Natural metrics and composition operators in generalized hyperbolic function spaces

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R E S E A R C H

Open Access

Natural metrics and composition operators in

generalized hyperbolic function spaces

A El-Sayed Ahmed

*

*_{Correspondence:}

[email protected] Faculty of Science, Mathematics Department, Sohag University, Sohag, 82524, Egypt

Current address: Faculty of Science, Mathematics Department, Taif University, Box 888, El-Hawiah, Taif, Saudi Arabia

Abstract

In this paper, we deﬁne some generalized hyperbolic function classes. We also introduce natural metrics in the generalized hyperbolic (p,

α

)-Bloch and in the generalized hyperbolicQ*(p,s) classes. These classes are shown to be complete metric spaces with respect to the corresponding metrics. Moreover, boundedness and compactness the composition operatorsCφacting from the generalized hyperbolic

(p,

α

)-Bloch class to the classQ*_(p,_{s) are characterized by conditions depending on an}

analytic self-map

φ

:D→D.

MSC: 47B38; 46E15

Keywords: hyperbolic classes; composition operators; (p,

α

)-Bloch space;Q*(p,s) classes

1 Introduction

LetD={z:|z|< }be the open unit disc of the complex planeC,∂Dits boundary. Let

H(D) denote the space of all analytic functions inDand letB(D) be the subset ofH(D) consisting of thosef ∈H(D) for which|f(z)|<  for allz∈D. Also,dA(z) be the normalized area measure onDso thatA(D)≡. The usualα-Bloch spacesBαandBα,are deﬁned as

the sets of thosef ∈H(D) for which

fBα=sup

z∈D

f(z) –|z|α<∞,

and

lim

|z|→f

₍_z₎_{ –}_|_z_|α

= ,

respectively. Now, we will give the following deﬁnition:

Deﬁnition . The (p,α)-Bloch spacesBp,α andBp,α, are deﬁned as the sets of those f∈H(D) for which

fBp,α=

p

supz∈D

f(z)

p

–_f₍_z₎_{ –}|_z|α_<_∞_,

and

lim

|z|→ f(z)

p

–_f₍_z₎_{ –}|_z|α_{= ,} where  <p,α<∞.

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Remark . The definition of (p,α)-Bloch spaces is introduced in the present paper for the first time. One should note that, if we putp=  in Definition ., we will obtain the spacesBαandBα,.

Remark . (p,α)-Bloch space is very useful in some calculations in this paper and it can be also used to study some other operators like integral operators (see []).

If (X,d) is a metric space, we denote the open and closed balls with centerxand radiusr>  byB(x,r) :={y∈X:d(y,x) <r}andB¯(x,r) :={y∈X:d(y,x) =r}, respectively. The well-known hyperbolic derivative is deﬁned byf*₍_z_{) =} |f(z)|

–|f(z)| off ∈B(D) and the hyperbolic distance is given byρ(f(z), ) :=_log(+_–|_|f_f(₍_zz)₎|_|) betweenf(z) and zero.

A functionf ∈B(D) is said to belong to the hyperbolicα-Bloch classBα* if

f_B* α=sup

z∈D

f*(z) –|z|α<∞.

The little hyperbolic Bloch-type classB*α,consists of allf∈Bα* such that

lim

|z|→f

*₍_z₎_{ –}_|_z_|α

= .

The Schwarz-Pick lemma impliesB*

α=B(D) for allα≥ withfB*α≤, and therefore, the hyperbolicα-Bloch classes are of interest only when  <α< .

It is obvious thatB*

αis not a linear space since the sum of two functions inB(D) does

not necessarily belong toB(D).

Now, let  <p<∞, we deﬁne the hyperbolic derivative byf_p*(z) = p_|f(z)|

p

 –_|_f(z)|

–|f(z)|p off ∈ B(D). Whenp= , we obtain the usual hyperbolic derivative as deﬁned above.

A functionf ∈B(D) is said to belong to the generalized hyperbolic (p,α)-Bloch class

B*

p,αif

f_B*

p,α=sup z∈Df

*

p(z)

 –|z|α<∞.

The little generalized (p,α)-hyperbolic Bloch-type classB*

p,α,consists of allf∈Bp*,αsuch

that

lim

|z|→f *

p(z)

 –|z|α= .

Let the Green’s function ofDbe deﬁned asg(z,a) =log_|_ϕ

a(z)|, whereϕa(z) =

a–z

–az¯ is the Möbius transformation related to the pointa∈D. For  <p,s<∞, the hyperbolic class

Q*₍_p_,_s_{) consists of those functions}_f _∈_B₍_D_{) for which}

fp_Q*₍_p_,_s₎=sup a∈D

D

f_p*(z)gs(z,a)dA(z) <∞.

Moreover, we say thatf ∈Q*₍_p_,_s_{) belongs to the class}_Q*₍_p_,_s_{, ) if}

lim

|a|→

D

f_p*(z)gs(z,a)dA(z) = .

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Remark . The Schwarz-Pick lemma implies that B*

p,α = B(D) for all α ≥  with

f_B*

p,α≤ and therefore, the generalized hyperbolic (p,α)-classes are of interest only when  <α< . AlsoQ*(p,s) =B(D) for alls> , and hence, the generalized hyperbolic

Q(p,s)-classes will be considered when ≤s≤.

For any holomorphic self-mappingφofD, the symbolφ induces a linear composition operatorCφ(f) =f◦φfromH(D) orB(D) into itself. The study of a composition operator

Cφ acting on the spaces of analytic functions has engaged many analysts for many years

(see,e.g., [–, , , ] and others).

Yamashita was probably the ﬁrst to consider systematically hyperbolic function classes. He introduced and studied hyperbolic Hardy, BMOA and Dirichlet classes in [–] and others. More recently, Smith studied inner functions in the hyperbolic little Bloch-class [], and the hyperbolic counterparts of theQpspaces were studied by Li in [] and Li

et al.in []. Further, hyperbolicQp classes and composition operators were studied by Pérez-Gonzálezet al.in [].

In this paper, we will study the generalized hyperbolic (p,α)-Bloch classesB*

p,αand the

hyperbolicQ*₍_p_,_s_{) type classes. We will also give some results to characterize Lipschitz}

continuous and compact composition operators mapping from the generalized hyperbolic (p,α)-Bloch classB*

p,αtoQ*(p,s) classes by conditions depending on the symbolφ only.

Thus, the results are generalizations of the recent results of Pérez-González, Rättyä and Taskinen [].

Recall that a linear operatorT :X→Yis said to be bounded if there exists a constant

C>  such thatT(f)Y≤CfXfor all mapsf ∈X. By elementary functional analysis, a linear operator between normed spaces is bounded if and only if it is continuous, and the boundedness is trivially also equivalent to the Lipschitz-continuity. Moreover,T:X→Y

is said to be compact if it takes bounded sets inXto sets inYwhich have compact closure. For Banach spacesXandYcontained inB(D) orH(D),T:X→Yis compact if and only if for each bounded sequence (xn)∈X, the sequence (Txn)∈Y contains a subsequence converging to a functionf ∈Y.

Throughout this paper,C stands for absolute constants which may indicate diﬀerent constants from one occurrence to the next.

The following lemma follows by standard arguments similar to those outlined in []. Hence we omit the proof.

Lemma . Assumeφis a holomorphic mapping fromDinto itself. Let <p,s<∞, and

 <α<∞. Then Cφ:B*p,α→Q*(p,s)is compact if and only if for any bounded sequence

(fn)n∈N∈Bp*,αwhich converges to zero uniformly on compact subsets ofDas n→ ∞, we

havelimn→∞CφfnQ*₍_p_,_s₎= .

Using the standard arguments similar to those outlined in Lemma  of [], we have the following lemma:

Lemma . Let <α<∞, then there exist two functions f,g∈B*

p,α such that for some

constant C,

_f*

p(z)+gp*(z) –|z|

α

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2 Natural metrics inB*

p,αandQ*(p,s)classes

In this section we introduce natural metrics on generalized hyperbolicα-Bloch classes

B*

p,αand the classesQ*(p,s).

Let  <p,s<∞, and  <α< . First, we can ﬁnd a natural metric inB*_p_,α (see []) by

deﬁning

df,g;B*_p_,α

:=d_B*

p,α(f,g) +f–gBp,α+f() –g()

p

_, _()

where

d_B*

p,α(f,g) :=sup z∈D

f(z)|f(z)| p

–

 –|f(z)|p –

g(z)|g(z)|p–  –|g(z)|p

 –|z|α.

Forf,g∈Q*(p,s), deﬁne their distance by

df,g;Q*(p,s):=dQ*(f,g) +f –g_Q₍_p_,_s₎+f() –g()

p

_,

where

dQ*(f,g) :=

p

supz∈D

D

f(z)|f(z)| p

–

 –|f(z)|p –

g(z)|g(z)|p–  –|g(z)|p

gs(z,a)dA(z)

  .

Now, we give a characterization of the complete metric spaced(·,·;B*

p,α).

Proposition . The classB*

p,αequipped with the metric d(·,·;Bp*,α)is a complete metric

space. Moreover,B_p*_,α,is a closed (and therefore complete) subspace ofB*p,α.

Proof Clearlyd(f,g;B*

p,α)≥,d(f,g,Bp*,α) =d(g,f;B*p,α). Also,

df,h;B_p*_,α

≤df,g;B*_p_,α

+dg,h;B_p*_,α

.

Moreover,d(f,f;B*_p_,α) =  for allf,g,h∈B*p,α.

It follows from the presence of the usual (p,α)-Bloch term thatd(f,g;B*

p,α) =  implies

f =g. Hence, (B*

p,α,d) is a metric space. Let (fn)∞n= be a Cauchy sequence in the metric

space (B*

p,α,d), that is, for anyε> , there is anN=N(ε)∈Nsuch that

dfn,fm;B_p*_,α

<ε

for alln,m>N. Since (fn)⊂B(D), the family (fn) is uniformly bounded and hence normal inD. Therefore, there existf ∈B(D) and a subsequence (fnj)

∞

j=such thatfnj converges to f uniformly on compact subsets, and by the Cauchy formula, the same also holds for the derivatives. Letm>N. Then the uniform convergence yields

f(z)|f(z)| p

–

 –|f(z)|p –

f_m(z)|fm(z)|p–  –|fm(z)|p

 –|z|α

= lim n→∞

fn(z)|fn(z)|

p

–

 –|fn(z)|p –

f_m(z)|fm(z)|p–  –|fm(z)|p

 –|z|α≤ lim n→∞d

fn,fm;Bp*,α

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for allz∈D, and it follows that

f_B*

p,α≤ fmB*p,α+ε.

Thus,f ∈B*

p,αas desired. Moreover, () and the completeness of the usual (p,α)-Bloch

im-ply that (fn)∞_n_=converges tof with respect to the metricd. The second part of the assertion

follows by ().

Next, we give a characterization of the complete metric spaced(·,·;Q*(p,s)).

Proposition . The class Q*₍_p_,_s₎_{equipped with the metric d}₍_·_,_·_;_Q*₍_p_,_s₎₎_{is a complete} metric space. Moreover, Q*₍_p_,_s_{, )}_{is a closed (and therefore complete) subspace of Q}*₍_p_,_s₎_.

Proof Forf,g,h∈Q*₍_p_,_s_{), then clearly}

• d(f,g;Q*₍_p_,_s₎₎_≥__,

• d(f,f;Q*₍_p_,_s_{)) = }_,

• d(f,g;Q*₍_p_,_s_{)) = }_implies_f ₌_g_,

• d(f,g;Q*(p,s)) =d(g,f;Q*(p,s)),

• d(f,h;Q*(p,s))≤d(f,g;Q*(p,s)) +d(g,h;Q*(p,s)).

Hence,dis metric onQ*₍_p_,_s_).

For the completeness proof, let (fn)∞_n_= be a Cauchy sequence in the metric space (Q*₍_p_,_s_),_d_{), that is, for any}_ε_{>  there is an}_N₌_N₍_ε₎_∈_N_{such that}_d₍_f_n,_f_m;_Q*₍_p_,_s_{)) <}_ε_,

for alln,m>N. Sincefn∈B(D) such thatfnconverges tof uniformly on compact subsets ofD. Letm>Nand  <r< . Then Fatou’s lemma yields

D(,r) |f(z)|

p

–f(z)  –|f(z)|p –

|fm(z)|

p

–f m(z)  –|fm(z)|p

gs(z,a)dA(z)

=

D(,r)

lim n→∞

|fn(z)|

p

–_f n(z)  –|fn(z)|p –

|fm(z)|p–_f m(z)  –|fm(z)|p

gs(z,a)dA(z)

≤ lim

n→∞

D

|fn(z)|

p

–f n(z)  –|fn(z)|p –

|fm(z)|

p

–f m(z)  –|fm(z)|p

gs(z,a)dA(z)≤ε,

and by lettingr→–_{, it follows that}

D

f_p*(z)gs(z,a)dA(z)≤ε+ 

D

|fm(z)|

p

–f m(z)  –|fm(z)|p

gs(z,a)dA(z). ()

This yields

fp_Q*₍_p_,_s₎≤ε+ fm_Q*₍_p_,_s₎,

and thusf ∈Q*(p,s). We also ﬁnd thatfn→f with respect to the metric ofQ*(p,s). The

second part of the assertion follows by ().

3 Lipschitz continuous and compactness ofCφ

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(i) Cφ:B_p*_,α→Q*(p,s)is bounded;

(ii) Cφ:Bp*,α→Q*(p,s)is Lipschitz continuous;

(iii) sup_a_∈_D_D |φ(z)|

(–|φ(z)|p₎αgs(z,a)dA(z) <∞.

Proof First, assume that (i) holds, then there exists a constantCsuch that

CφfQ*₍_p_,_s₎≤Cf_B*

p,α, for allf ∈B

*

p,α.

For givenf ∈B*

p,α, the functionft(z) =f(tz), where  <t< , belongs toB*p,αwith the

prop-ertyft_B*

p,α≤ fB*p,α. Letf,gbe the functions from Lemma . such that



( –|z|₎α ≤f

*

p(z)+gp*(z),

for allz∈D, so that

|φ(z)|

( –|φ(z)|)α ≤(f◦φ)

*₍_z_{) + (}_g_◦_φ₎*₍_z_).

Thus,

D

|tφ(z)|

( –|tφ(z)|₎αg

s₍_z_,_a₎_dA₍_z₎

≤C

D

(f◦tφ)*_p(z)+(g◦tφ)*_p(z)gs(z,a)dA(z)

≤CCφ

f_B*

p,α+g



B*

p,α

.

This estimate together with the Fatou’s lemma implies (iii). Conversely, assuming that (iii) holds and thatf ∈B*

p,α, we see that

sup a∈D

D

(f ◦φ)*_p(z)gs(z,a)dA(z)

=sup a∈D

D

f_p*φ(z)φ(z)gs(z,a)dA(z)

≤ f_B*

p,αsup_a_∈_D

D

|φ(z)|

( –|φ(z)|p₎αg

s₍_z_,_a₎_dA₍_z_).

Hence, it follows that (i) holds.

(ii)⇐⇒(iii). Assume ﬁrst thatCφ:B*p,α→Q*(p,s) is Lipschitz continuous, that is, there

exists a positive constantCsuch that

df ◦φ,g◦φ;Q*(p,s)≤Cdf,g;B*_p_,α

, for allf,g∈B_p*_,α.

Takingg= , this implies

f◦φ_Q*₍_p_,_s₎≤C

f_B*

p,α+fBp,α+f()

p

_, _{for all}_f ∈B*

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The assertion (iii) forα=  follows by choosingf(z) =zin (). If  <α< , then



pf(z) p

 ₌

z

 _f₍_t₎p

_f₍_t₎_dt₊_f_()

p



≤ fBp,α

|z|

 dx

( –x₎α +f()

p



≤ fBα

( –α)+f()

p

_,

this yields



pf

φ()–gφ()

p

 ≤f–gBp,α

( –α) +f() –g()

p

_.

Moreover, Lemma . implies the existence off,g∈B*

p,αsuch that

_f*

p(z) +g*p(z) –|z|

α

≥C> , for allz∈D. ()

Combining () and (), we obtain

f_B*

p,α+gB*p,α+fBp,α+gBp,α+f()

p

 ₊_g_()

p



≥C

D

|φ(z)|

( –|φ(z)|p₎αg

s₍_z_,_a₎_dA₍_z₎

for which the assertion (iii) follows. Assume now that (iii) is satisﬁed, we have

df ◦φ,g◦φ;Q*(p,s)=d_Q*

(p,s)(f◦φ,g◦φ) +f◦φ–g◦φQ(p,s)

+fφ()–gφ()

p



≤d_B*

p,α(f,g)

sup a∈D

D

|φ(z)|

( –|φ(z)|p₎αg

s₍_z_,_a₎_dA₍_z₎

 

+f –gBp,α

sup a∈D

D

|φ(z)|

( –|φ(z)|p₎αg

s₍_z_,_a₎_dA₍_z₎

 

+f–gBp,α

( –α) +f() –g()

p



≤Cdf,g;B*_p_,α

.

ThusCφ:Bp*,α→Q*(p,s) is Lipschitz continuous and the proof is completed.

Remark . We know that a composition operator Cφ :B*_p_,α→Q*(p,s) is said to be

bounded if there is a positive constantCsuch thatCφfQ*₍_p_,_s₎≤Cf_B*

p,αfor allf ∈B

*

p,α.

Theorem . shows that Cφ :Bp*,α →Q*(p,s) is bounded if and only if it is

Lipschitz-continuous, that is, if there exists a positive constantCsuch that

df ◦φ,g◦φ;Q*(p,s)≤Cdf,g;B*_p_,α

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By elementary functional analysis, a linear operator between normed spaces is bounded if and only if it is continuous, and the boundedness is trivially also equivalent to the Lipschitz-continuity. So, our result for composition operators in hyperbolic spaces is the correct and natural generalization of the linear operator theory.

Recall that a composition operatorCφ:B*p,α→Q*(p,s) is compact if it maps any ball in

B*

p,αonto a precompact set inQ*(p,s).

The following observation is sometimes useful.

Proposition . Let <p<∞,≤s≤and <α≤. Assume thatφis a holomorphic mapping fromDinto itself. If Cφ:Bp*,α→Q*(p,s)is compact, it maps closed balls onto

compact sets.

Proof IfB⊂B*

p,αis a closed ball andg∈Q*(p,s) belongs to the closure ofCφ(B), we can

ﬁnd a sequence (fn)∞n=⊂Bsuch thatfn◦φconverges tog∈Q*(p,s) asn→ ∞. But (fn)∞n=is

a normal family, hence it has a subsequence (fnj)∞j=converging uniformly on the compact

subsets ofDto an analytic functionf. As in earlier arguments of Proposition . in [], we get a positive estimate which shows thatf must belong to the closed ballB. On the other hand, also the sequence (fnj◦φ)∞j= converges uniformly on compact subsets to an

analytic function, which isg∈Q*(p,s). We getg=f◦φ,i.e.,gbelongs toCφ(B). Thus, this

set is closed and also compact.

Compactness of composition operators can be characterized in full analogy with the linear case.

Theorem . Let <p<∞,≤s≤, and <α≤. Assume thatφ is a holomorphic mapping fromDinto itself. Then the following statements are equivalent:

(i) Cφ:Bp*,α→Q*(p,s)is compact.

(ii)

lim r→–sup

a∈D

|φ|≥rj

|φ(z)|

( –|φ(z)|p₎αg

s₍_z_,_a₎_dA₍_z_{) = .}

Proof We ﬁrst assume that (ii) holds. LetB:=B¯(g,δ)⊂B_p*_,α, whereg∈B*p,αandδ> , be a

closed ball, and let (fn)∞_n_=⊂Bbe any sequence. We show that its image has a convergent subsequence inQ*₍_p_,_s_{), which proves the compactness of}_C

φby deﬁnition.

Again, (fn)∞n=⊂B(D) is a normal family, hence there is a subsequence (fnj)∞j=which

con-verges uniformly on the compact subsets ofDto an analytic functionf. By the Cauchy formula for the derivative of an analytic function, also the sequence (f_n_j)∞_j_=converges uni-formly tof. It follows that also the sequences (fnj◦φ)

∞

j=and (fnj◦φ)

∞

j=converge uniformly

on the compact subsets ofDtof◦φandf◦φ, respectively. Moreover,f∈B⊂B*_p_,αsince

for any ﬁxedR,  <R< , the uniform convergence yields

sup

|z|≤R

f(z)|f(z)| p

–

 –|f(z)|p –

g(z)|g(z)|p–  –|g(z)|p

 –|z|α

+sup

|z|≤R

f(z) –g(z)

p

–_f₍_z_{) –}_g₍_z₎_{ –}|_z|α

+f() –g()

p

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=lim j→∞_|sup_z_|≤_R

fnj(z)|fnj(z)| p

–

 –|fnj(z)|p

–g

₍_z₎_|_g₍_z₎_|p

–

 –|g(z)|p

 –|z|α

+sup

|z|≤R

fnj(z) –g(z) p

–_f nj(z) –g

₍_z₎_{ –}_|_z_|α

+f() –g()

p

–≤_δ_.

Hence,d(f,g;B*

p,α)≤δ.

Letε> . Since (ii) is satisﬁed, we may ﬁxr,  <r< , such that

sup a∈D

|φ(z)|≥r

|φ(z)|p–_|_φ₍_z₎_|

( –|φ(z)|p₎α g

s₍_z_,_a₎_dA₍_z₎_≤_ε_.

By the uniform convergence, we may ﬁxN∈Nsuch that

fnj◦φ() –f◦φ()≤ε, for allj≥N. ()

The condition (ii) is known to imply the compactness ofCφ:Bp,α→Q(p,s), hence

pos-sibly to passing once more to a subsequence and adjusting the notations, we may assume that

fnj◦φ–f◦φQ(p,s)≤ε, for allj≥N, for someN∈N. ()

Now let

I(a,r) =sup a∈D

|φ(z)|≥r

(fnj◦φ) *

p(z) – (f ◦φ)*p(z)



gs(z,a)dA(z),

and

I(a,r) =sup a∈D

|φ(z)|≤r

(fnj◦φ) *

p(z) – (f ◦φ)*p(z)



gs(z,a)dA(z).

Since (fnj)

∞

j=⊂Bandf ∈B, it follows that

I(a,r) =sup a∈D

|φ(z)|≥r

(fnj◦φ) *

p(z) – (f◦φ)*p(z)



gs(z,a)dA(z)

≤p

supa∈D

|φ(z)|≥r

L(fnj,f,φ)g

s₍_z_,_a₎_dA₍_z₎

≤d_B*

α(fnj,f)sup

a∈D

|φ(z)|≥r

|φ(z)| ( –|φ(z)|p₎αg

s₍_z_,_a₎_dA₍_z_),

where

L(fnj,f,φ) =

|(fnj◦φ)(z)| p

–₍_f_n

j◦φ)(z)

 –|(fnj◦φ)(z)|p

–|(f◦φ)(z)|

p

–₍_f ◦_φ₎₍_z₎  –|(f◦φ)(z)|p

.

Hence,

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On the other hand, by the uniform convergence on compact subsets ofD, we can ﬁnd an

N∈Nsuch that for allj≥N,

L(fnj,f,φ) =

|(fnj◦φ)(z)| p

–_f nj(φ(z))

 –|fnj(φ(z))|p

–|(f◦φ)(z)|

p

–f(_φ(z))  –|f(φ(z))|p

≤ε

for allz∈Dwith|φ(z)| ≤r. Hence, for suchj, we obtain

I(a,r) =sup a∈D

|φ(z)|≤r

(fnj◦φ) *

p(z) – (f◦φ)*p(z)



gs(z,a)dA(z)

≤sup

a∈D

|φ(z)|≤r

L(fnj,f,φ)φ(z) 

gs(z,a)dA(z)

≤ε

sup a∈D

|φ(z)|≤r

|φ(z)|

( –|φ(z)|p₎αg

s₍_z_,_a₎_dA₍_z₎

 

≤Cε,

hence,

I(a,r)≤Cε, ()

whereCis the bound obtained from (iii) of Theorem .. Combining (), (), () and (), we deduce thatfnj→f inQ*(p,s).

As for the converse direction, letfn(z) :=_nα–znfor alln∈N,n≥.

f_B*

p,α =

p

supa∈D

nαp|z| αp

–( –|z|)α  – –p_np(α–)_|_z_|np

≤p–+ sup

a∈Dn αp

|_z| αp

–_{ –}|_z|α_. _()

The functionrnp–_{( –}_r₎α_{attains its maximum at the point}_r_{=  –} α

α+αp–. For simplicity, we see that () has the upper bound

p–+ nα

 – α

α+n– 

n–

α α+n– 

α

≤p–+ .

Then the sequence (fn)∞n=belongs to the ballB¯(, (p–+ ))⊂B*p,α.

Suppose thatCφ maps the closed ballB¯(, (p–+ ))⊂Bp*,α into a compact subset of

Q*(p,s); hence, there exists an unbounded increasing subsequence (nj)∞_j_= such that the image subsequence (Cφfnj)j∞=converges with respect to the norm. Since both (fn)∞n=and

(Cφfnj)∞j= converge to the zero function uniformly on compact subsets ofD, the limit of

the latter sequence must be zero. Hence,

nα_j–φnj

Q*₍_p_,_s₎→, asj→ ∞. ()

Now letrj=  –_n_j. For all numbersa,rj≤a< , we have the estimate

nα_janj–

 –anj ≥



e( –a)α

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Using (), we deduce

nα_j–φnj

Q*₍_p_,_s₎≥

p

supa∈D

|φ|≥rj

nαj(φ(z))nj–|φnj(z)|

p

–_φ₍_z₎  –|φnj₍_z₎|p

gs(z,a)dA(z)

≥ Cp

esup

a∈D

|φ|≥rj

|φ(z)|

( –|φ(z)|p₎αg

s₍_z_,_a₎_dA₍_z_). _()

From () and (), the condition (ii) follows. This completes the proof.

For  <p<∞and ≤s<∞, we deﬁne the weighted Dirichlet-classD(p,s) consists of those functionsf ∈H(D) for which

D

_f₍_z₎p–

f(z) –|z|sdA(z) <∞.

For  <p<∞ and ≤s<∞, the generalized hyperbolic weighted Dirichlet-class

D*₍_p_,_s_{) consists of those functions}_f _∈_B₍_D_{) for which}

D

f_p*(z) –|z|sdA(z) <∞.

The proof of Proposition . implies the following corollary:

Corollary . For f,g∈D*(p,s). Then,D*(p,s)is a complete metric space with respect to the metric deﬁned by

df,g;D*(p,s):=d_D*₍_p_,_s₎(f,g) +f–g_D₍_p_,_s₎+f() –g()

p

_,

where

d_D*₍_p_,_s₎(f,g) :=

p

supz∈D

D

f(z)|f(z)| p

–

 –|f(z)|p –

g(z)|g(z)|p–  –|g(z)|p

 –|z|sdA(z)

  .

Moreover, the proofs of Theorems . and . yield the following result:

Theorem . Let <p<∞,– <s≤, and <α≤. Assume thatφ is a holomorphic mapping fromDinto itself. Then the following statements are equivalent:

(i) Cφ:Bp*,α→D*(p,s)is Lipschitz continuous;

(ii) Cφ:B_p*_,α→D*(p,s)is compact;

(iii)

D

|φ(z)|

( –|φ(z)|p₎α

 –|z|sdA(z) <∞.

(12)

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doi:10.1186/1029-242X-2012-185