# Compact Weighted Composition Operators and Multiplication Operators

Authors: Sei-Ichiro Ueki; Luo Luo
Journal: Abstract and Applied Analysis
 (2008)

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## Abstract

We estimate the essential norm of a compact weighted composition operatoruCφ acting between different Hardy spaces of the unit ball in ℂN. Also we will discuss a compact multiplication operator between Hardy spaces.


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## Body

## 1. Introduction

LetN be a fixed
integer.
Let BN denote the unit
ball of ℂN and
let H(BN) denote the
space of all holomorphic functions in BN. For each p,1≤p<∞, the Hardy spaceHp(BN) is defined
by(1.1)Hp(BN)={f∈H(BN):sup⁡0<r<1∫∂BN|f(rζ)|pdσ(ζ)<∞},∥f∥p=[∫∂BN|f∗(ζ)|pdσ(ζ)]1/p,where dσ is  the normalized Lebesgue measure on the boundary ∂BN of BN.For a given holomorphic self-mapφ of BN and holomorphic
function u in BN, the weighted composition operatoruCφ is defined by uCφf=u(f◦φ). In particular, if u is the constant
function 1, then uCφ becomes the composition
operatorCφ. In the special case that φ is the identity
mapping of BN, uCφ is called the
multiplication operator and is denoted by Mu.LetX and Y be Banach
spaces. For a bounded linear operator T:X→Y, the essential norm∥T∥e,X→Y is defined to
be the distance from T to the set of
the compact operators 𝒦, namely,(1.2)∥T∥e,X→Y=inf⁡{∥T−𝒦∥:𝒦iscompactfromXintoY},  where ∥⋅∥ denotes the
usual operator norm. Clearly, T is compact if
and only if ∥T∥e,X→Y=0. Thus, the essential norm is closely related to the
compactness problem of concrete operators. Many mathematicians have studied the
essential norm of various concrete operators. For these studies about
composition operators on Hardy spaces of the unit disc, refer to [1–4]. In this paper, our objects are weighted composition
operators between Hardy spaces of the unit ball BN. Several authors have also studied weighted
composition operators on various analytic function spaces. For more information
about weighted composition operators, refer to [5–10].Recently, Contreras and Hernández-Díaz [11, 12] have characterized the
compactness of uCφ from Hp(B1) into Hq(B1)(1<p≤q<∞) in terms of the
pull-back measure. Here, B1 denotes the
open unit disc in the complex plane. But they have not given the estimate for
the essential norm of uCφ. The essential norm of uCφ:Hp(B1)→Hq(B1) has been
studied by C⌣uc⌣ković and Zhao
[13, 14]. In the
higher-dimensional case,  Ueki [15] characterized the
boundedness and compactness of uCφ:Hp(BN)→Hq(BN), in terms of the pull-back measure and the integral
operator. The purpose of this paper is to estimate the essential norm of uCφ:Hp(BN)→Hq(BN). The following theorem is our main result.Main 1  Theorem.
Let1<p≤q<∞. IfuCφis a bounded
weighted composition operator fromHp(BN)intoHq(BN), then(1.3)∥uCφ∥e,Hp→Hqq∼limsup⁡|w|→1−∫∂BN|u∗(ζ)|q{1−|w|2|1−〈φ∗(ζ),w〉|2}qN/pdσ(ζ)∼limsup⁡t→0+sup⁡ζ∈∂BNμφ,u(S(ζ,t))tqN/p,where μφ,u is the
pull-back measure induced by φ and u, S(ζ,t) is the Carleson
set of BN¯, and the notation ∼ means that the
ratios of two terms are bounded below and above by constants dependent upon the
dimension N and other
parameters.The one variable case of the first estimate for∥uCφ∥e in above
theorem may be found in the work [14] by C⌣uc⌣ković   and Zhao. In
the case p=q=2 and u=1, Choe [1] and Luo [16] showed that the essential norm ∥Cφ∥e is comparable
to the value lim⁢sup⁡t→0+sup⁡ζ∈∂BN(μφ(S(ζ,t))/tN).We give the proof of main theorem in Section3. The
ideas of our proofs are based on the method which Choe or Luo used in their
papers. In Section 4, we have a discussion on the compact multiplication
operator between different Hardy spaces.Throughout the paper, the symbolC denotes a
positive constant, possibly different at each occurrence, but always
independent of the function f and other
parameters r or t.


## 2. Carleson-type Measures

For eachu∈Hq(BN), we can define a finite positive Borel measure μφ,u on BN¯ by(2.1)μφ,u(E)=∫φ∗−1(E)|u∗|qdσ(∀BorelsetsEofBN¯),where φ∗ denotes the radial
limit map of the mapping φ considered as a
map of ∂BN→BN¯. A change-of-variable
formula from measure theory shows that(2.2)∫BN¯gdμφ,u=∫∂BN|u∗|q(g◦φ∗)dσ,for each nonnegative measurable
function g on BN¯. This type of pull-back measure played an important
role in past studies of composition operators on Hardy spaces of BN.For eachζ∈∂BN and t>0, let(2.3)S(ζ,t)={z∈BN¯:|1−〈z,ζ〉|<t},B(ζ,t)=S(ζ,t)∩BN,Q(ζ,t)=S(ζ,t)∩∂BN.It is well known that σ(Q(ζ,t)) is comparable
to tN ([17, page 67]).The proof of the following lemma is essentially the
same as that of Power's theorem in [18].Lemma 2.1.
Let1≤α<∞. Suppose that μ is a positive
Borel measure on BN and that there
exists a constant C>0 such
that(2.4)μ(B(ζ,t))≤CtαN(ζ∈∂BN,⁢t>0).Then there exists a constant K>0 such
that(2.5)[∫BN|f|pαdμ]1/pα≤K∥f∥Hp(f∈Hp(BN)).Proof.
Fixf∈Hp(BN) and s>0. By the same argument as in the proof of theorem in
[18, pages 14-15], it
follows from (2.4) that there exists a constant C>0 such
that(2.6)μ({z∈BN:|f(z)|≥s})≤C[σ({ζ∈∂BN:Mf(ζ)≥s})]α,where Mf is the
admissible maximal function of f which is
defined by(2.7)Mf(ζ)=sup⁡{|f(z)|:z∈ℂn,|1−〈z,ζ〉|<1−|z|2},for ζ∈∂BN. By (2.6), we have(2.8)∫BN|f|pαdμ=pα∫0∞μ{|f|>s}spα−1ds≤Cpα∫0∞σ{Mf≥s}αspα−1ds.Since f∈Hp(BN), it follows from [17, Theorem 5.6.5] that(2.9)σ{Mf≥s}α−1spα−p≤[∫∂BN{Mf(ζ)}pdσ(ζ)]α−1≤C∥f∥Hpp(α−1).By (2.8) and (2.9), we
have(2.10)∫BN|f|pαdμ≤C∥f∥Hpp(α−1)p∫0∞σ{Mf≥s}sp−1ds≤C∥f∥Hpp(α−1)∫∂BN{Mf(ζ)}pdσ(ζ)≤C∥f∥Hppα.This completes the proof.Lemma 2.2.
Let1≤α<∞. Suppose that μ is a positive
Borel measure on ∂BN such
that(2.11)μ(Q(ζ,t))≤CtαN(ζ∈∂BN,⁢t>0),for some constant C>0.
(a)
Ifα=1, then there exist a g∈L∞(∂BN) and a constant C'>0 (C' is the product of C and a constant
depending only on the dimension N) such that dμ=gdσ and ∥g∥L∞≤C'.(b)
Ifα>1, then μ≡0 for all Borel
sets of ∂BN.Proof.
Part (a) is completely analogous to
[19, page 238, Lemma 1.3]. So we only prove part (b). Combining σ(Q(ζ,t))∼tN with (2.11), we
have(2.12)μ(Q(ζ,t))σ(Q(ζ,t))≤CtN(α−1)for all ζ∈∂BN and t>0. Hence we see that the maximal functionMμ of the positive
measure μ satisfies Mμ(ζ)<∞ for all ζ∈∂BN. By [17, page 70, Theorem 5.2.7], we obtain dμ=gdσ for some g∈L1(∂BN). By (2.12), we have(2.13)0≤1σ(Q(ζ,t))∫Q(ζ,t)gdσ=μ(Q(ζ,t))σ(Q(ζ,t))≤CtN(α−1)for all ζ∈∂BN and t>0. Letting t→0+, we see that g=0 a.e. on ∂BN, and so μ≡0. This completes the proof of part (b).Combining Lemma2.1 with
Lemma 2.2 and using the same
argument as in [19, page 239], we obtain the following lemma.Lemma 2.3.
Let1<p≤q<∞. Suppose that μ is a positive
Borel measure on BN¯ such
that(2.14)μ(S(ζ,t))≤CtqN/p(ζ∈∂BN,⁢t>0),for some constant C>0. Then, there exists a constant K>0 such that(2.15)[∫BN¯|f∗|qdμ]1/q≤K∥f∥Hp,for all f∈Hp(BN). Here, the notation f∗ denotes the
function defined on BN¯ by f∗=f in BN and f∗=lim⁡r→1−fr a.e. [σ] on ∂BN.Remark 2.4.
In Lemma2.3 (or in Lemma 2.1), we
see that the constant K of (2.15) (or
(2.5)) can be chosen to be the product of C and a positive
constant depending only on p,q, and the dimension N.In order to prove the main theorem, we need some
results. These are the extensions of [19, Corollary 1.4 and Lemma 1.6] to the case of weighted
composition operators uCφ.Proposition 2.5.
Let1<p≤q<∞. Suppose that φ:BN→BN is a
holomorphic map and u∈Hq(BN)∖{0} such that uCφ:Hp(BN)→Hq(BN) is bounded.
Then φ∗ cannot carry a
set of positive σ-measure
in ∂BN into a set of σ-measure 0 in ∂BN.Proof.
Suppose
thatE,F⊂∂BN and φ∗(E)⊂F with σ(E)>0 and σ(F)=0. Put μ=μφ,u|∂BN. As in the case of composition operators, it is well
known that the boundedness of uCφ:Hp(BN)→Hq(BN) implies(2.16)μ(S(ζ,t))≤CtqN/p(ζ∈∂BN,⁢t>0),for some positive constant C (see [15]). By Lemma 2.2, we see that μ≡0 (if p<q) or μ is absolutely
continuous with respect to dσ (if p=q). Thus we
have(2.17)0≥μ(φ∗(E))≡∫φ∗−1(φ∗(E))|u∗|qdσ≥∫E|u∗|qdσ.That is, u∗=0 a.e. on E. Hence [17, page 83, Theorem 5.5.9]
gives that u≡0 in BN. This contradicts u≢0.Lemma 2.6.
Let1<p≤q<∞ and f∈Hp(BN). Suppose that φ:BN→BN is a
holomorphic map and u∈Hq(BN)∖{0} such that uCφ:Hp(BN)→Hq(BN) is bounded.
Then u∗(f◦φ)∗=u∗(f∗◦φ∗) a.e. [σ] on ∂BN. Here the notation f∗ is used as in
Lemma 2.3.Proof.
Sinceφ∗ cannot carry a
set of positive measure in ∂BN into a set of
measure 0 in ∂BN (by Proposition
2.5) and since the radial limit of φ, f and ψ exist on a set
of full measure in ∂BN, we have lim⁡r→1−u∗(fr◦φ∗)=u∗(f∗◦φ∗) a.e. [σ] on ∂BN.
On the other hand, sincefr is in the ball
algebra A(BN) and fr→f as r→1− in Hp(BN), the boundedness of uCφ shows
that(2.18)0≤∫∂BN|u∗(ζ)(f◦φ)∗(ζ)−u∗(ζ)(f∗◦φ∗)(ζ)|qdσ(ζ)≤lim⁢inf⁡r→1−∫∂BN|u∗(ζ)(f◦φ)∗(ζ)−u∗(ζ)(fr◦φ)∗(ζ)|qdσ(ζ)=lim⁢inf⁡r→1−∥uCφf−uCφfr∥Hqq=0.This implies that u∗(f◦φ)∗=u∗(f∗◦φ∗) a.e. [σ] on ∂BN.


## 3. Weighted Composition Operators between Hardy Spaces

Theorem 3.1.
Let1<p≤q<∞. If uCφ is a bounded
weighted composition operator from Hp(BN) into Hq(BN), then(3.1)∥uCφ∥e,Hp→Hqq∼lim⁢sup⁡|w|→1−∫∂BN|u∗(ζ)|q{1−|w|2|1−〈φ∗(ζ),w〉|2}qN/pdσ(ζ)∼lim⁢sup⁡t→0+⁢sup⁡ζ∈∂BNμφ,u(S(ζ,t))tqN/p.Proof  of the lower estimates.
For eachw∈BN, we define the function fw on BN¯ by(3.2)fw(z)={1−|w|2(1−〈z,w〉)2}N/p.Then the functions {fw:w∈BN} belong to the ball algebra A(BN) and form a
bounded sequence of Hp(BN). Take a compact operator 𝒦:Hp(BN)→Hq(BN) arbitrarily.
Since the bounded sequence {fw} converges to 0 uniformly on
compact subsets of BN as |w|→1−, we have ∥𝒦fw∥Hq→0 as |w|→1−. Thus we obtain(3.3)∥uCφ−𝒦∥Hp→Hq≥C⁢lim⁢sup⁡|w|→1−∥(uCφ−𝒦)fw∥Hq≥C⁢lim⁢sup⁡|w|→1−∥uCφfw∥Hq.By the definition of fw, we also see that(3.4)∥uCφfw∥Hqq=∫∂BN|u∗(ζ)|q{1−|w|2|1−〈φ∗(ζ),w〉|2}qN/pdσ(ζ).Combining this with (3.3), we
get(3.5)∥uCφ−𝒦∥Hp→Hqq≥C⁢lim⁢sup⁡|w|→1−∫∂BN|u∗(ζ)|q{1−|w|2|1−〈φ∗(ζ),w〉|2}qN/pdσ(ζ).Since this holds for every
compact operator 𝒦, it follows that(3.6)∥uCφ∥e,Hp→Hqq≥C⁢lim⁢sup⁡|w|→1−∫∂BN|u∗(ζ)|q{1−|w|2|1−〈φ∗(ζ),w〉|2}qN/pdσ(ζ).
Furthermore, we putw=(1−t)ζ for each t,0<t<1 and ζ∈∂BN in the
definition of fw. Since we see that |f(1−t)ζ(z)|≥Ct−qN/p for all z∈S(ζ,t), we have(3.7)Csup⁡ζ∈∂BNμφ,u(S(ζ,t))tqN/p≤sup⁡ζ∈∂BN∫S(ζ,t)|f(1−t)ζ|qdμφ,u≤sup⁡ζ∈∂BN∥uCφf(1−t)ζ∥Hqq.Letting t→0+, we get(3.8)C⁢lim⁢sup⁡t→0+⁢sup⁡ζ∈∂BNμφ,u(S(ζ,t))tqN/p≤lim⁢sup⁡t→0+⁢sup⁡ζ∈∂BN∥uCφf(1−t)ζ∥Hqq.Combining this with (3.6), we
obtain(3.9)C⁢lim⁢sup⁡t→0+⁢sup⁡ζ∈∂BNμφ,u(S(ζ,t))tqN/p≤∥uCφ∥e,Hp→Hqq,completing the proof of the
lower estimates.To prove the upper estimates, we need some technical
results about the polynomial approximation off∈Hp(BN). Recall that a holomorphic function f in BN has the
homogeneous expansion(3.10)f(z)=∑k=0∞∑|γ|=kc(γ)zγ,where γ=(γ1,…,γN) is a
multi-index, |γ|=γ1+⋯+γN, and zγ=z1γ1⋯zNγN. For the homogeneous expansion
of f and any integer n≥1, let(3.11)Rnf(z)=∑k=n∞∑|γ|=kc(γ)zγ,and Kn=I−Rn, where If=f is the identity
operator.Proposition 3.2.
Suppose thatX is a Banach
space of holomorphic functions in BN with the
property that the polynomials are dense in X. Then ∥Knf−f∥X→0 as n→∞ if and only if sup⁡{∥Kn∥:n≥1}<∞.Proof.
We see
that [20, Proposition 1] also holds if we replace the unit disc with the unit ball. So we omit the
proof of this proposition.Proposition 3.3.
If1<p<∞, then ∥Knf−f∥Hp→0 as n→∞ for each f∈Hp(BN).Proof.
For eachf∈Hp(BN) and r,0<r<1, the slice function (fr)ζ(ζ∈∂BN) of fr is in the disc
algebra A(𝔻). Here, fr denotes the
dilated function of f, that is fr(z)=f(rz). Hence [20, Corollary 3 and Proposition 1] implies that there is
a positive constant Cp such
that(3.12)12π∫−ππ|Kn(fr)ζ(eiθ)|pdθ≤Cp12π∫−ππ|(fr)ζ(eiθ)|pdθ,for every integer n≥1. Since Kn(fr)ζ(eiθ)=Knf(reiθζ), integration by slices (see [17, page 15,
Proposition 1.4.7.]) shows(3.13)∫∂BN|Knf(rζ)|pdσ(ζ)≤Cp∫∂BN|f(rζ)|pdσ(ζ),that is, ∥Kn∥≤Cp1/p for every integer n≥1. By Proposition 3.2, we see ∥Knf−f∥Hp→0 as n→∞. This completes the proof of the proposition.Corollary 3.4.
If1<p<∞, then Rn converges to 0 pointwise in Hp(BN) as n→∞. Moreover, sup⁡{∥Rn∥:n≥1}<∞.Proof.
SinceRnf=f−Knf, Proposition 3.3 shows that ∥Rnf∥p→0 as n→∞. Furthermore, the principle of uniform boundedness
implies that sup⁡n≥1∥Rn∥<∞.Lemma 3.5.
Let1<p<∞. For each f∈Hp(BN) and n≥1,(3.14)|Rnf(z)|≤∥f∥Hp∑k=n∞Γ(N+k)k!Γ(N)|z|k.Proof.
LetKw be the
reproducing kernel for H2(BN) and
let C[f] be the
Cauchy-Szegö projection. Then, the orthogonality of monomials ζα implies
that(3.15)Rnf(z)=C[Rnf](z)=∫∂BNRnf(ζ)Kz(ζ)¯dσ(ζ)=∫∂BNf(ζ)RnKz(ζ)¯dσ(ζ).Hölder's inequality and the
expansion of Kz(w) give(3.16)|Rnf(z)|≤∫∂BN|f(ζ)||RnKz(ζ)|dσ(ζ)≤{∫∂BN|f(ζ)|pdσ(ζ)}1/p{∫∂BN|RnKz(ζ)|qdσ(ζ)}1/q≤∥f∥Hp∑k=n∞Γ(N+k)k!Γ(N)|z|k.This completes the proof.The following lemma is well known in the case of
functional Hilbert spaces (cf. [4, 21]). As in the proof of [21,  Lemma 3.16], an elementary argument verifies Lemma 3.6.Lemma 3.6.
Let1<p≤q<∞. If uCφ is bounded from Hp(BN) into Hq(BN), then(3.17)∥uCφ∥e,Hp→Hq≤lim⁢inf⁡n→∞∥uCφRn∥Hp→Hq.Let us prove the upper estimates for the essential norm
ofuCφ.Proof  of the upper estimates.
For
the sake of convenience, we set(3.18)M1=lim⁢sup⁡|w|→1−∫∂BN|u∗(ζ)|q{1−|w|2|1−〈φ∗(ζ),w〉|2}(qN/p)dσ(ζ),(3.19)M2=lim⁢sup⁡t→0+sup⁡ζ∈∂BNμφ,u(S(ζ,t))tqN/p,(3.20)D(ζ,t)={z∈BN¯:|z|>1−t,z|z|∈Q(ζ,t)}.
By the notation
(3.18), for given ε>0, we can choose an R1,⁢0<R1<1 such
that(3.21)∫∂BN|u∗(ζ)|q{1−|w|2|1−〈φ∗(ζ),w〉|2}qN/pdσ(ζ)<M1+ε,for w∈BN with |w|≥R1. For each ζ∈∂BN and t,0<t≤1−R1≡t1, we put w1=(1−t)ζ. Since the function fw1(z)={(1−|w1|2)/(1−〈z,w1〉)2}N/p satisfies |fw1(z)|p>4−Nt−N for all z∈S(ζ,t), the inequality (3.21) implies that(3.22)μφ,u(S(ζ,t))tqN/p<C∫S(ζ,t)|fw1(z)|qdμφ,u(z)<C(M1+ε)for all ζ∈∂BN and all t,0<t≤t1.
By the notation (3.19), we can also choose a t2,0<t2<1, so that(3.23)sup⁡ζ∈∂BNμφ,u(S(ζ,t))tqN/p<M2+εfor all t,0<t≤t2. Let μ1 and μ2 be the
restrictions of μφ,u to BN¯∖(1−t1)BN¯ and BN¯∖(1−t2)BN¯, respectively. We claim that μj(j=1,2) also satisfies
the Carleson measure condition(3.24)μj(S(ζ,t))≤C(Mj+ε)tqN/pfor all ζ∈∂BN and t>0. By (3.22) or
(3.23), these conditions are true for all t,0<t≤tj. Hence, we assume that t>tj. For a finite cover {Q(wk,tj/3)}, where wk∈Q(ζ,t) of the set Q¯(ζ,t)={z∈∂BN:|1−〈z,ζ〉|≤t}, the covering property implies that there exists a
disjoint subcollection Γ of {Q(wk,tj/3)} so
that(3.25)Q(ζ,t)⊂⋃ΓQ(wk,tj).Furthermore, we obtain card(Γ)≤C(t/tj)N. By the notation (3.20),
we have(3.26)μj(S(ζ,t))≤μj(D(ζ,t))≤∑Γμj(D(wk,tj))≤∑Γμj(S(wk,2tj))≤C(ttj)N(Mj+ε)tjqN/p=C(Mj+ε)tNtj(q/p−1)N≤C(Mj+ε)tqN/p,where the constant C depends only on p,q, and the dimension N.
Now, we take a functionf∈Hp(BN) with ∥f∥Hp≤1. By Lemma 2.6, we have(3.27)∥uCφRnf∥Hqq=∫∂BN|u∗(Rnf∗◦φ∗)|qdσ=∫BN¯|Rnf∗|qdμφ,u=∫BN¯|Rnf∗|qdμj+∫(1−tj)BN¯|Rnf|qdμφ,ufor all integers n≥1. Condition (3.24) and
Lemma 2.3 implies
that(3.28)∫BN¯|Rnf∗|qdμj≤C(Mj+ε)∥Rnf∥Hpq≤C⁢sup⁡n≥1∥Rn∥q(Mj+ε).On the other hand,
by Lemma 3.5,
we have(3.29)∫(1−tj)BN¯|Rnf|qdμφ,u≤∥f∥Hpq{∑k=n∞Γ(N+k)k!Γ(N)|1−tj|k}q∥u∥Hqq.The boundedness of uCφ implies that u∈Hq(BN) and the
convergence of the series ∑(Γ(N+k)/k!Γ(N))|1−tj|k implies
that(3.30)∑k=n∞Γ(N+k)k!Γ(N)|1−tj|k⟶0asn⟶∞.So we obtain(3.31)∫(1−tj)BN¯|Rnf|qdμu,φ⟶0asn⟶∞.Combining
(3.27), (3.28), and
(3.31) with Lemma 3.6, we have(3.32)∥uCφ∥e,Hp→Hqq≤lim⁢inf⁡n→∞∥uCφRn∥Hp→Hqq≤C⁢sup⁡n≥1∥Rn∥q(Mj+ε).Since Corollary 3.4 implies that sup⁡n≥1∥Rn∥<∞, and ε>0 was arbitrary,
we conclude that(3.33)∥uCφ∥e,Hp→Hqq≤{C⁢lim⁢sup⁡|w|→1−∫∂BN|u∗(ζ)|q{1−|w|2|1−〈φ∗(ζ),w〉|2}qN/pdσ(ζ),Clim⁢sup⁡t→0+⁢sup⁡ζ∈∂BNμφ,u(S(ζ,t))tqN/p,which were to be proved.Corollary 3.7 (see [15]).
Suppose that1<p≤q<∞. For the bounded weighted composition operator uCφ:Hp(BN)→Hq(BN), the following conditions are equivalent:
(a)
uCφ:Hp(BN)→Hq(BN) is compact;(b)
u and φ satisfy(3.34)lim⁡|w|→1−∫∂BN|u∗(ζ)|q{1−|w|2|1−〈φ∗(ζ),w〉|2}qN/pdσ(ζ)=0;(c)
u and φ satisfy(3.35)lim⁡t→0+sup⁡ζ∈∂BNμφ,u(S(ζ,t))tqN/p=0.


## 4. Multiplication Operators between Hardy Spaces

In this section, we consider the compact
multiplication operatorMu between Hardy
spaces. As a consequence of Theorem 3.1, we obtain the following
results.Corollary 4.1.
Suppose that1<p≤q<∞. For the bounded multiplication operator Mu:Hp(BN)→Hq(BN), the following inequality holds:(4.1)∥Mu∥e,Hp→Hqq∼lim⁢sup⁡|w|→1−∫∂BN|u∗(ζ)|q{1−|w|2|1−〈ζ,w〉|2}qN/pdσ(ζ).Furthermore, Mu:Hp(BN)→Hq(BN) is compact if
and only if(4.2)lim⁡|w|→1−∫∂BN|u∗(ζ)|q{1−|w|2|1−〈ζ,w〉|2}qN/pdσ(ζ)=0.By using Corollary4.1, we can completely characterize
the compactness of a multiplication operator Mu from Hp(BN) into Hq(BN).Theorem 4.2.
Suppose that1<p≤q<∞. Then Mu:Hp(BN)→Hq(BN) is compact if
and only if u=0 in BN.Proof.
Ifu≡0, then Mu is compact.
Thus, we only prove that the compactness of Mu implies u≡0. The boundedness of Mu implies that u∈Hq(BN). Hence, the Poisson representation for u gives
that(4.3)u(w)=∫∂BNu∗(ζ)P(w,ζ)dσ(ζ)(w∈BN),where P(w,ζ) is the Poisson
kernel. Hölder's inequality shows that(4.4)|u(w)|≤∫∂BN|u∗(ζ)|P(w,ζ)dσ(ζ)≤{∫∂BN|u∗(ζ)|qP(w,ζ)q/pdσ(ζ)}1/q{∫∂BNP(w,ζ)(1−1/p)q'dσ(ζ)}1/q',where 1/q+1/q'=1. By the assumption 1<p≤q<∞, we see that(4.5)s≡(1−1p)q'=q(p−1)p(q−1)≤pq−pp(q−1)=1,and so we have(4.6)∫∂BNP(w,ζ)(1−1/p)q'dσ(ζ)≤{∫∂BN{P(w,ζ)s}1/sdσ(ζ)}s=1.Inequality (4.4) and Corollary 4.1 give that lim⁡|w|→1−|u(w)|=0. Since u∈Hq(BN), this implies that u has a K-limit 0 on a set of
positive σ-measure
in ∂BN. Hence [17, page 83, Theorem 5.5.9] shows that u≡0. This completes the proof.



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*Source: 196498-2008-03-25.xml*