2), 139–149 (2007)

13. Zhu, X.: Essential norm and compactness of the product of diﬀerentiation and composition operators on Bloch type spaces. Math. Inequal. Appl.19(1), 325–334 (2016)

14. Stevi´c, S.: Composition operators from the Hardy space to the Zygmund-type space on the upper half-plane. Abstr. Appl. Anal.2009, Article ID 161528 (2009)

15. Stevi´c, S.: Composition operators from the Hardy space to thenth weighted-type space on the unit disk and the half-plane. Appl. Math. Comput.215, 3950–3955 (2010)

16. Stevi´c, S.: Essential norm of some extensions of the generalized composition operators betweenkth weighted-type spaces. J. Inequal. Appl.2017, Article ID 220 (2017)

17. Matache, V.: Composition operators onHp_{of the upper half-plane. Proc. An. Univ. Timisoara Ser. Stiint. Mat.}₂₇_{, 63–66}

(1989)

18. Matache, V.: Compact composition operators onHp_{of a half-plane. Proc. Am. Math. Soc.}₁₂₇_{, 1483–1491 (1999)}

19. Sharma, S.D.: Compact and Hilbert–Schmidt composition operators on Hardy spaces of the upper half-plane. Acta Sci. Math.46(1–4), 197–202 (1983)

20. Singh, R.K., Sharma, S.D.: Non-compact composition operators. Bull. Aust. Math. Soc.21, 125–130 (1980) 21. Sharma, S.D., Sharma, A.K., Ahmed, S.: Composition operators between Hardy and Bloch-type spaces of the upper

half-plane. Bull. Korean Math. Soc.44, 475–482 (2007)

22. Sharma, A.K., Ueki, S.: Compact composition operators on the Bloch space and the growth space of the upper half-plane. Mediterr. J. Math.14(2), Article ID 76 (2017)

23. Stevi´c, S., Sharma, A.K.: Weighted composition operators between growth spaces of the upper half-plane. Util. Math.

(18)

24. Stevi´c, S., Sharma, A.K.: Weighted composition operators between Hardy and growth spaces on the upper half-plane. Appl. Math. Comput.217, 4928–4934 (2011)

25. Stevi´c, S., Sharma, A.K., Sharma, S.D.: Weighted composition operators from weighted Bergman spaces to weighted-type spaces on the upper half-plane. Abstr. Appl. Anal.2011, Article ID 989625 (2011)

26. Li, S., Stevi´c, S.: Composition followed by diﬀerentiation betweenH∞andα-Bloch spaces. Houst. J. Math.35, 327–340 (2009)

27. Ohno, S.: Products of composition and diﬀerentiation between Hardy spaces. Bull. Aust. Math. Soc.73(2), 235–243 (2006)

28. Yang, W.: Products of composition and diﬀerentiation operators fromQK(p,q) spaces to Bloch-type spaces. Abstr.

Appl. Anal.2009, Article ID 741920 (2009)

29. Zhou, J., Zhu, X.: Product of diﬀerentiation and composition operators on the logarithmic Bloch space. J. Inequal. Appl.2014, Article ID 453 (2014)

30. Stevi´c, S.: Weighted diﬀerentiation composition operators from mixed-norm spaces to weighted-type spaces. Appl. Math. Comput.211, 222–233 (2009)

31. Yang, W.: Generalized weighted composition operators from theF(p,q,s) space to the Bloch-type space. Appl. Math. Comput.218, 4967–4972 (2012)

32. Yu, Y., Liu, Y.: The essential norms of a generalized composition operators between Bloch type spaces andQKtype

spaces. Complex Anal. Oper. Theory6(6), 1231–1240 (2012)

33. Zhu, X.: Essential norm of generalized weighted composition operators on Bloch-type spaces. Appl. Math. Comput.

274, 133–142 (2016)

34. Li, S., Stevi´c, S.: Products of composition and integral type operators fromH∞to the Bloch space. Complex Var. Elliptic Equ.53(5), 463–474 (2008)

35. Li, H., Li, S.: Norm of an integral operator on some analytic function spaces on the unit disk. J. Inequal. Appl.2013, Article ID 342 (2013)

36. Ueki, S.I.: On the Li–Stevi´c integral type operators from weighted Bergman spaces intoβ-Zygmund spaces. Integral Equ. Oper. Theory74(1), 137–150 (2012)

37. Yang, W.: Volterra composition operators fromF(p,q,s) spaces to Bloch-type spaces. Bull. Malays. Math. Sci. Soc.34(2), 267–277 (2011)

38. Yu, Y., Liu, Y.: On Li–Stevi´c integral-type operators between diﬀerent weighted Bloch-type spaces. J. Inequal. Appl.

2008, Article ID 780845 (2008)

39. Zhu, X.: Volterra composition operators from weighted-type spaces to Bloch–type spaces and mixed norm spaces. Math. Inequal. Appl.14(1), 223–233 (2011)

40. Zhu, X.: Generalized Riemann–Stieltjes operators between diﬀerent weighted Bergman spaces. J. Comput. Anal. Appl.

14(3), 426–432 (2012)

41. Jiang, Z.J.: On a product operator from weighted Bergman–Nevanlinna spaces to weighted Zygmund spaces. J. Inequal. Appl.2014, Article ID 404 (2014)

42. Liu, Y., Yu, Y.: On a Stevi´c–Sharma operator from Hardy spaces to the logarithmic Bloch spaces. J. Inequal. Appl.2014, Article ID 22 (2014)

43. Sharma, A.K.: Products of multiplication, composition and diﬀerentiation between weighted Bergman–Nevanlinna and Bloch-type spaces. Turk. J. Math.35, 275–291 (2011)

44. Stevi´c, S.: On operatorPgϕfrom the logarithmic Bloch-type space to the mixed-norm space on unit ball. Appl. Math.

Comput.215, 4248–4255 (2010)

45. Stevi´c, S., Sharma, A.K., Bhat, A.: Products of multiplication composition and diﬀerentiation operators on weighted Bergman spaces. Appl. Math. Comput.217, 8115–8125 (2011)

46. Stevi´c, S., Sharma, A.K., Bhat, A.: Essential norm of products of multiplication composition and diﬀerentiation operators on weighted Bergman spaces. Appl. Math. Comput.218, 2386–2397 (2011)