# Multipliers in Holomorphic Mean Lipschitz Spaces on the Unit Ball

Authors: Hong Rae Cho
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

For1≤p≤∞ and s>0, let Λsp be holomorphic mean Lipschitz spaces on the unit ball in ℂn. It is shown that, if s>n/p, the space Λsp is a multiplicative algebra. If s>n/p, then the space Λsp is not a multiplicative algebra. We give some sufficient conditions for a holomorphic function to be a pointwise multiplier of Λn/pp.


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## Body

## 1. Introduction

LetX and Y be two function spaces. We call φ a pointwise multiplier from X to Y if φf∈Y for every f∈X. The collection of all pointwise multipliers from X to Y is denoted by ℳ(X→Y). When X=Y, we let ℳ(X)=ℳ(X→X).Multipliers arise in the theory of differential equations. Coefficients of differential operators can be naturally considered as multipliers. The same is true for symbols of more general pseudodifferential operators.To give some motivations for our study, we recall studies on multipliers of Sobolev spaces. Strichartz [1] was the first who studied on multipliers of Sobolev spaces. Let Ω be a bounded domain in ℝn with Lipschitz boundary. Let 1<p<∞. For s>0, let Ws,p(Ω) be the Sobolev spaces over Ω. Given f and g in Ws,p(Ω), one cannot in general expect that their product fg will belong to Ws,p(Ω). However, if s>n/p, then there exists a constant K depending on s,p,n, and Ω, such that [1–3]
(1.1)‖fg‖Ws,p(Ω)≤K‖f‖Ws,p(Ω)‖g‖Ws,p(Ω).
This implies that Ws,p(Ω)⊂ℳ(Ws,p(Ω)). Since Ws,p(Ω) contains constant functions, ℳ(Ws,p(Ω))⊂Ws,p(Ω). Thus, we have
(1.2)M(Ws,p(Ω))=Ws,p(Ω)ifs>np.In the complex case, multipliers on Hardy-Sobolev spaces on the unit ball inℂn were studied by [4, 5] for n=1 and [6] for n≥1. Let 𝔹n={z∈ℂn:|z|<1} denote the open unit ball in ℂn. Let 1≤p<∞ and m be a positive integer. Let Hmp(𝔹n) be the Hardy-Sobolev space of order m. In papers [4–6], it was proved that
(1.3)M(Hmp(Bn))=Hmp(Bn)ifm>np.
Complete characterization of multipliers on other Hardy-Sobolev spaces of nonregular cases m≤n/p remains open, but Beatrous and Burbea [7] gave some sufficient conditions for functions to be pointwise multipliers in these nonregular cases. Ortega and Fàbrega [8] introduced a family of nonisotropic tent-Sobolev spaces to characterize multipliers in some Hardy-Sobolev spaces of nonregular cases. Usually, characterization of multipliers of nonregular cases is difficult.Many authors have studied properties of multipliers for several function spaces (see [9–12] for Dirichlet-type spaces, [13, 14] for Bloch-type spaces, [4] for Bergman-Sobolev spaces, [15] for mixed norm spaces, [16] for Qs spaces, [17] for Fs∞,q spaces, and [18] for the BMO space).For pointsz=(z1,…,zn) and w=(w1,…,wn) in ℂn, we write
(1.4)〈z,w〉=z1w¯1+⋯+znw¯n,|z|=|z1|2+⋯+|zn|2.
Let 𝕊n={ζ∈ℂn:|ζ|=1} denote the unit sphere in ℂn. The normalized Lebesgue measure on 𝕊n will be denoted by dσ. Let H(𝔹n) denote the space of all holomorphic functions in 𝔹n. Given 0<r<1, 0<p<∞, and f∈H(𝔹n), we define
(1.5)Mp(r,f)=[∫Sn|f(rζ)|pdσ(ζ)]1/p.
When p=∞, we write
(1.6)M∞(r,f)=sup⁡{|f(rζ)|:ζ∈Sn}.
For 0<p≤∞, the Hardy space Hp consists of all functions f∈H(𝔹n) such that
(1.7)‖f‖Hp=sup⁡0<r<1Mp(r,f)<∞.
See [19] for basic information about the Hardy spaces.We denote byℛf the radial derivative of f in H(𝔹n) defined by
(1.8)Rf=∑j=1nzj∂∂zjf.
We consider the space Λsp of holomorphic functions on 𝔹n such that
(1.9)[∫Sn|Rmf(rζ)|pdσ]1/p≲1(1-r)m-s
for m>s. We define the norm of Λsp as follows:
(1.10)‖f‖Λsp=‖f‖Hp+sup⁡0<r<1(1-r)m-sMp(r,Rmf).
It can be shown that the norm is independent of the choice of m; see [20]. When p=∞, this is exactly the classical holomorphic Lipschitz space Λs; see [19].We adapt the first orderHp mean variation defined as follows:
(1.11)ωp(t,f)=sup⁡{(∫Sn|f(Uζ)-f(ζ)|pdσ(ζ))1/p:U∈U,‖U-I‖≤t},
where 𝒰 denotes the group of all unitary operators on ℂn, I denotes the identity of 𝒰, and ∥U-I∥:=supζ∈S|Uζ-ζ|. Then, we have
(1.12)‖f‖Λsp≈‖f‖Hp+sup⁡0<t<1ωp(t,f)ts
for 0<s<1 (see [20]). This justifies our usage of the term holomorphic mean Lipschitz space for Λsp with 0<s<1. Now, for s≥1, we consider the second-order Hp mean variation defined as follows:
(1.13)ωp*(t,f)=sup⁡{(∫Sn‍|f(Uζ)-2f(ζ)+f(U-1ζ)|pdσ(ζ))1/p:U∈U,‖U-I‖≤t}.
It was shown in [21] that, if 0<s<2,1≤p<∞, and f∈Hp,   then
(1.14)‖f‖Λsp≈‖f‖Hp+sup⁡0<t<1ωp*(t,f)ts.Theorem 1.1.
Let1≤p<∞ and s>0.  (i)
Ifs>n/p (regular), then Λsp is a multiplicative algebra.(ii)
Ifs≤n/p (nonregular), then Λsp is not a multiplicative algebra.By (ii) of Theorem1.1, the space Λn/pp is not a multiplicative algebra. We give some sufficient conditions for a holomorphic function to be a pointwise multiplier of Λn/pp as follows. We do not know if our sufficient condition is also necessary.Theorem 1.2.
Let1≤p<q<∞. Then, Λn/pq⊂ℳ(Λn/pp).Throughout the paper, we writeX≲Y or Y≳X for nonnegative quantities X and Y whenever there is a constant C>0 (independent of the parameters in X and Y) such that X≤CY. Similarly, we write X≈Y if X≲Y and Y≲X.


## 2. Auxiliary Embedding Results

The ball algebraA(𝔹n) is the class of all functions f:𝔹n→ℂ that are continuous on the closed ball 𝔹¯n and that are holomorphic in its interior 𝔹n.Proposition 2.1.
Let1≤p<∞ and s>0.  (i)
There is a function in the ball algebraA(𝔹n) that is not in Λsp.(ii)
Ifs≤n/p, then there is a function in Λsp that is not in H∞.(iii)
Ifs>n/p, then Λsp⊂Λs-n/p⊂A(𝔹n).Remark 2.2.
By (ii) and (iii), we can see thatΛsp⊂H∞ if and only if s>n/p.Proof.
(i) Let(pk) be a sequence of Ryll and Wojtaszczyk [22] homogeneous polynomials in the unit sphere of H∞(𝔹n) such that pk has degree k and
(2.1)‖pk‖H2≥π2n.
Let
(2.2)f=∑1k2p2k.
This function was constructed in [7]. In fact, it was shown in [7] that this function is not contained in any Hardy-Sobolev space. Thus, the result of (i) of Proposition 2.1 follows, since every mean Lipschitz space is contained in a Hardy-Sobolev space.
Since the series converges uniformly on𝔹¯n, its sum f is therefore in the ball algebra. It is enough to prove that f∉Λs1 for 0<s<1. If f∈Λs1, then
(2.3)|∫𝕊nℛf(rζ)p2k(ζ)¯dσ(ζ)|≤∫𝕊n|ℛf(rζ)|dσ(ζ)≲1(1-r)1-s,0<r<1.
However, since the polynomials pk are orthogonal, for any 0<r<1,
(2.4)|∫SnRf(rζ)p2k(ζ)¯dσ(ζ)|=|∫Sn2kk2p2k(rζ)p2k(ζ)¯dσ(ζ)|=∫Sn‍2kk2r2k|p2k(ζ)|2dσ(ζ)=2kk2r2k‖p2k‖H22≥2kk2r2kπ4n.
Take
(2.5)rk=1-12k.
Then,
(2.6)|∫SnRf(rkζ)p2k(ζ)¯dσ(ζ)|≥11-rk1(log⁡2(1/(1-rk)))2rk2kπ4n=1(1-rk)1-sck,
where
(2.7)ck=1(1-rk)s1(log⁡2(1/(1-rk)))2rk2kπ4n⟶∞ask⟶∞.
By (2.3) and (2.6), it is a contradiction. Thus, f∉Λs1 for 0<s<1.
(ii) It is clear, if we consider the function(2.8)f(z)=log⁡(1-〈z,ζ〉)
with ζ∈𝕊n.
(iii) Letf∈Λsp. Let m be the greatest integer less than s and α=s-m. Let 0<ρ<1. By the Cauchy's integral formula, we have
(2.9)Rm+1f(ρz)=∫SnRm+1f(ρζ)(1-〈z,ζ〉)ndσ(ζ),z∈Bn.
Making a change of variables and replacing ρz by z, we get
(2.10)Rm+1f(z)=ρ∫|ζ|=ρRm+1f(ζ)(ρ2-〈z,ζ〉)ndσ(ζ),|z|<ρ.
By (2.10) and Hölder's inequality, we have
(2.11)|Rm+1f(z)|≲Mp(ρ,Rm+1f)1(ρ2-|z|2)n/p.
Take ρ=(1+|z|)/2. Then, we obtain
(2.12)|Rm+1f(z)|≲Mp(ρ,Rm+1f)1(1-|z|2)n/p≲1(1-ρ)1-α1(1-|z|2)n/p≲1(1-|z|2)m+1-(s-n/p).
Therefore, f∈Λs-n/p⊂A(𝔹n) if s>n/p.Proposition 2.3.
Lets>0 and 1≤p,q<∞.   (i)
Λsp⊂Hq,0≤n(1/p-1/q)<s.(ii)
Λn/pp⊂⋂0<q<∞Hq.Proof.
(i) Let1/2<r<1 and α=s-m, where m is the greatest integer less than s. By the fundamental theorem of calculus and Minkowski's inequality, we have
(2.13)Mq(r,f)≲sup⁡|z|<1/2|f(z)|+∫0rMq(t,Rf)dt.
Applying this repeatedly and using Fubini's theorem, we obtain
(2.14)Mq(r,f)≲sup⁡|z|<1/2|f(z)|+∫0r(r-t)mMq(t,Rm+1f)dt≲sup⁡|z|<1/2|f(z)|+∫01(1-t)mMq(t,Rm+1f)dt.
Let0<ρ<1. By (2.11), we have
(2.15)|Rm+1f(z)|≲Mp(ρ,Rm+1f)(ρ2-|z|2)-n/p,|z|<ρ.
For 0<r<1, we take ρ=(1+r)/2. Then,
(2.16)M∞(r,Rm+1f)≲Mp(ρ,Rm+1f)(1-r)-n/p.
For q≥p, we have
(2.17)Mq(r,Rm+1f)=(∫S|Rm+1f(rζ)|p|Rm+1f(rζ)|q-pdσ(ζ))1/q≲M∞(r,Rm+1f)1-p/qMp(r,Rm+1f)p/q≲Mp(ρ,Rm+1f)1-p/q(1-r)-n/p(1-p/q)Mp(r,Rm+1f)p/q≲‖f‖Λsp(1-r)n(1/q-1/p)-1+α.
By (2.14) and (2.17), we have
(2.18)Mq(r,f)≲‖f‖Λsp∫011(1-t)n(1/p-1/q)+1-α-mdt<∞,
since n(1/p-1/q)+1-α-m<s+1-α-m=1. Thus, we get the result.
(ii) Ifs=n/p in (2.18), then, for any p<q<∞, we have
(2.19)Mq(r,f)≲‖f‖Λn/pp∫0r1(1-t)1-n/qdt≲‖f‖Λn/pp.Remark 2.4.
The obvious question: IsΛn/pp contained in BMOA? In the case n=1, this was proved by Bourdon et al. in [23]. However, their method does not work in higher dimensions.
We have some observations by a Carleson measure. There is a characterization forBMOA functions by a Carleson measure such that f∈BMOA if and only if (1-|z|2)|ℛf(z)|2dV(z) is a Carleson measure, where dV is the volume measure on 𝔹n (see [19]). Even though we cannot prove that (1-|z|2)|ℛf(z)|2dV(z) is a Carleson measure for f∈Λn/pp, we have some weak results such that (1-|z|2)q-1|ℛf(z)|qdV(z) is a Carleson measure for q with max{p,n}<q. For the proof, let f∈Λn/pp and 1≤p<∞. By (2.17), we have Λn/pp⊂Λn/qq. This means that
(2.20)Mq(r,Rf)≲1(1-r)1-n/q.
Hence, we have
(2.21)∫Bn(1-|z|2)q-1|Rf(z)|q(1-|a|2)n|1-〈a,z〉|2ndV(z)≲∫01(1-r)q-1(1-|a|2)n(1-r|a|)2nMqq(r,Rf)dr≲(1-|a|2)n∫01(1-r)n-11(1-r|a|)2ndr≲1.
Thus, (1-|z|2)q-1|ℛf(z)|qdV(z) is a Carleson measure (see [19]). However, the embedding problem Λn/pp⊂BMOA is still open.


## 3. Regular Cases

We need an elementary variant of Hölder's inequality.Lemma 3.1.
Let0<p,q,δ<∞ with 1/p=1/q+1/δ. Then, for f∈Lq and g∈Lδ, the product fg is in Lp and
(3.1)‖fg‖Lp≤‖f‖Lq‖g‖Lδ.Theorem 3.2.
Ifs>n/p, then Λsp is a multiplicative algebra.Proof.
Letm be the greatest integer less than s and α=s-m. Let f,g∈Λsp. We will prove that
(3.2)Mp(r,Rm+1(fg))≲1(1-r)1-α.
We note that
(3.3)Rm+1(fg)=∑j=0m+1cjRjfRm+1-jg.
Since s>n/p, by (iii) of Proposition 2.1, we have Λsp⊂A(𝔹n)⊂H∞.
Let0≤k≤m+1. Then,
(3.4)M(m+1)p/k(r,Rkf)≤C‖f‖H∞1-k/(m+1)Mp(r,Rm+1f)k/(m+1),
applied to the sphere of radius r (see [24], Theorem 1 on page 69, for such inequalities). Thus, for 0≤j≤m+1, we have
(3.5)M(m+1)p/j(r,ℛjf)≤C∥f∥H∞1-j/(m+1)Mp(r,ℛm+1f)j/(m+1),M(m+1)p/(m+1-j)(r,Rm+1-jg)≤C‖g‖H∞1-(m+1-j)/(m+1)Mp(r,Rm+1g)(m+1-j)/(m+1).
Therefore, by a variant of Hölder's inequality,
(3.6)Mp(r,Rm+1(fg))≲∑j=0m+1M(m+1)p/j(r,Rjf)M(m+1)p/(m+1-j)(r,Rm+1-jg)≲‖f‖H∞1-j/(m+1)‖g‖H∞1-(m+1-j)/(m+1)Mp(r,Rm+1f)j/(m+1)Mp(r,Rm+1g)(m+1-j)/(m+1)≲‖f‖H∞1-j/(m+1)‖g‖H∞1-(m+1-j)/(m+1)‖f‖Λspj/(m+1)‖g‖Λsp(m+1-j)/(m+1)1(1-r)1-α≲‖f‖Λsp‖g‖Λsp1(1-r)1-α.
That is,
(3.7)‖fg‖Λsp≲‖f‖Λsp‖g‖Λsp.


## 4. Nonregular Cases

A functionφ is a multiplier for Λsp if the multiplication operator Mφf=φf is continuous from Λsp to itself. The space of those multipliers will be denoted by ℳ(Λsp).Letφ∈ℳ(Λsp). Since 1∈Λsp, we have φ=φ·1∈Λsp. Thus,
(4.1)M(Λsp)⊂Λsp.
Hence,
(4.2)M(Λsp)⊂H(Bn).The following is a special case of Lemma5.1 in [7].Lemma 4.1 (see [7]).
Lets>0. Then, φ∈ℳ(Λsp) is bounded pointwise by its multiplier norm.Corollary 4.2.
Lets>0. Then, we have
(4.3)M(Λsp)⊊H∞.Proof.
Letφ∈ℳ(Λsp). Since 1∈Λsp, we have φ=φ·1∈Λsp. Thus,
(4.4)M(Λsp)⊂Λsp.
Hence,
(4.5)M(Λsp)⊂H(Bn).
Thus, by Lemma 4.1, we get ℳ(Λsp)⊂H∞. By (i) of Proposition 2.1, there is a function ψ∈A(𝔹n)∖Λsp⊂H∞∖Λsp. Since ℳ(Λsp)⊂Λsp, it follows that ψ∉ℳ(Λsp). Thus, we get the result.Theorem 4.3.
Let0<s≤n/p. Then, Λsp is not a multiplicative algebra.Proof.
IfΛsp is multiplicative, then
(4.6)Λsp=M(Λsp)⊊H∞.
By (ii) of Proposition 2.1, this is a contradiction.


## 5. Some Sufficient Conditions for a Nonregular Case

Lemma 5.1.
Letf∈Λsp. Let m be the integer part of s and α=s-m. Let 0<r<1. Then, for q>p, we have
(5.1)Mq(r,Rm+1f)≲‖f‖Λsp(1-r)n(1/q-1/p)-1+α.Proof.
We proved this inequality at (2.17) in the proof of (i) of Proposition 2.3.Lemma 5.2.
Let0≤l<k and 1/2<r<1. Then, we have
(5.2)Mp(r,Rlf)≲sup⁡|z|<1/2|f(z)|+∫0r(r-t)k-l-1Mp(t,Rkf)dt.Proof.
We proved this inequality at (2.14) in the proof of (i) of Proposition 2.3.Theorem 5.3.
Let1≤p<q<∞. Then, Λn/pq⊂ℳ(Λn/pp).Proof.
Letφ∈Λn/pq and f∈Λn/pp. Let m be the integer part of n/p and α=n/p-m. It is enough to prove that
(5.3)Mp(r,Rm+1(φf))≲1(1-r)1-α,0<r<1.
We note that(5.4)Rm+1(φf)=∑j=0m+1cj(Rjφ)(Rm+1-jf).
Thus,
(5.5)Mp(r,Rm+1(φf))≲∑j=0m+1Mp(r,(Rjφ)(Rm+1-jf)).
Letm=0. Then,
(5.6)Mp(r,R(φf))≤Mp(r,(Rφ)f)+Mp(r,φ(Rf)).
By Proposition 2.1, we have
(5.7)φ∈Λn/pq⊂Λn/p-n/q⊂H∞,f∈Λn/pp⊂⋂δ>0‍Hδ.
Choose δ>1 with 1/p=1/q+1/δ. Then,
(5.8)Mp(r,(Rφ)f)≤Mq(r,Rφ)Mδ(r,f)≲Mq(r,Rφ)‖f‖Hδ≲Mq(r,Rφ)‖f‖Λn/pp≲‖φ‖Λn/pq‖f‖Λn/pp1(1-r)1-α.
Also,
(5.9)Mp(r,φ(Rf))≲‖φ‖H∞Mp(r,Rf)≲‖φ‖H∞‖f‖Λn/pp1(1-r)1-α.
Thus,
(5.10)‖φf‖Λn/pp≲‖φ‖Λn/pq‖f‖Λn/pp.
Now, letm≥1.
Ifj=0, then, since φ∈H∞ and f∈Λn/pp,
(5.11)Mp(r,φRm+1f)≲‖φ‖H∞Mp(r,Rm+1f)≲‖φ‖H∞‖f‖Λn/pp1(1-r)1-α.
Ifj=m+1, we choose δ>p such that
(5.12)1p=1q+1δ.
Then,
(5.13)Mp(r,(Rm+1φ)f)≤Mq(r,Rm+1φ)Mδ(r,f).
By (ii) of Proposition 2.3, Λn/pp⊂⋂δ>0‍Hδ. By a variant of Hölder's inequality, we have
(5.14)Mp(r,(Rm+1φ)f)≲Mq(r,Rm+1φ)‖f‖Hδ≲Mq(r,Rm+1φ)‖f‖Λn/pp≲‖φ‖Λn/pq‖f‖Λn/pp1(1-r)1-α.
Now, we consider the case1≤j≤m. Since Λn/pq is decreasing in the parameter q, it is enough to consider for q sufficiently near p so that j>n/p-n/q. We choose μj>0 such that
(5.15)nμj=j-(np-nq).
Then,
(5.16)n(1q-1μj)=np-j>0.
Since p<q<μj, we can choose δj>0 such that
(5.17)1p=1μj+1δj,j=1,2,…,m.
By a variant of Hölder's inequality, we have(5.18)Mp(r,(Rjφ)(Rm+1-jf))≤Mμj(r,Rjφ)Mδj(r,Rm+1-jf).
By Lemmas 5.2 and 5.1,
(5.19)Mμj(r,Rjφ)≲sup⁡|z|<1/2|φ(z)|+∫0r(r-t)m-jMμj(t,Rm+1φ)dt≲‖φ‖Λsq∫0r(r-t)m-j(1-t)n(1/q-1/μj)+1-αdt.
Here,
(5.20)∫0r(r-t)m-j(1-t)n(1/q-1/μj)+1-αdt≲∫0r11-tdt≲log⁡(11-r).
Moreover, by Lemmas 5.2 and 5.1 again, we have
(5.21)Mδj(r,Rm+1-jf)≲sup⁡|z|<1/2|f(z)|+∫0r(r-t)j-1Mδj(t,Rm+1f)dt≲‖f‖Λn/pp∫0r(r-t)j-1(1-t)n(1/p-1/δj)+1-αdt≲‖f‖Λn/pp1(1-r)1-α-(n/p-n/q).
Thus, it follows that
(5.22)Mp(r,(Rjφ)(Rm+1-jf))≲‖φ‖Λn/pp‖f‖Λn/pp1(1-r)1-αcr,
where
(5.23)cr=log⁡(11-r)(1-r)n/p-n/q⟶0asr⟶1.
Thus,
(5.24)Mp(r,(Rjφ)(Rm+1-jf))≲1(1-r)1-α.
This completes the proof.



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