# The Multiplication Operator fromF(p,q,s) Spaces to nth Weighted-Type Spaces on the Unit Disk

Authors: Yongmin Liu; Yanyan Yu
Journal: Journal of Function Spaces and Applications
 (2012)

---

## Abstract

LetH(𝔻) be the space of analytic functions on 𝔻 and u∈H(𝔻). The boundedness and compactness of the multiplication operator Mu from F(p,q,s),(orF0(p,q,s)) spaces to nth weighted-type spaces on the unit disk are investigated in this paper.


---

## Body

## 1. Introduction

LetH(𝔻) denote the space of all analytic functions in the open unit disc 𝔻 of the finite-complex plane ℂ, ∂𝔻 the boundary of 𝔻, ℕ0 the set of all nonnegative integers and ℕ the set of all positive integers. Let μ(z) be a positive continuous function on 𝔻 (weight) such that μ(z)=μ(|z|) and n∈ℕ0. The nth weighted-type spaces on the unit disk 𝔻, denoted by 𝒲μ(n)(𝔻) which were introduced in [1], consist of all f∈H(𝔻) such that(1.1)bWμ(n)(D)(f)=sup⁡z∈Dμ(z)|f(n)(z)|<∞.
For n=0, the space becomes the weighted-type space Hμ∞(𝔻) (see, e.g., [2–4]), for n=1 the Bloch-type space ℬμ(𝔻) and for n=2 the Zygmund-type space 𝒵μ(𝔻). For μ(z)=1-|z|2, we obtain correspondingly the classical weighted-type space, the Bloch space ℬ(𝔻)=ℬ, and the Zygmund space 𝒵(𝔻)=𝒵. Some information on Zygmund-type spaces on the unit disk and some operators on them, for example, in [5–9] and on the unit ball, can be found, for example, in [10, 11]. From now on, we will assume that n∈ℕ. Set(1.2)‖f‖Wμ(n)(D)=∑j=0n-1|f(j)(0)|+bWμ(n)(D)(f).
With this norm, the nth weighted-type space becomes a Banach space.The littlenth weighted-type space, denoted by 𝒲μ,0(n)(𝔻), is a closed subspace of 𝒲μ(n)(𝔻) consisting of those f for which(1.3)lim⁡|z|→1μ(z)|f(n)(z)|=0.A positive continuous functionϕ on [0,1) is called a normal if there exist positive numbers a,b,0<a<b, and t0∈[0,1), such that(1.4)ϕ(t)(1-t2)adecreasesfort0≤t<1andlim⁡t→1-ϕ(t)(1-t2)a=0,ϕ(t)(1-t2)bincreasesfort0≤t<1andlim⁡t→1-ϕ(t)(1-t2)b=∞
(see [12]).For0<p,s<∞,-2<q<∞, a function f∈H(𝔻) is said to belong to the general function space F(p,q,s)=F(p,q,s)(𝔻) if(1.5)‖f‖F(p,q,s)p=|f(0)|p+sup⁡z∈D∫D|f′(z)|p(1-|z|2)q(1-|φa(z)|2)sdA(z)<∞,
where φa(z)=(a-z)/(1-a¯z),a∈𝔻. An f∈H(𝔻) is said to belong to F0(p,q,s)=F0(p,q,s)(𝔻) if(1.6)lim⁡|a|→1∫D|f′(z)|p(1-|z|2)q(1-|φa(z)|2)sdA(z)=0.
The space F(p,q,s) was introduced by Zhao in [13]. We can get many function spaces if we take some specific parameters of p,q,s; for example (see [13]), F(p,q,s)=ℬ(2+q)/p and F0(p,q,s)=ℬ0(2+q)/p for s>1; F(p,q,s)⊂ℬ(2+q)/p and F0(p,q,s)⊂ℬ0(2+q)/p for 0<s≤1; F(2,0,s)=Qs and F0(2,0,s)=Qs,0; F(2,0,1)=BMOA and F0(2,0,1)=VMOA. Since for q+s≤-1, F(p,q,s) is the space of constant functions, we assume that q+s>-1.The multiplication operatorMu is defined by Muf=uf. It is interesting to provide a function theoretic characterization of when u induces a bounded or compact composition operator on various spaces (see, e.g., [14–20]). Yu and Liu in [21] studied the boundedness and compactness of the operator DMu from mixed-norm spaces to the Bloch-type space. Stević in [22] studied the boundedness and compactness of the product of the differentiation and composition operator from the space of bounded analytic functions, the Bloch space and the little Bloch space to nth weighted-type spaces on the unit disk. Zhang and Xiao in [23] studied the bounded and compact-weighted composition operator uCφ from the F(p,q,s) space to the Bloch-type space in the unit disc. Zhang and Zeng in [24] studied the boundedness and compactness of weighted differentiation composition operators from weighted bergman space to nth weighted-type spaces on the unit disk. Ye in [25] studied the boundedness and compactness of the weighted composition operator uCφ from F(p,q,s) into the logarithmic Bloch space ℬlog on the unit disk. Yang in [26] studied the boundedness and compactness of weighted differentiation composition operators from the F(p,q,s) space to the Bloch-type space. Yang in [27] studied the boundedness and compactness of the composition operator from the F(p,q,s) space to nth weighted-type spaces on the unit disk. Zhou and Chen in [28] studied the weighted composition operator from the F(p,q,s) space to the Bloch-type space in the unit ball. Zhu in [29] studied the weighted composition operator from the F(p,q,s) space to Fμ∞ space in the unit ball. Stević in [30, 31] studied the boundedness and compactness of the integral operators between F(p,q,s) spaces and Bloch-type spaces in the unit ball. This paper focuses on the boundedness and compactness of the operators Mu from F(p,q,s)(orF0(p,q,s)) to nth weighted-type spaces on the unit disk.From now on, we will always assume that0<p, s<∞, -2<q<∞, q+s>-1, μ(z)=μ(|z|) is normal and n∈ℕ. Further, for the sake of simplicity, C will always denote an independent constant, which can be different from one display to another.


## 2. Auxiliary Results

In this section we formulate some auxiliary results which will be used in the proofs of the main results.Lemma 2.1 (see [13, 27]).
Assume thatf∈F(p,q,s).  Then, for each n∈ℕ,  there is a positive constant C, independent of f such that ∥f∥ℬ(2+q)/p≤C∥f∥F(p,q,s) and
(2.1)|f(n)(z)|≤C‖f‖F(p,q,s)(1-|z|2)((2+q-p)/p)+n,z∈D.
Moreover, if f∈F0(p,q,s), then f∈ℬ0(2+q)/p.Lemma 2.2 (see [32, 33]).
Letα>0 and f∈ℬα. Then,
(2.2)|f(z)|≤{C‖f‖Bα,if0<α<1,Clog⁡21-|z|2‖f‖Bα,ifα=1,C(α-1)(1-|z|2)α-1‖f‖Bα,ifα>1.Lemma 2.3 (see [1]).
Assumea>0 and
(2.3)Dn(a)=|11⋯1a(a+1)⋯(a+n-1)a(a+1)(a+1)(a+2)⋯(a+n-1)(a+n)⋮⋮⋱⋮∏j=0n-2(a+j)∏j=0n-2(a+j+1)⋯∏j=0n-2(a+j+n-1)|.
Then, Dn(a)=∏j=0n-1j!.By standard arguments (see, e.g., [34] or Lemma  3 in [35]) the following lemma follows.Lemma 2.4.
Assume thatu∈H(𝔻). Then, Mu:F(p,q,s)(orF0(p,q,s))→𝒲μ(n)(𝔻) is compact if and only if Mu:F(p,q,s)(orF0(p,q,s))→𝒲μ(n)(𝔻) is bounded and for any bounded sequence {fk} in  F(p,q,s)(orF0(p,q,s)) which converges to zero uniformly on compact subsets of 𝔻 as k→∞, one has ∥Mufk∥𝒲μ(n)(𝔻)→0 as k→∞.


## 3. The Boundedness and Compactness ofMu from F(p,q,s) to 𝒲μ(n)(𝔻)

In this section, we characterize the boundedness and compactness ofMu:F(p,q,s)→𝒲μ(n)(𝔻).Theorem 3.1.
Assume thatu∈H(𝔻) and μ is normal.(a)
If2+q>p, then Mu:F(p,q,s)→𝒲μ(n)(𝔻) is bounded if and only if
(3.1)sup⁡z∈Dμ(|z|)|u(z)|(1-|z|2)((2+q-p)/p)+n<∞.(b)
If2+q<p, then Mu:F(p,q,s)→𝒲μ(n)(𝔻) is bounded if and only if (3.1) holds and u∈𝒲μ(n)(𝔻).(c)
If2+q=p,s>1, then Mu:F(p,q,s)→𝒲μ(n)(𝔻) is bounded if and only if (3.1) holds and
(3.2)sup⁡z∈Dμ(|z|)|u(n)(z)|log⁡21-|z|2<∞.Proof.
Let0<p, s<∞,-2<q<∞, q+s>-1. Assume that conditions (3.1) holds. Then for all z∈𝔻(3.3)|u(z)|≤C(1-|z|2)((2+q-p)/p)+nμ(|z|).
Since
(3.4)12π∫02π1|eiθ-z|2dθ=11-|z|2,∀z∈D,
let δz=(1+|z|)/2, then we have |z/δz|=2|z|/(1+|z|)<1, so
(3.5)12π∫02π1|δzeiθ-z|2dθ=1δz2-|z|2.
By the Cauchy integral formula and (3.3), we obtain
(3.6)|u′(z)|=12π|∫02πu(δzeiθ)(δzeiθ-z)2δzeiθdθ|≤C(1-δz2)((2+q-p)/p)+nμ(δz)12π∫02πδz|δzeiθ-z|2dθ=C(1-δz2)((2+q-p)/p)+nμ(δz)δzδz2-|z|2≤C(1-δz2)((2+q-p)/p)+nμ(δz)11-δz.
Note that
(3.7)12(1-|z|)≤1-δz2=(1+δz)(1-δz)≤(1-|z|),
and μ are normal, we have
(3.8)|u′(z)|≤C(1-|z|2)((2+q-p)/p)+n-1μ(|z|),
hence
(3.9)sup⁡z∈Dμ(|z|)|u′(z)|(1-|z|2)((2+q-p)/p)+n-1<∞.
Similarly, for j∈{2,3,…,n}, we have that
(3.10)sup⁡z∈Dμ(|z|)|u(j)(z)|(1-|z|2)((2+q-p)/p)+n-j<∞.
Hence, if 2+q>p, by Lemmas 2.1 and 2.2, the Leibnitz formula, (3.1), (3.9), and (3.10), we have that
(3.11)|μ(|z|)(Muf)(n)(z)|=μ(|z|)|(u(z)f(z))(n)|=μ(|z|)|∑j=0nCnju(j)(z)f(n-j)(z)|≤μ(|z|)|u(n)(z)f(z)|+C∑j=0n-1|Cnjμ(|z|)u(j)(z)|(1-|z|2)((2+q-p)/p)+n-j‖f‖F(p,q,s)≤Cμ(|z|)|u(n)(z)|(1-|z|2)(2+q-p)/p‖f‖B(2+q)/p+C∑j=0n-1|Cnjμ(|z|)u(j)(z)|(1-|z|2)((2+q-p)/p)+n-j‖f‖F(p,q,s)≤C‖f‖F(p,q,s),
for every z∈𝔻 and f∈F(p,q,s), if 2+q<p, using u∈𝒲μ(n)(𝔻), we have that
(3.12)|μ(|z|)(Muf)(n)(z)|≤Cμ(|z|)|u(n)(z)|‖f‖B(2+q)/p+C∑j=0n-1|Cnjμ(|z|)u(j)(z)|(1-|z|2)((2+q-p)/p)+n-j‖f‖F(p,q,s)≤C‖u‖Wμ(n)(D)‖f‖B(2+q)/p+C∑j=0n-1|Cnjμ(|z|)u(j)(z)|(1-|z|2)((2+q-p)/p)+n-j‖f‖F(p,q,s)≤C‖f‖F(p,q,s),
for every z∈𝔻 and f∈F(p,q,s), if 2+q=p, by (3.2), we have that
(3.13)|μ(|z|)(Muf)(n)(z)|≤Cμ(|z|)|u(n)(z)|log⁡21-|z|2‖f‖B+C∑j=0n-1|Cnjμ(|z|)u(j)(z)|(1-|z|2)((2+q-p)/p)+n-j‖f‖F(p,q,s)≤C‖f‖B+C∑j=0n-1|Cnjμ(|z|)u(j)(z)|(1-|z|2)n-j‖f‖F(p,q,s)≤C‖f‖F(p,q,s),
for every z∈𝔻 and f∈F(p,q,s). We also have that
(3.14)|(Muf)(0)|=|u(0)f(0)|≤C‖f‖F(p,q,s),
and for each s∈{1,2,…,n-1},
(3.15)|(Muf)(s)(0)|=|∑j=0sCsju(j)(0)f(s-j)(0)|≤C∑j=0s|Csju(j)(0)|‖f‖F(p,q,s).
Using (3.11), (3.12), (3.13), (3.14), and (3.15) it follows that the operator Mu:F(p,q,s)→𝒲μ(n)(𝔻) is bounded.
On the other hand, suppose thatMu:F(p,q,s)→𝒲μ(n)(𝔻) is bounded, that is, there exists a constant C such that(3.16)‖Muf‖Wμ(n)(D)≤C‖f‖F(p,q,s)
for all f∈F(p,q,s). Then, we can easily obtain u∈𝒲μ(n)(𝔻) by taking f(z)=1. For a fixed w∈𝔻, and constants c1,c2,…,cn, set
(3.17)fw(z)=∑j=1ncj(1-|w|2)j+1(1-w¯z)α+j,
where α=(2+q)/p. A straightforward calculation shows that, for l∈{1,…,n},
(3.18)fw(l)(z)=∑j=1ncj∏k=0l-1(α+j+k)(w¯)l(1-|w|2)j+1(1-w¯z)α+j+l,
so
(3.19)fw(l)(w)=(w¯)l(1-|w|2)α-1+l∑j=1ncj∏k=0l-1(α+j+k).
It is easy to see that fw∈F(p,q,s) for each w∈𝔻 and ∥fw∥F(p,q,s)≤C by using the same methods in [23]. By Lemma 2.3, using the same method in [22, 36], we can choose c1,c2,…,cn, the corresponding function is denoted by fw, such that
(3.20)fw(n)(w)=∑j=1ncj∏k=0l-1(α+j+k)(w¯)n(1-|w|2)α-1+n,fw(l)(w)=0,l∈{0,1,…,n-1}.
Therefore,
(3.21)C≥‖Mufw‖Wμ(n)(D)≥μ(|w|)|u(w)fw(n)(w)|=∑j=1ncj∏k=0l-1(α+j+k)μ(|w|)|w|n|u(w)|(1-|w|2)((2+q-p)/p)+n.
From this, we obtain
(3.22)sup⁡|w|>1/2μ(|w|)|u(w)|(1-|w|2)((2+q-p)/p)+n≤C.
Since μ is normal and u∈H(𝔻), we get
(3.23)sup⁡|w|≤1/2μ(|w|)|u(w)|(1-|w|2)((2+q-p)/p)+n≤Csup⁡|w|≤1/2μ(|w|)|u(w)|≤C,
which along with (3.22) implies that (3.1) is necessary for all case. Let 2+q=p,s>1, and Mu:F(p,q,s)→𝒲μ(n)(𝔻) be bounded. To prove (3.2), we set
(3.24)gw(z)=Alog⁡21-w¯z+B(log⁡(2/(1-w¯z)))2log⁡⁡(2/(1-|w|2)),z,w∈D.
By a direct calculation, we get
(3.25)|gw′(z)|≤|A||1-w¯z|+2|Blog⁡(2/(1-w¯z))||log⁡(2/(1-|w|2))||1-w¯z|≤C|1-w¯z|,
thus we have gw∈F(p,q,s) and supw∈𝔻∥gw∥F(p,q,s)≤C<∞ (see [19, 30]). On the other hand, we have that
(3.26)gw′(z)=Aw¯1-w¯z+2Bw¯log⁡⁡(2/(1-w¯z))(log⁡(2/(1-|w|2)))(1-w¯z),gw′′(z)=A(w¯)2(1-w¯z)2+2B(w¯)2(log⁡(2/(1-|w|2)))(1-w¯z)2+2B(w¯)2log⁡(2/(1-w¯z))(log⁡⁡(2/(1-|w|2)))(1-w¯z)2,gw′′′(z)=2A(w¯)3(1-w¯z)3+6B(w¯)3(log⁡⁡(2/(1-|w|2)))(1-w¯z)3+2⋅2B(w¯)3log⁡(2/(1-w¯z))(log⁡(2/(1-|w|2)))(1-w¯z)3,gw(4)(z)=3⋅2A(w¯)4(1-w¯z)4+22B(w¯)4(log⁡(2/(1-|w|2)))(1-w¯z)4+3!2B(w¯)4log⁡(2/(1-w¯z))(log⁡(2/(1-|w|2)))(1-w¯z)4,gw(5)(z)=4!A(w¯)5(1-w¯z)5+100B(w¯)5(log⁡(2/(1-|w|2)))(1-w¯z)5+4!2B(w¯)5log⁡⁡(2/(1-w¯z))(log⁡(2/(1-|w|2)))(1-w¯z)5,⋮gw(m)(z)=(m-1)!A(w¯)m(1-w¯z)m+AmB(w¯)m(log⁡(2/(1-|w|2)))(1-w¯z)m+2(m-1)!B(w¯)mlog⁡(2/(1-w¯z))(log⁡(2/(1-|w|2)))(1-w¯z)m.
Moreover, we have that
(3.27)gw(w)=(A+B)log⁡21-|w|2,gw′(w)=(A+2B)w¯1-|w|2,gw′′(w)=(A+2B)(w¯)2(1-|w|2)2+2B(w¯)2(log⁡(2/(1-|w|2)))(1-|w|2)2,gw′′′(w)=2(A+2B)(w¯)3(1-|w|2)3+6B(w¯)3(log⁡(2/(1-|w|2)))(1-|w|2)3,gw(4)(w)=6(A+2B)(w¯)4(1-|w|2)4+22B(w¯)4(log⁡(2/(1-|w|2)))(1-|w|2)4,gw(5)(w)=24(A+2B)(w¯)5(1-|w|2)5+100B(w¯)5(log⁡(2/(1-|w|2)))(1-|w|2)5,⋮gw(m)(w)=(m-1)!(A+2B)(w¯)m(1-|w|2)m+AmB(w¯)m(log⁡(2/(1-|w|2)))(1-|w|2)m.
Taking A=2, B=-1, if n=1, we have
(3.28)μ(|w|)|u′(w)log⁡21-|w|2|≤‖Mugw‖Wμ(n)(D)≤C‖Mu‖<∞,
if n≠1, we have
(3.29)‖Mugw‖Wμ(n)(D)≥μ(|w|)|∑j=0nCnju(j)(w)gw(n-j)(w)|≥μ(|w|)|u(n)(w)gw(w)|-μ(|w|)|∑j=0n-1Cnju(j)(w)gw(n-j)(w)|≥μ(|w|)|u(n)(w)gw(w)|-μ(|w|)×|∑j=0n-1Cnju(j)(w)An-jB(w¯)n-j(log⁡(2/(1-|w|2)))(1-|w|2)n-j|.
From (3.29), (3.1), (3.9), and (3.10) we have
(3.30)μ(|w|)|u(n)(w)log⁡21-|w|2|≤‖Mugw‖Wμ(n)(D)+C|∑j=0n-1μ(|w|)u(j)(w)(1-|w|2)n-j|≤C‖Mu‖+C<∞.
Using (3.28) and (3.30), it is easy to get that (3.2) holds, finishing the proof of the theorem.Theorem 3.2.
Assume thatu∈H(𝔻) and μ is normal.(a)
If2+q>p, then Mu:F(p,q,s)→𝒲μ(n)(𝔻) is compact if and only if
(3.31)lim⁡|z|→1μ(|z|)|u(z)|(1-|z|2)((2+q-p)/p)+n=0.(b)
If2+q<p, then Mu:F(p,q,s)→𝒲μ(n)(𝔻) is compact if and only if (3.31) holds and u∈𝒲μ(n)(𝔻).(c)
If2+q=p,s>1, then Mu:F(p,q,s)→𝒲μ(n)(𝔻) is compact if and only if (3.31) holds and
(3.32)lim⁡|z|→1μ(|z|)|u(n)(z)|log⁡21-|z|2=0.Proof.
Assume that conditions (3.31) hold. By Theorem 3.1, Mu:F(p,q,s)→𝒲μ(n)(𝔻) is bounded. For any bounded sequence {fk} in F(p,q,s) with fk→0 uniformly on compact subsets of 𝔻 establish the assertion, it suffices, in view of Lemma 2.4, to show that
(3.33)‖Mufk‖Wμ(n)(D)⟶0ask⟶∞.
We assume that ∥fk∥F(p,q,s)≤1. From (3.31) and (3.32), given ϵ>0, there exists a δ∈(0,1), when δ<|z|<1, we have
(3.34)μ(|z|)|u(z)|(1-|z|2)((2+q-p)/p)+n<ϵ,(3.35)μ(|z|)|u(n)(z)|log⁡21-|z|2<ϵ.
By (3.34) and the Cauchy integral formula, we obtain
(3.36)|u′(z)|=12π|∫02πu(δzeiθ)(δzeiθ-z)2δzeiθdθ|<ϵ(1-δz2)((2+q-p)/p)+nμ(δz)12π∫02πδz|δzeiθ-z|2dθ=ϵ(1-δz2)((2+q-p)/p)+nμ(δz)δzδz2-|z|2≤ϵC(1-|z|2)((2+q-p)/p)+n-1μ(|z|),
when δ<|z|<1. Hence,
(3.37)μ(|z|)|u′(z)|(1-|z|2)((2+q-p)/p)+n-1<Cϵ,
when τ<|z|<1. Similarly, for j∈{2,…,n}, we have that
(3.38)μ(|z|)|u(j)(z)|(1-|z|2)((2+q-p)/p)+n-j<Cϵ,
when δ<|z|<1. Since fk→0 uniformly on compact subsets of 𝔻, Cauchy’s estimate gives that fk(j) converges to 0 uniformly on compact subsets of 𝔻 for each j∈{1,…,n}, there exists a K0∈ℕ such that k>K0 implies that
(3.39)∑j=0n-1∑m=0j-1Cjm|u(m)(0)fk(j-m)(0)|+sup⁡|z|≤δμ(|z|)|∑j=0nCnju(j)(z)fk(n-j)(z)|<Cϵ.
If 2+q>p, from (3.34), (3.38), (3.39), and Lemmas 2.1 and 2.2, we have
(3.40)‖Mufk‖Wμ(n)(D)=∑j=0n-1|(ufk)(j)(0)|+sup⁡z∈Dμ(|z|)|(ufk)(n)(z)|≤∑j=0n-1∑m=0j-1Cjm|u(m)(0)fk(j-m)(0)|+sup⁡|z|≤δμ(|z|)|∑j=0nCnju(j)(z)fk(n-j)(z)|+sup⁡|z|>δμ(|z|)|∑j=0nCnju(j)(z)fk(n-j)(z)|<Cϵ+Csup⁡|z|>δ|μ(|z|)u(n)(z)fk(z)|+Csup⁡|z|>δ∑j=0n-1|Cnjμ(|z|)u(j)(z)(1-|z|2)((2+q-p)/p)+n-j|‖fk‖F(p,q,s)≤Cϵ+Csup⁡|z|>δ|μ(|z|)u(n)(z)(1-|z|2)(2+q-p)/p|‖fk‖B(2+q)/p+Csup⁡|z|>δ∑j=0n-1|Cnjμ(|z|)u(j)(z)(1-|z|2)((2+q-p)/p)+n-j|≤Cϵ+Csup⁡|z|>δ|μ(|z|)u(n)(z)(1-|z|2)(2+q-p)/p|+Csup⁡|z|>δ∑j=0n-1|Cnjμ(|z|)u(j)(z)(1-|z|2)((2+q-p)/p)+n-j|<(2+C∑j=0nCnj)Cϵ,
when k>K0. If 2+q<p, then fk∈F(p,q,s)⊂ℬ(2+q)/p,k∈ℕ. By [37, Lemma 3.2], we have
(3.41)lim⁡k→∞sup⁡z∈D|fk(z)|=0.
By (3.38), (3.39), Lemma 2.1, and u∈𝒲μ(n)(𝔻), we get
(3.42)‖Mufk‖Wμ(n)(D)=∑j=0n-1|(ufk)(j)(0)|+sup⁡z∈Dμ(|z|)|(ufk)(n)(z)|<Cϵ+Csup⁡|z|>δ|μ(|z|)u(n)(z)fk(z)|+Csup⁡|z|>δ∑j=0n-1|Cnjμ(|z|)u(j)(z)(1-|z|2)((2+q-p)/p)+n-j|‖fk‖F(p,q,s)≤Cϵ+Csup⁡z∈D|fk(z)|+Csup⁡|z|>δ∑j=0n-1|Cnjμ(|z|)u(j)(z)(1-|z|2)((2+q-p)/p)+n-j|≤Cϵ+Cϵ+Csup⁡|z|>δ∑j=0n-1|Cnjμ(|z|)u(j)(z)(1-|z|2)((2+q-p)/p)+n-j|<(2+C∑j=0nCnj)Cϵ,
when k>K0. If 2+q=p, from (3.35), (3.38), (3.39), and Lemmas 2.1 and 2.2, we get
(3.43)‖Mufk‖Wμ(n)(D)=∑j=0n-1|(ufk)(j)(0)|+sup⁡z∈Dμ(|z|)|(ufk)(n)(z)|<Cϵ+Csup⁡|z|>δ|μ(|z|)u(n)(z)fk(z)|+Csup⁡|z|>δ∑j=0n-1|Cnjμ(|z|)u(j)(z)(1-|z|2)((2+q-p)/p)+n-j|‖fk‖F(p,q,s)<Cϵ+Csup⁡|z|>δ|μ(|z|)u(n)(z)|log⁡21-|z|2‖fk‖B+Csup⁡|z|>δ∑j=0n-1|Cnjμ(|z|)u(j)(z)(1-|z|2)((2+q-p)/p)+n-j|≤Cϵ+Csup⁡|z|>δ|μ(|z|)u(n)(z)|log⁡21-|z|2‖fk‖F(p,q,s)+Csup⁡|z|>δ∑j=0n-1|Cnjμ(|z|)u(j)(z)(1-|z|2)n-j|<(2+C∑j=0nCnj)Cϵ,
when k>K0. It follows that the operator Mu:F(p,q,s)→𝒲μ(n)(𝔻) is compact.
Conversely, assume thatMu:F(p,q,s)→𝒲μ(n)(𝔻) is compact. Taking f=1, we get u∈𝒲μ(n)(𝔻). Let {zk} be a sequence in 𝔻 such that |zk|→1 as k→∞. Taking the test functions hzk, where hzk is defined by (3.17), we will write
(3.44)hk(z)=fzk(z).
We obtain that hk∈F(p,q,s),
(3.45)sup⁡k∈N‖hk‖F(p,q,s)≤C,
and for t∈{1,…,n-1},
(3.46)hk(n)(zk)=∑j=1ncj∏k=0l-1(α+j+k)(zk¯)n(1-|zk|2)((2+q-p)/p)+n,hk(t)(zk)=0.
For |z|=r<1, we have
(3.47)|hk(z)|=|∑j=1ncj(1-|zk|2)j+1(1-zk¯z)α+j|≤∑j=1ncj(1-|zk|2)j+1(1-r)α+j⟶0ask⟶∞,
that is, hk converges to 0 uniformly on compact subsets of 𝔻, using the compactness of Mu:F(p,q,s)→𝒲μ(n)(𝔻), we obtain
(3.48)μ(zk)|u(zk)||zk|n(1-|zk|2)((2+q-p)/p)+n≤C‖Muhk‖Wμ(n)(D)⟶0ask⟶∞,
and consequently (3.31) holds.
Assume that2+q=p,s>1 and Mu:F(p,q,s)→𝒲μ(n)(𝔻) is compact, we only need to prove (3.32) holds. To do this, let {zk} be a sequence in 𝔻 such that |zk|→1 as k→∞. Set
(3.49)ak=log⁡21-|zk|2,Jk(z)=Eak(log⁡21-zk¯z)2+Fak2(log⁡21-zk¯z)3.
Since
(3.50)|Jk′(z)|≤|2Eak(log⁡21-zk¯z)zk¯1-zk¯z|+|3Fak2(log⁡21-zk¯z)2zk¯1-zk¯z|≤C|1-zk¯z|,
we have Jk∈F(p,q,s) and supk∈ℕ∥Jk∥F(p,q,s)≤C (see [19, 30]), and Jk→0 uniformly on compact subsets of 𝔻. Since Mu:F(p,q,s)→𝒲μ(n)(𝔻) is compact, we have
(3.51)‖MuJk‖Wμ(n)(D)⟶0ask⟶∞.
We also have that
(3.52)Jk′(z)=2Eak(log⁡21-zk¯z)zk¯1-zk¯z+3Fak2(log⁡21-zk¯z)2zk¯1-zk¯z,Jk′′(z)=2Eak(zk¯)2(1-zk¯z)2+2Eak(log⁡21-zk¯z)(zk¯)2(1-zk¯z)2+3⋅2Fak2(log⁡21-zk¯z)(zk¯)2(1-zk¯z)2+3Fak2(log⁡21-zk¯z)2(zk¯)2(1-zk¯z)2,Jk′′′(z)=2⋅2Eak(zk¯)3(1-zk¯z)3+2Eak(zk¯)3(1-zk¯z)3+4Eak(log⁡21-zk¯z)(zk¯)3(1-zk¯z)3+3⋅2Fak2(zk¯)3(1-zk¯z)3+3⋅2⋅2Fak2(log⁡21-zk¯z)(zk¯)3(1-zk¯z)3+3⋅2Fak2(log⁡21-zk¯z)(zk¯)3(1-zk¯z)3+6Fak2(log⁡21-zk¯z)2(zk¯)3(1-zk¯z)3,=6Eak(zk¯)3(1-zk¯z)3+4Eak(log⁡21-zk¯z)(zk¯)3(1-zk¯z)3+6Fak2(zk¯)3(1-zk¯z)3+18Fak2(log⁡21-zk¯z)(zk¯)3(1-zk¯z)3+6Fak2(log⁡21-zk¯z)2(zk¯)3(1-zk¯z)3,Jk(4)(z)=22Eak(zk¯)4(1-zk¯z)4+3!2Eak(log⁡21-zk¯z)(zk¯)4(1-zk¯z)4+36Fak2(zk¯)4(1-zk¯z)4+66Fak2(log⁡21-zk¯z)(zk¯)4(1-zk¯z)4+3!3Fak2(log⁡21-zk¯z)2(zk¯)4(1-zk¯z)4,⋮Jk(m)(z)=bmEak(zk¯)m(1-zk¯z)m+(m-1)!2Eak(log⁡21-zk¯z)(zk¯)m(1-zk¯z)m+dmFak2(zk¯)m(1-zk¯z)m+cmFak2(log⁡21-zk¯z)(zk¯)m(1-zk¯z)m+(m-1)!3Fak2(log⁡21-zk¯z)2(zk¯)m(1-zk¯z)m.
Therefore,
(3.53)Jk(zk)=(E+F)ak,Jk′(zk)=(2E+3F)zk¯1-|zk|2,Jk′′(zk)=(2E+3F)(zk¯)2(1-|zk|2)2+2(E+3F)ak(zk¯)2(1-|zk|2)2,Jk′′′(zk)=2(2E+3F)(zk¯)3(1-|zk|2)3+6(E+3F)ak(zk¯)3(1-|zk|2)3+6Fak2(zk¯)3(1-|zk|2)3,Jk(4)(zk)=3!(2E+3F)(zk¯)4(1-|zk|2)4+22(E+3F)ak(zk¯)4(1-|zk|2)4+36Fak2(zk¯)4(1-|zk|2)4,⋮Jk(m)(zk)=(m-1)!(2E+3F)(zk¯)m(1-|zk|2)m+bmE+cmFak(zk¯)m(1-|zk|2)m+dmFak2(zk¯)m(1-|zk|2)m.
Taking E=3 and F=-2, if n=1, we have
(3.54)μ(|zk|)|u′(zk)log⁡21-|zk|2|≤‖MuJk‖Wμ(n)(D)⟶0ask⟶∞,
if n≠1, we have
(3.55)‖MuJk‖Wμ(n)(D)≥μ(|zk|)|∑j=0nCnju(j)(zk)Jk(n-j)(zk)|≥μ(|zk|)|u(n)(zk)Jk(zk)|-μ(|zk|)|∑j=0n-1Cnju(j)(zk)Jk(n-j)(zk)|≥μ(|zk|)|u(n)(zk)log⁡21-|zk|2|-μ(|zk|)|∑j=0n-1Cnju(j)(zk)Jk(n-j)(zk)|.
From (3.31), (3.35), (3.37), and (3.38), we have that for sufficiently large k:
(3.56)μ(|zk|)|u(n)(zk)log⁡21-|zk|2|≤‖MuJk‖Wμ(n)(D)+μ(|zk|)|∑j=0n-1Cnju(j)(zk)Jk(n-j)(zk)|≤‖MuJk‖Wμ(n)(D)+C∑j=0n-1(μ(|zk|)|u(j)(zk)|ak(1-|zk|2)n-j+μ(|zk|)|u(j)(zk)|ak2(1-|zk|2)n-j)≤‖MuJk‖Wμ(n)(D)+C∑j=0n-1μ(|zk|)|u(j)(zk)|(1-|zk|2)n-j<Cϵ.
Using (3.54) and (3.56), it is easy to get that (3.32) holds. From which we obtain the desired result.


## 4. The Boundedness ofMu from F(p,q,s) (orF0(p,q,s)) to 𝒲μ,0(n)(𝔻)

In this section, we characterize the boundedness ofMu:F(p,q,s)(orF0(p,q,s))→𝒲μ,0(n)(𝔻).Theorem 4.1.
Assume thatu∈H(𝔻) and μ is normal. Then,(1)
Mu:F(p,q,s)→𝒲μ,0(n)(𝔻) is bounded if and only if Mu:F(p,q,s)→𝒲μ(n)(𝔻) is bounded and
(4.1)lim⁡|z|→1μ(z)|u(n)(z)|=0.(2)
Mu:F0(p,q,s)→𝒲μ,0(n)(𝔻) is bounded if and only if Mu:F0(p,q,s)→𝒲μ(n)(𝔻) is bounded and (4.1) holds.Proof.
(1) Assume thatMu:F(p,q,s)→𝒲μ(n)(𝔻) is bounded and condition (4.1) holds. Since
(4.2)u(z)-u(0)=∫01xu′(zx)dx,
it follows that
(4.3)μ(z)|u(z)|≤μ(z)|u(0)|+μ(z)|∫01xu′(zx)dx|≤μ(z)|u(0)|+μ(z)|∫01/2xu′(zx)dx|+μ(z)|∫1/21xu′(zx)dx|≤μ(z)|u(0)|+μ(z)max⁡|z|≤1/2|u′(z)|+μ(z)|∫1/21xu′(zx)dx|.
Since μ is normal, by the monotonicity of μ(t)/(1-t2)a, for t0≤t1<t<1, we have
(4.4)μ(t)=(1-t2)aμ(t)(1-t2)a≤(1-t2)aμ(t1)(1-t12)a<μ(t1),
that is, μ is decreasing on [t0,1), and for any ϵ>0, there is a τ>0 such that
(4.5)0<μ(|z|)<ϵ(1-|z|2)a,(τ<|z|<1),
which implies lim|z|→1μ(|z|)=0. Since for 1/2<x<1 and 2t0<|z|<1, we have μ(|z|)≤μ(x|z|). From (4.3), it follows that
(4.6)μ(|z|)|u(|z|)|≤μ(|z|)|u(0)|+μ(|z|)max⁡|z|≤1/2|u′(z)|+∫1/21μ(x|z|)|u′(zx)|dx.
For j∈{2,3,…,n}, by applying formula (4.6) to the function u(j-1), we get
(4.7)μ(|z|)|u(j-1)(z)|≤μ(|z|)|u(j-1)(0)|+μ(|z|)max⁡|z|≤1/2|u(j)(z)|+∫1/21μ(|z|x)|u(j)(zx)|dx,
when 2t0<|z|<1. It follows from (4.1), (4.6), and (4.7) that lim|z|→1μ(|z|)|u(j-1)(z)|=0 for j∈{1,3,…,n}. Since, for each polynomial p, we have
(4.8)|μ(|z|)(Mup)(n)(z)|=μ(|z|)|(u(z)p(z))(n)|=μ(|z|)|∑j=0nCnju(j)(z)p(n-j)(z)|≤C∑j=0n|Cnjμ(|z|)u(j)(z)|‖p(n-j)‖∞,
hence, Mup∈𝒲μ,0(n)(𝔻). Since the set of all polynomials is dense in F(p,q,s), we have that for every f∈F(p,q,s) there is a sequence of polynomials {pk} such that
(4.9)lim⁡k→∞‖pk-f‖F(p,q,s)=0.
From this and since the operator Mu:F(p,q,s)→𝒲μ(n)(𝔻) is bounded, we have that
(4.10)‖Mupk-Muf‖Wμ(n)(D)≤‖Mu‖‖pk-f‖F(p,q,s)⟶0ask⟶∞.
Since 𝒲μ,0(n)(𝔻) is a closed subspace of 𝒲μ(n)(𝔻), therefore, we have Mu(F(p,q,s)⊂𝒲μ,0(n)(𝔻), from which the boundedness of Mu:F(p,q,s)→𝒲μ,0(n)(𝔻) follows.
On the other hand, assume thatMu:F(p,q,s)→𝒲μ,0(n)(𝔻) is bounded, then Mu:F(p,q,s)→𝒲μ(n)(𝔻) is bounded. By taking the function given by f(z)=1, we obtain
(4.11)μ(|z|)|u(n)(z)|=|μ(|z|)(Muf)(n)(z)|⟶0(as|z|⟶1),
as desired.
(2) The proof is similar to that of the case (1). We leave the details to the interested reader.



---

*Source: 343194-2012-05-15.xml*