Weighted Composition Operators and

Volume 2009, Article ID 952831,21pages doi:10.1155/2009/952831

Research Article

Weighted Composition Operators and

Integral-Type Operators between Weighted Hardy Spaces on the Unit Ball

Stevo Stevi ´c

and Sei-Ichiro Ueki

1Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11000 Beograd, Serbia

2Faculty of Engineering, Ibaraki University, Hitachi 316-8511, Japan

Correspondence should be addressed to Stevo Stevi´c,sstevic@ptt.rs Received 2 May 2009; Revised 3 June 2009; Accepted 8 June 2009 Recommended by Leonid Berezansky

We study the boundedness and compactness of the weighted composition operators as well as integral-type operators between weighted Hardy spaces on the unit ball.

Copyrightq2009 S. Stevi´c and S.-I. Ueki. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Let B denote the open unit ball of the n-dimensional complex vector space Cⁿ, ∂B its boundary, and letHBdenote the space of all holomorphic functions onB. For 0< p < ∞ andα≥0 we define the weighted Hardy spaceH_α^pBas follows:

H_α^pB

f∈HB : sup

0<r<1

1−r^α

∂B|frζ|^p dσζ<∞

, 1.1

where dσ is the normalized Lebesgue measure on∂B see, also 1, as well as 2, for an equivalent definition of the space. Note that forα 0 the weighted Hardy space becomes the Hardy spaceH^pB. We define the norm · _Hα^pon this space as follows:

f^p

H^p_α sup

0<r<1

1−r^α

∂B

frζ^pdσζ. 1.2

With this norm H_α^pB is a Banach space when 1 ≤ p < ∞. For a related space on the unit polydisk; see3. In this paper, we investigate two types of operators acting between weighted Hardy spaces.

Letϕbe a holomorphic self-map ofBandu∈HB. Thenϕanduinduce a weighted composition operator uCϕonHBwhich is defined byuCϕfuf◦ϕ. This type of operators has been studied on various spaces of holomorphic functions inCⁿ, by many authors; see, for example,4, recent papers5–17, and the references therein.

Letg∈HDandϕbe a holomorphic self-map of the open unit diskDin the complex plane. Products of integral and composition operators onHDwere introduced by S. Li and S. Stevi´c in a private communicationsee18–21, as well as papers22and23for closely related operatorsas follows:

C_ϕJ_gfz _ϕz

0

fζgζdζ, 1.3

J_gC_ϕfz _z

0

f ϕζ

gζdζ. 1.4

In24the first author of this paper has extended the operator in1.4in the unit ball settings as followssee also25,26. Assumeg ∈ HB,g0 0,andϕis a holomorphic self-map ofB, then we define an operator on the unit ball as follows:

P_ϕ^g f

z ₁

0

f ϕtz

gtzdt

t , f ∈HB, z∈B. 1.5

Ifn 1, theng ∈ HDandg0 0, so thatgz zg₀z,for someg₀ ∈ HD. By the change of variableζtz, it follows that

P_ϕ^gfz ₁

0

f ϕtz

tzg₀tzdt t

_z

0

f ϕζ

g₀ζdζ. 1.6

Thus the operator1.5is a natural extension of operatorJgCϕin1.4. For related operators see27–33as well as the references therein.

In this paper we study the boundedness and compactness of the weighted composition operators as well as the integral-type operatorP_ϕ^g, between different weighted Hardy spaces on the unit ball.

Throughout this paper, constants are denoted byC, they are positive and may differ from one occurrence to the other. The notationa bmeans that there is a positive constantC such thata≤Cb. Moreover, if botha bandb ahold, then one says thatab.

2. Weighted Composition Operators

This section is devoted to studying weighted composition operators between weighted Hardy spaces. Weighted composition operators between different Hardy spaces on the unit ball were previously studied in15,34, while the composition operators on the unit ball were studied in35,36. For the case of the unit disk see also37.

Before we formulate the main results in this section we quote several auxiliary results which will be used in the proofs of these ones.

Lemma 2.1. Let 0< p <∞andα≥0. Suppose thatu∈HBandϕis a holomorphic self-map ofB.

Then for eachf∈HB

uCϕf

H^pα ≤lim inf

R→1⁻ uCϕfR

Hα^p, 2.1

whereuCϕfRz uzfRϕz.

Proof. Fixr∈0,1. Fatou’s lemma shows that

1−r^α

∂B|urζf ϕrζ

|^pdσζ≤1−r^αlim inf

R→1⁻

∂B|urζf

Rϕrζ

|^pdσζ

lim inf

R→1⁻ 1−r^α

∂B

urζf

Rϕrζ^pdσζ

≤lim inf

R→1⁻ uCϕf_RH^p

α.

2.2

Hence we have the desired inequality.

Recall that anf∈HBhas the homogeneous expansion

fz ^∞

k0|^γ|^k c

γ

z^γ, 2.3

where γ γ1, . . . , γn is a multi-index, |γ| γ1 · · · γn and z^γ z1γ1· · ·znγn. For the homogeneous expansion offand any integerj ≥1, let

R_jfz ^∞

k0|^γ|^k c

γ

z^γ, 2.4

andK_j I−R_jwhereIf f is the identity operator. Note thatK_jis compact operator on H_α^pBfor eachj ∈N.

Lemma 2.2. If 1< p <∞, thenR_jconverges to 0 pointwise in the Hardy spaceH^pBasj → ∞.

Proof. See34, Corollary 3.4.

Lemma 2.2and the uniform boundedness principle show that{Rj}is an uniformly bounded sequence inH^pB.

The following lemma is proved similar to4, Lemma 3.16. We omit its proof.

Lemma 2.3. IfuC_ϕis bounded fromH_α^pBintoH_β^qB, then

uCϕ_e,H^p

αB→H^q_βB≤lim inf

j→ ∞ uCϕRj_H^p

αB→H_β^qB, 2.5

where · _e,H^p_αB→H^q

βB and · _H^p_αB→H^q

βB denote the essential norm and the operator norm, respectively.

Lemma 2.4. Let 0< p≤q <∞. Suppose thatμis a positive Borel measure onBwhich satisfies μBζ, t≤C1t^qn/p ζ∈∂B, t >0, 2.6

for some positive constantC₁. Then there exists a positive constantC₂ which depends only onp, q, and the dimensionnsuch that

B|f|^qdμ≤C1C2f^q_Hp, 2.7 for anyf ∈H^pB. HereBζ, t {z∈B:|1− z, ζ|< t}.

Proof. See38, page 13, Theoremor34, Lemma 2.1.

Let 0< q < ∞. For eachr ∈0,1, a holomorphic self-mapϕofBandu∈HB, we define a positive Borel measureμ^r_u,ϕonBby

μ^r_u,ϕE

^ϕr−1E|ur|^qdσ, 2.8

for all Borel setsEof B. By the change of variables formula from measure theory, we can verify

Bg dμ^r_u,ϕ

∂B|urζ|^q g◦ϕ

rζdσζ, 2.9

for each nonnegative measurable functionginB.

Theorem 2.5. Let 0 < p ≤ q < ∞andα, β ≥ 0. Suppose thatu ∈ HBandϕis a holomorphic self-map ofB. ThenuCϕ :Hα^pB → H_β^qBis bounded if and only if

sup

w∈Bsup

0<r<1

1−r^β

∂B|urζ|^q

1− |w|² 1−

ϕrζ, w²

_qαn/p

dσζ<∞. 2.10

Proof. Forw∈Bwe put

fwz

1− |w|² 1− z, w²

_αn/p

. 2.11

Then we see that f_w ∈ H_α^pB and moreover sup_w∈Bfw_H^p

α ≤ C. By a straightforward calculation, we have

uCϕf_w^q_Hq β

sup

0<r<1

1−r^β

∂B|urζ|^q

1− |w|² 1−

ϕrζ, w²

_qαn/p

dσζ, 2.12

for allw∈B. Hence ifuC_ϕ :H_α^pB → H_β^qBis bounded, thenuandϕsatisfy the condition

sup

w∈Bsup

0<r<1

1−r^β

∂B|urζ|^q

1− |w|² 1−

ϕrζ, w²

_qαn/p

dσζ≤CuCϕ^q_Hp

αB→H^q_βB<∞.

2.13 Next we assume

M:sup

w∈Bsup

0<r<1

1−r^β

∂B|urζ|^q

1− |w|² 1−

ϕrζ, w²

_qαn/p

dσζ<∞. 2.14

Fixr ∈0,1andR ∈0,1, respectively. Forζ ∈∂Bandt,0 < t≤ tR 1−R, we put w 1−tζandw_R 1−t_Rζ. Since the functionf_wz, which is defined by2.11for this w, satisfies

|fwz|^q >4^−qαn/p t^−qn/p1−R^−qα/p 2.15 for allz∈Bζ, t, we have

μ^r_u,ϕBζ, t

t^qn/p ≤4^qαn/p1−R^qα/p

Bζ,t|fwz|^qdμ^r_u,ϕz

≤4^qαn/p1−R^qα/p

∂B|urζ|^q

1− |w|² 1−

ϕrζ, w²

_qαn/p

dσζ

≤4^qαn/p1−R^qα/p 1−r^β M.

2.16

By the same argument, the functionfwRzgives the following estimate:

μ^r_u,ϕBζ,2tR≤9^qαn/p1−R^qα/p

1−r^β MtRqn/p. 2.17

Now we need to prove that there exists a positive constantCsuch that

μ^r_u,ϕBζ, t≤C1−R^qα/p

1−r^β Mt^qn/p, 2.18 for allζ∈∂Bandt >0. By the estimate2.16, we see that the inequality2.18is true for all t ∈0, tR. Thus we assumet > tR. By the same argument as in36, pages 241-242, proof of Theorem 1.1, we see that the inequality2.17shows that there exists a positive constantC_n which depends only on the dimensionnsuch that

μ^r_u,ϕBζ, t≤Cn

t t_R

n

9^qαn/p1−R^qα/p

1−r^β MtRqn/p

Cn9^qαn/p1−R^qα/p

1−r^β MtⁿtRq/p−1n

≤Cn9^qαn/p1−R^qα/p

1−r^β Mt^qn/p.

2.19

Hence forCmax{4^qαn/p, C_n9^qαn/p}, we have the inequality in2.18.

Forf ∈ H_α^pBthe dilate functionf_R belongs to the ball algebra, and sof_R is in the Hardy spaceH^pB. HenceLemma 2.4gives

B|fRz|^qdμ^r_u,ϕz≤CC1−R^qα/p

1−r^β MfR^q_Hp, 2.20 for some positive constantCand allR∈0,1. This implies that

1−r^β

∂B

uCϕfRrζ^qdσζ≤CCM

1−R^α

∂B

fRζ^pdσζ q/p

, 2.21

and so we have

uCϕfR^q_Hq β

≤CCMf^q_Hp

α, 2.22

for allR∈0,1. ByLemma 2.1we have uCϕf^q_Hq

β

≤CCf^q_Hp α

×sup

w∈Bsup

0<r<1

1−r^β

∂B|urζ|^q

1− |w|² 1−

ϕrζ, w²

_qαn/p

dσζ. 2.23

This completes the proof.

The following proposition is proved in a standard way; see, for example, the proofs of the corresponding results in4,32,33,39. Hence we omit its proof.

Proposition 2.6. Let 0< p, q <∞andα, β≥0. Suppose thatu∈HBandϕis a holomorphic self- map ofBwhich induce the bounded operatoruCϕ:H_α^pB → H_β^qB. ThenuCϕ :H_α^pB → H_β^qB is compact if and only if for every bounded sequence{fj}_j∈NinH_α^pBwhich converges to 0 uniformly on compact subsets ofB,{uCϕfj}_j∈Nconverges to 0 inH_β^qB.

In the proof ofTheorem 2.8, we need the following lemma.

Lemma 2.7. Let 1< p < ∞,α≥ 0, andf_wbe the family of test functions defined in2.11. Then f_w → 0 weakly inH_α^pBas|w| → 1⁻.

Proof. The family{fw}_w∈Bis bounded inH_α^pBandfw → 0 uniformly on compact subsets ofBas|w| → 1⁻. By the definitions of the spaceH_α^pB and the norm · _Hα^p, we see that H_α^pBis a subspace of the weighted Bergman spaceA^p_αBand

f_A^p_α ≤C α, p, n

f_Hα^p

f∈H_α^pB

, 2.24

for some positive constantCα, p, nwhich depends onα, p,andn. This inequality implies that the family{fw}_w∈B is also bounded inA^p_αB. Note also that the family converges to 0 uniformly on compact subsets ofBas|w| → 1⁻. Hencef_w → 0 weakly inA^p_αBas|w| → 1⁻. In order to prove thatf_w → 0 weakly inH_α^pBas|w| → 1⁻, we take an arbitrary bounded linear functionalΛonH_α^pB. By the Hahn-Banach theorem,Λcan be extended to a bounded linear functionalΛ onA^p_αBso thatΛf w Λfwfor allw∈B. Sincef_w → 0 weakly inA^p_αBas|w| → 1⁻, we haveΛfw Λf w → 0 as|w| → 1⁻, and sof_w → 0 weakly inH_α^pBas|w| → 1⁻.

Theorem 2.8. Let 1 < p ≤ q < ∞andα, β ≥ 0. Suppose thatu ∈ HBandϕis a holomorphic self-map ofBsuch thatuC_ϕ:H_α^pB → H_β^qBis bounded. Then theqth power of the essential norm uCϕ_e,H^p

αB→H_β^qBis comparable to

lim sup

|w| →1⁻

sup

0<r<1

1−r^β

∂B|urζ|^q

1− |w|² 1−

ϕrζ, w²

_qαn/p

dσζ. 2.25

HenceuC_ϕ:H_α^pB → H_β^qBis compact if and only if

|w| →lim1⁻sup

0<r<1

1−r^β

∂B|urζ|^q

1− |w|² 1−

ϕrζ, w²

_qαn/p

dσζ 0. 2.26

Proof. To prove a lower estimate

uCϕ^q_e,Hp

αB→H^q_βB≥lim sup

|w| →1⁻

sup

0<r<1

1−r^β

∂B|urζ|^q

1− |w|² 1−

ϕrζ, w²

_qαn/p

dσζ, 2.27

we consider the test functionsfwdefined in2.11. The family{fw}_w∈Bis bounded inH_α^pB, say byL, andfw → 0 uniformly on compact subsets ofBas|w| → 1⁻. Thus byLemma 2.7 we have thatf_w → 0 weakly inH_α^pBas|w| → 1⁻, so thatKfw_H^q

β → 0 as|w| → 1⁻ for every compact operatorK:H_α^pB → H_β^qB. Hence

LuCϕ− K_H^p_αB→H^q

βB≥lim sup

|w| →1⁻

uCϕ− K fw_H^q

β

≥lim sup

|w| →1⁻

uCϕfw_H^q

β. 2.28

This inequality and2.12give the lower estimate foruCϕ^q_e,Hp

αB→H_β^qB. Next we prove an upper estimate. Takef ∈H_α^pBwithf_H^p

α ≤1. Fixε >0 and put

M₁:lim sup

|w| →1⁻

sup

0<r<1

1−r^β

∂B|urζ|^q

1− |w|² 1−

ϕrζ, w²

_qαn/p

dσζ. 2.29

Then we can chooseR0∈0,1such that

sup

0<r<1

1−r^β

∂B|urζ|^q

1− |w|² 1−

ϕrζ, w²

_qαn/p

dσζ< M1ε, 2.30

forw ∈Bwith|w|> R₀. Fixr ∈0,1andR ∈R0,1. By the same argument as in the proof of inequality2.20inTheorem 2.5, we obtain that

B

R_jf

Rz^qdμ^r_u,ϕz≤C1−R^qα/p

1−r^β M1εRjf_R^q_Hp, 2.31

where the positive constantCis independent ofr,Rand a positive integerj. SincefRis in the ball algebra,Lemma 2.2gives

Rjf_R^q_Hp Rj

f_R

^q_Hp ≤sup

j≥1Rj^q_HpB→H^pBfR^q_Hp. 2.32

Combining this with2.31, we have

1−r^β

∂B|uCϕ

RjfR

rζ|^qdσζ≤CM1ε

1−R^α

∂B

fRζ^pdσζ _q/p

≤CM1εf^q_Hp α,

2.33

and so we have

uCϕ

RjfR

^q_Hq β

≤CM1εf^q_Hp

α. 2.34

LettingR → 1⁻, byLemma 2.1, we obtain uCϕRjf^q_Hq β

≤CM1εf^q_Hp

α. 2.35

Sinceε >0 is arbitrary, this estimate andLemma 2.3imply uCϕ^q_e,Hp

αB→H_β^qB≤lim inf

j→ ∞ uCϕRj^q_Hp

αB→H_β^qB

≤Clim sup

|w| →1⁻

sup

0<r<1

1−r^β

∂B|urζ|^q

1− |w|² 1−

ϕrζ, w²

_qαn/p

dσζ,

2.36

which completes the proof.

Remark 2.9. In the above proof, we usedLemma 2.2. This lemma required the assumption 1<

p <∞. Hence we cannot have an upper estimate foruCϕ_e,H^p

αB→H_β^qBin the case 0< p≤1.

However,Proposition 2.6shows that the compactness ofuC_ϕ:H_α^pB → H_β^qB 0< p≤q <

∞is equivalent to

|w| →lim1⁻sup

0<r<1

1−r^β

∂B|urζ|^q

1− |w|² 1−

ϕrζ, w²

_qαn/p

dσζ 0. 2.37

3. Integral-Type Operators

Here we study the boundedness and compactness of the integral-type operatorsP_ϕ^gbetween weighted Hardy spaces on the unit ball.

Forf ∈ HBwith the Taylor expansionfz

|γ|≥0aγz^γ, letRfz

|γ|≥0|γ|aγz^γ be the radial derivative off.

The following lemma was proved in24 see also25.

Lemma 3.1. Assume thatϕis a holomorphic self-map ofB,g∈HBandg0 0.Then for every f∈HBone holds

R P_ϕ^g

f

z f ϕz

gz. 3.1

A positive continuous functionωon the interval0,1is called normal40if there is aδ∈0,1andaandb, 0< a < bsuch that

ωr

1−r^a is decreasing onδ,1and lim

r→1

ωr 1−r^a 0, ωr

1−r^b is increasing on δ,1and lim

r→1

ωr 1−r^b ∞.

3.2

If it is said that a functionω:B → 0,∞is normal, it is also assume that it is radial.

Lemma 3.2. Assume that 0 < q ≤ ∞, mis a positive integer and ω is normal. Then for every f∈HB

sup

0<r<1

ωrMq

f, r

f0sup

0<r<1

1−r^mωrMq

R^mf, r

, 3.3

where

Mq

f, r

∂B|frζ|^qdσζ 1/q

, M_∞ f, r

sup

ζ∈∂B

frζ. 3.4

Proof. The proof of the lemma in the case 1≤q≤ ∞can be found in27, Theorem 2. However, due to an overlook, the proof for the caseq∈0,1has a gap. Hence we will give a correct proof here in the case.

We may assume thatf0 0, otherwise we can consider the functionshz fz− f0.Also we may assume thatδ0, to avoid some minor technical difficulties.

By27, Lemma 1, for each fixedq ∈ 0,1, there is a positive constantCdepending only onqand the dimensionnsuch that

M_q f, r

≤ C r

_r

0

r−t^q−1M_q^q Rf, t

dt 1/q

, 3.5

for everyr ∈0,1andf∈HBsuch thatf0 0.

From3.5and the fact thatωis normal, we have

sup

0≤r<1ωrMq

f, r

≤ Csup

0≤r<1ωr1 r

_r

0

r−t^q−1M^q_q Rf, t

dt 1/q

≤ Csup

0≤r<11−r^a1 r

r 0

r−t^q−1 ω^qt 1−t^aqM_q^q

Rf, t dt

1/q

≤ Csup

0≤r<11−r^a1 r

_r

0

r−t^q−1 1−t^aqqdt

_1/q sup

0≤t<11−tωtMq

Rf, t

Csup

0≤r<11−r^a ₁

0

1−u^q−1 1−ur^aqqdu

_1/q sup

0≤t<11−tωtMq

Rf, t .

3.6

By40, page 291, Lemma 6there exists a positive constantCsuch that ₁

0

1−u^q−1

1−ur^aqqdu≤ C

1−r^aq, 3.7

for everyr ∈0,1. Combining this with3.6, we have

sup

0≤r<1ωrMq

f, r

≤ Csup

0≤r<11−r^a 1

1−r^aq 1/q

sup

0≤t<11−tωtMq

Rf, t Csup

0≤t<11−tωtMq

Rf, t .

3.8

The reverse inequality is proved by the following inequality:

1−rMq

Rf, r

≤CMq

f,1r

2

3.9

and the fact thatωr ω1r/2forωnormalsee27. Hence, we obtain the result for the casem1.

Form≥ 2 it should be only noticed that1−r^mωris still normal, thatR^mf0 0 for every integerm≥1, and use the method of induction.

Theorem 3.3. Let 0< p ≤ q <∞andα, β > 0. Suppose thatg ∈HBwithg0 0 andϕis a holomorphic self-map ofB. ThenP_ϕ^g:H_α^pB → H_β^qBis bounded if and only if

sup

w∈Bsup

0<r<1

1−r^βq

∂B

grζ^q

1− |w|² 1−

ϕrζ, w²

_qαn/p

dσζ<∞. 3.10

Proof. Takef ∈H_α^pBwithf_Hα^p≤1. Since the function1−r^β/qforβ >0 and 0< q <∞is normal,Lemma 3.2gives

sup

0<r<1

1−r^β/qM_q P_ϕ^gf, r

P_ϕ^gf0sup

0<r<1

1−r^β/q1M_q R

P_ϕ^gf , r

. 3.11

The assumptiong0 0 impliesP_ϕ^gf0 0,andLemma 3.1showsRPϕ^gf gCϕf. Hence we obtain

sup

0<r<1

1−r^βM^q_q P_ϕ^gf, r

sup

0<r<1

1−r^βqM^q_q

gCϕf, r

, 3.12

and so we obtainPϕ^gf^q_H^q

β gCϕf^q_Hq

βq. This implies that the boundedness ofP_ϕ^g:H_α^pB → H_β^qBis equivalent to the boundedness ofgC_ϕ :H_α^pB → H_βq^q B. SoTheorem 2.5shows that the condition

sup

w∈Bsup

0<r<1

1−r^βq

∂B

grζ^q

1− |w|² 1−

ϕrζ, w²

_qαn/p

dσζ<∞ 3.13

is a necessary and sufficient condition for the boundedness ofP_ϕ^g : H_α^pB → H_β^qB. This completes the proof.

The next proposition is proved similar toProposition 2.6.

Proposition 3.4. Let 0 < p, q < ∞, andα, β > 0. Suppose thatg ∈ HBwithg0 0 andϕ is a holomorphic self-map of Bwhich induce the bounded operator P_ϕ^g : H_α^pB → H_β^qB. Then P_ϕ^g :H_α^pB → H_β^qBis compact if and only if for every bounded sequence{fj}_j∈NinH_α^pBwhich converges to 0 uniformly on compact subsets ofB,{Pϕ^gfj}_j∈Nconverges to 0 inH_β^qB.

Theorem 3.5. Let 0 < p ≤ q < ∞and α, β > 0. Suppose that g ∈ HBwithg0 0 andϕ is a holomorphic self-map of Bwhich induce the bounded operator P_ϕ^g : H_α^pB → H_β^qB. Then P_ϕ^g :H_α^pB → H_β^qBis compact if and only if

|w| →lim1⁻sup

0<r<1

1−r^βq

∂B

grζ^q

1− |w|² 1−

ϕrζ, w²

_qαn/p

dσζ 0. 3.14

Proof. First we assume that condition3.14holds. Take a bounded sequence{fj}_j∈N⊂H_α^pB which converges to 0 uniformly on compact subsets of B.Theorem 2.8and the remark in Section 2show thatgC_ϕ :H_α^pB → H_βq^q Bis compact. ThusProposition 2.6implies that

jlim→ ∞gCϕf_jH^q

βq0. 3.15

From3.15and sincePϕ^gfj^q_H^q

β gCϕfj^q_Hq

βq, we have thatPϕ^gfj^q_H^q

β → 0 asj → ∞. By Proposition 3.4, we see thatP_ϕ^g :H_α^pB → H_β^qBis compact.

To prove the necessity of the condition in 3.14, we consider the family of test functionsf_wwhich is defined by2.11. Hence we have

Pϕ^gf_w^q_H^q

β gCϕf_w^q_Hq βq

sup

0<r<1

1−r^βq

∂B

grζ^q

1− |w|² 1−

ϕrζ, w²

_qαn/p

dσζ,

3.16

for allw ∈ B. Since {fw}_w∈B is a bounded sequence inH_α^pB andf_w → 0 uniformly on compact subsets ofB as|w| → 1⁻, the compactness ofP_ϕ^g and Proposition 3.4show that Pϕ^gfw^q_H^q

β → 0 as |w| → 1⁻. This fact along with3.16 implies the condition in 3.14, finishing the proof of the theorem.

Theorem 3.6. Let 1< p ≤ q <∞andα, β > 0. Suppose thatg ∈HBwithg0 0 andϕis a holomorphic self-map ofBwhich induce the bounded operatorP_ϕ^g :H_α^pB → H_β^qB. Then theqth power of the essential norm ofP_ϕ^gis comparable to

lim sup

|w| →1⁻

sup

0<r<1

1−r^βq

∂B

grζ^q

1− |w|² 1−

ϕrζ, w²

_qαn/p

dσζ. 3.17

Proof. To prove a lower estimate, we take an arbitrary compact operator K : H_α^pB → H_β^qB. SinceLemma 2.7implies that the family of functionsfwdefined by2.11converges to 0 weakly inH_α^pBas|w| → 1⁻, we obtain

CP_ϕ^g− K_H^p

αB→H_β^qB≥lim sup

|w| →1⁻

P_ϕ^gfw_H^q

β− Kfw_H^q

β

≥lim sup

|w| →1⁻

P_ϕ^gfw_H^q

β. 3.18

Combining this with3.16, we have

CPϕ^gq_e,H^p

αB→H_β^qB≥lim sup

|w| →1⁻

sup

0<r<1

1−r^βq

∂B

grζ^q

1− |w|² 1−

ϕrζ, w²

_qαn/p

dσζ, 3.19 which is a lower estimate.

By some modification ofLemma 2.3and the application of Lemmas3.1and3.2, we get Pϕ^gq_e,H^p

αB→H_β^qB≤lim inf

j→ ∞ sup

f_H^p

α≤1Pϕ^gRjf^q_H^q

β

≤Clim inf

j→ ∞ sup

f_H^p

α≤1gCϕRjf^q_Hq βq.

3.20

As in the proof ofTheorem 2.8, we obtain lim inf

j→ ∞ sup

f_H^p

α≤1gCϕRjf^q_Hq

βq≤Clim sup

|w| →1⁻

sup

0<r<1

1−r^βq

×

∂B

grζ^q

1− |w|² 1−

ϕrζ, w²

_qαn/p

dσζ,

3.21

and so we have an upper estimate forPϕ^gq_e,Hp

αB→H_β^qB.

4. The Case P

: H

B → H

B

Whenp∞andα >0, we define the weighted-type spaceH_α^∞Bas follows:

H_α^∞B

f∈HB: sup

0<r<1

1−r^αM_∞ f, r

<∞

. 4.1

It is easy to see thatf ∈ H_α^∞Bif and only if sup_z∈B1− |z|^α|fz| < ∞, so we define the normf_H∞

α onH_α^∞Bby this supremum.

Furthermore we consider the subspaceH_α,0^∞Bdefined by

H_α,0^∞B

f ∈HB: lim

r→1⁻1−r^αM_∞ f, r

0

. 4.2

Theorem 4.1. Letα, β >0. Suppose thatg ∈HBwithg0 0 andϕis a holomorphic self-map ofB. ThenP_ϕ^g:H_α^∞B or H_α,0^∞B → H_β^∞Bis bounded if and only if

sup

z∈B

1− |z|^β1gz

1−ϕz^α <∞. 4.3 In this case, the operator normPϕ^g_H∞

αB or H_α,0^∞B→H_β^∞Bis comparable to the above supremum.

Proof. By the definition of the spaceH_α^∞B,f∈H_α^∞Bsatisfies the growth condition fw≤1− |w|^−αf_H∞

α w∈B, 4.4

so it follows fromLemma 3.1andLemma 3.2that P_ϕ^gf_H∞

β sup

z∈B1− |z|^β1gCϕfz≤ f_H∞

αsup

z∈B

1− |z|^β1gz

1−ϕz^α , 4.5 for everyf ∈H_α^∞B.

Hence we obtain Pϕ^g_H∞

αB^{or H}α,0^∞B^→Hβ^∞B≤Csup

z∈B

1− |z|^β1gz

1− |ϕz|^α . 4.6 Now we prove the reverse inequality. Forw∈B, we put

f_wz 1

1− z, w^α. 4.7 Note thatf_w∈H_α,0^∞Bfor eachw∈Band moreover sup_w∈Bfw_H∞

α ≤1.

Whenϕz/0, we have Pϕ^g_H∞

α,0B→H_β^∞B≥ Pϕ^gf_t^ϕz/|^ϕz|

H^∞_β sup

w∈B1− |w|^β1gCϕf_t^ϕz/|^ϕz|w

≥1− |z|^β1gzf_t^ϕz/|^ϕz| ϕz

1− |z|^β1gz 1−tϕz^α ,

4.8

for allt∈0,1. Lettingt → 1⁻in4.8, we have

Pϕ^g_H∞

α,0B→H_β^∞B≥C1− |z|^β1gz

1−ϕz^α . 4.9

For the constant function 1∈H_α,0^∞Bwe obtain Pϕ^g1_H^q

β sup

w∈B1− |w|^β1gC_ϕ1w≥1− |z|^β1gz. 4.10 Inequality4.10shows that the estimate in4.9also holds whenϕz 0.

Hence, from4.9we obtain

Pϕ^g_H∞

α,0B→H_β^∞B≥Csup

z∈B

1− |z|^β1gz

1−ϕz^α , 4.11

which along with the obvious inequality Pϕ^g_H∞

αB→H_β^∞B≥ Pϕ^g_H∞

α,0B→H^∞_βB 4.12

completes the proof of the theorem.

For the compactness ofP_ϕ^g :H_α^∞B or H_α,0^∞B → H_β^∞B, we can also prove the following proposition which is similar toProposition 2.6.