Multipliers in Holomorphic Mean Lipschitz Spaces on the Unit Ball

Abstract

For 1≤𝑝≤∞ and 𝑠>0, let Λ^𝑝_𝑠 be holomorphic mean Lipschitz spaces on the unit ball in ℂ^𝑛. It is shown that, if 𝑠>𝑛/𝑝, the space Λ^𝑝_𝑠 is a multiplicative algebra. If 𝑠>𝑛/𝑝, then the space Λ^𝑝_𝑠 is not a multiplicative algebra. We give some sufficient conditions for a holomorphic function to be a pointwise multiplier of Λ^𝑝_𝑛/𝑝.

1. Introduction

Let 𝑋 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 ℝ^𝑛 with Lipschitz boundary. Let 1<𝑝<∞. For 𝑠>0, let 𝑊^𝑠,𝑝(Ω) be the Sobolev spaces over Ω. Given 𝑓 and 𝑔 in 𝑊^𝑠,𝑝(Ω), one cannot in general expect that their product 𝑓𝑔 will belong to 𝑊^𝑠,𝑝(Ω). However, if 𝑠>𝑛/𝑝, then there exists a constant 𝐾 depending on 𝑠,𝑝,𝑛, and Ω, such that [1–3] 𝑓𝑔_{𝑊𝑠,𝑝(Ω)}≤𝐾𝑓_{𝑊𝑠,𝑝(Ω)}𝑔_{𝑊𝑠,𝑝(Ω)}. (1.1) This implies that 𝑊^𝑠,𝑝(Ω)⊂ℳ(𝑊^𝑠,𝑝(Ω)). Since 𝑊^𝑠,𝑝(Ω) contains constant functions, ℳ(𝑊^𝑠,𝑝(Ω))⊂𝑊^𝑠,𝑝(Ω). Thus, we have ℳ𝑊^𝑠,𝑝(Ω)=𝑊^𝑠,𝑝(Ω)𝑛if𝑠>𝑝. (1.2)

In the complex case, multipliers on Hardy-Sobolev spaces on the unit ball in ℂ^𝑛 were studied by [4, 5] for 𝑛=1 and [6] for 𝑛≥1. Let 𝔹^𝑛={𝑧∈ℂ^𝑛∶|𝑧|<1} denote the open unit ball in ℂ^𝑛. Let 1≤𝑝<∞ and 𝑚 be a positive integer. Let 𝐻^𝑝_𝑚(𝔹^𝑛) be the Hardy-Sobolev space of order 𝑚. In papers [4–6], it was proved that ℳ𝐻^𝑝_𝑚𝔹^𝑛=𝐻^𝑝_𝑚𝔹^𝑛𝑛if𝑚>𝑝. (1.3) 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 𝑄_𝑠 spaces, [17] for 𝐹_𝑠^∞,𝑞 spaces, and [18] for the BMO space).

For points 𝑧=(𝑧₁,…,𝑧_𝑛) and 𝑤=(𝑤₁,…,𝑤_𝑛) in ℂ^𝑛, we write ⟨𝑧,𝑤⟩=𝑧₁𝑤₁+⋯+𝑧_𝑛𝑤_𝑛,|𝑧|=𝑧₁²+⋯+𝑧_𝑛². (1.4) Let 𝕊^𝑛={𝜁∈ℂ^𝑛∶|𝜁|=1} denote the unit sphere in ℂ^𝑛. The normalized Lebesgue measure on 𝕊^𝑛 will be denoted by 𝑑𝜎. Let 𝐻(𝔹^𝑛) denote the space of all holomorphic functions in 𝔹^𝑛. Given 0<𝑟<1, 0<𝑝<∞, and 𝑓∈𝐻(𝔹^𝑛), we define 𝑀_𝑝𝑟,𝑓=_𝕊𝑛𝑓(𝑟𝜁)^𝑝𝑑𝜎(𝜁)^1/𝑝. (1.5) When 𝑝=∞, we write 𝑀_∞𝑟,𝑓=sup𝑓(𝑟𝜁)∶𝜁∈𝕊^𝑛. (1.6) For 0<𝑝≤∞, the Hardy space 𝐻^𝑝 consists of all functions 𝑓∈𝐻(𝔹^𝑛) such that 𝑓_𝐻𝑝=sup_0<𝑟<1𝑀_𝑝𝑟,𝑓<∞. (1.7) See [19] for basic information about the Hardy spaces.

We denote by ℛ𝑓 the radial derivative of 𝑓 in 𝐻(𝔹^𝑛) defined by ℛ𝑓=^𝑛_𝑗=1𝑧_𝑗𝜕𝜕𝑧_𝑗𝑓. (1.8) We consider the space Λ^𝑝_𝑠 of holomorphic functions on 𝔹^𝑛 such that _𝕊𝑛ℛ^𝑚𝑓(𝑟𝜁)^𝑝𝑑𝜎^1/𝑝≲1(1−𝑟)^𝑚−𝑠 (1.9) for 𝑚>𝑠. We define the norm of Λ^𝑝_𝑠 as follows: 𝑓_Λ^𝑝_𝑠=𝑓_𝐻𝑝+sup_0<𝑟<1(1−𝑟)^𝑚−𝑠𝑀_𝑝𝑟,ℛ^𝑚𝑓. (1.10) 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 Λ_𝑠; see [19].

We adapt the first order 𝐻^𝑝 mean variation defined as follows: 𝜔_𝑝𝑡,𝑓=sup_𝕊𝑛𝑓(𝑈𝜁)−𝑓(𝜁)^𝑝𝑑𝜎(𝜁)^1/𝑝∶𝑈∈𝒰,‖𝑈−𝐼‖≤𝑡, (1.11) where 𝒰 denotes the group of all unitary operators on ℂ^𝑛, 𝐼 denotes the identity of 𝒰, and ‖𝑈−𝐼‖∶=sup_𝜁∈𝑆|𝑈𝜁−𝜁|. Then, we have 𝑓_Λ^𝑝_𝑠≈𝑓_𝐻𝑝+sup_0<𝑡<1𝜔_𝑝𝑡,𝑓𝑡^𝑠 (1.12) for 0<𝑠<1 (see [20]). This justifies our usage of the term holomorphic mean Lipschitz space for Λ^𝑝_𝑠 with 0<𝑠<1. Now, for 𝑠≥1, we consider the second-order 𝐻^𝑝 mean variation defined as follows: 𝜔^∗_𝑝𝑡,𝑓=sup_𝕊𝑛𝑓(𝑈𝜁)−2𝑓(𝜁)+𝑓𝑈⁻¹𝜁^𝑝𝑑𝜎(𝜁)^1/𝑝∶𝑈∈𝒰,‖𝑈−𝐼‖≤𝑡. (1.13) It was shown in [21] that, if 0<𝑠<2,1≤𝑝<∞, and 𝑓∈𝐻^𝑝,   then 𝑓_Λ^𝑝_𝑠≈𝑓_𝐻𝑝+sup_0<𝑡<1𝜔^∗_𝑝𝑡,𝑓𝑡^𝑠. (1.14)

Theorem 1.1. Let 1≤𝑝<∞ and 𝑠>0. (i) If 𝑠>𝑛/𝑝 (regular), then Λ^𝑝_𝑠 is a multiplicative algebra. (ii) If 𝑠≤𝑛/𝑝 (nonregular), then Λ^𝑝_𝑠 is not a multiplicative algebra.

By (ii) of Theorem 1.1, the space Λ^𝑝_𝑛/𝑝 is not a multiplicative algebra. We give some sufficient conditions for a holomorphic function to be a pointwise multiplier of Λ^𝑝_𝑛/𝑝 as follows. We do not know if our sufficient condition is also necessary.

Theorem 1.2. Let 1≤𝑝<𝑞<∞. Then, Λ^𝑞_𝑛/𝑝⊂ℳ(Λ^𝑝_𝑛/𝑝).

Throughout the paper, we write 𝑋≲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 algebra 𝐴(𝔹^𝑛) is the class of all functions 𝑓∶𝔹^𝑛→ℂ that are continuous on the closed ball 𝔹^𝑛 and that are holomorphic in its interior 𝔹^𝑛.

Proposition 2.1. Let 1≤𝑝<∞ and 𝑠>0. (i)There is a function in the ball algebra 𝐴(𝔹^𝑛) that is not in Λ^𝑝_𝑠. (ii) If 𝑠≤𝑛/𝑝, then there is a function in Λ^𝑝_𝑠 that is not in 𝐻^∞. (iii) If 𝑠>𝑛/𝑝, then Λ^𝑝_𝑠⊂Λ_{𝑠−𝑛/𝑝}⊂𝐴(𝔹^𝑛).

Remark 2.2. By (ii) and (iii), we can see that Λ^𝑝_𝑠⊂𝐻^∞ if and only if 𝑠>𝑛/𝑝.

Proof. (i) Let (𝑝_𝑘) be a sequence of Ryll and Wojtaszczyk [22] homogeneous polynomials in the unit sphere of 𝐻^∞(𝔹^𝑛) such that 𝑝_𝑘 has degree 𝑘 and 𝑝_𝑘𝐻2≥√𝜋2^𝑛. (2.1) Let 1𝑓=𝑘²𝑝_2𝑘. (2.2) 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 on 𝔹^𝑛, its sum 𝑓 is therefore in the ball algebra. It is enough to prove that 𝑓∉Λ¹_𝑠 for 0<𝑠<1. If 𝑓∈Λ¹_𝑠, then ℛ𝑓(_𝕊𝑛𝑟𝜁)𝑝_2𝑘(𝜁)𝑑𝜎(𝜁)≤_𝕊𝑛((𝜁)≲1ℛ𝑓𝑟𝜁)𝑑𝜎(1−𝑟)^1−𝑠,0<𝑟<1. (2.3) However, since the polynomials 𝑝_𝑘 are orthogonal, for any 0<𝑟<1, ℛ𝑓(_𝕊𝑛𝑟𝜁)𝑝_2𝑘(𝜁)𝑑𝜎(𝜁)=_𝕊𝑛2^𝑘𝑘²𝑝_2𝑘(𝑟𝜁)𝑝_2𝑘(𝜁)𝑑𝜎(𝜁)=_𝕊𝑛2^𝑘𝑘²𝑟^2𝑘𝑝_2𝑘(𝜁)²(𝜁)=2𝑑𝜎_𝑘𝑘²𝑟^2𝑘𝑝_2𝑘²_𝐻2≥2^𝑘𝑘²𝑟^2𝑘𝜋4^𝑛. (2.4) Take 𝑟_𝑘1=1−2^𝑘. (2.5) Then, ℛ𝑓𝑟_𝕊𝑛𝑘𝜁𝑝_2𝑘(𝜁)(𝜁≥1𝑑𝜎)1−𝑟_𝑘1log₂1/1−𝑟_𝑘²𝑟^2𝑘_𝑘𝜋4^𝑛=11−𝑟_𝑘^1−𝑠𝑐_𝑘, (2.6) where 𝑐_𝑘=11−𝑟_𝑘^𝑠1log₂1/1−𝑟_𝑘²𝑟^2𝑘_𝑘𝜋4^𝑛⟶∞as𝑘⟶∞. (2.7) By (2.3) and (2.6), it is a contradiction. Thus, 𝑓∉Λ¹_𝑠 for 0<𝑠<1.
(ii) It is clear, if we consider the function 𝑓(𝑧)=log1−𝑧,𝜁 (2.8) with 𝜁∈𝕊^𝑛.
(iii) Let 𝑓∈Λ^𝑝_𝑠. 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 ℛ^𝑚+1𝑓𝜌𝑧=_𝕊𝑛ℛ^𝑚+1𝑓𝜌𝜁1−𝑧,𝜁^𝑛𝑑𝜎(𝜁),𝑧∈𝔹^𝑛. (2.9) Making a change of variables and replacing 𝜌𝑧 by 𝑧, we get ℛ^𝑚+1𝑓(𝑧)=𝜌_𝜁=𝜌ℛ^𝑚+1𝑓(𝜁)𝜌²−𝑧,𝜁^𝑛𝑑𝜎(𝜁),|𝑧|<𝜌. (2.10)
By (2.10) and Hölder's inequality, we have ℛ^𝑚+1𝑓(𝑧)≲𝑀_𝑝𝜌,ℛ^𝑚+1𝑓1𝜌²−|𝑧|^2𝑛/𝑝. (2.11) Take 𝜌=(1+|𝑧|)/2. Then, we obtain ℛ^𝑚+1𝑓(𝑧)≲𝑀_𝑝𝜌,ℛ^𝑚+1𝑓11−|𝑧|^2𝑛/𝑝≲11−𝜌^1−𝛼11−|𝑧|^2𝑛/𝑝≲11−|𝑧|^{2𝑚+1−(𝑠−𝑛/𝑝)}. (2.12) Therefore, 𝑓∈Λ_{𝑠−𝑛/𝑝}⊂𝐴(𝔹^𝑛) if 𝑠>𝑛/𝑝.

Proposition 2.3. Let 𝑠>0 and 1≤𝑝,𝑞<∞. (i)Λ^𝑝_𝑠⊂𝐻^𝑞,0≤𝑛(1/𝑝−1/𝑞)<𝑠. (ii)Λ^𝑝_𝑛/𝑝⊂⋂_0<𝑞<∞𝐻^𝑞.

Proof. (i) Let 1/2<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 𝑀_𝑞𝑟,𝑓≲sup_|𝑧|<1/2𝑓(𝑧)+^𝑟₀𝑀_𝑞𝑡,ℛ𝑓𝑑𝑡. (2.13) Applying this repeatedly and using Fubini's theorem, we obtain 𝑀_𝑞𝑟,𝑓≲sup_|𝑧|<1/2𝑓(𝑧)+^𝑟₀(𝑟−𝑡)^𝑚𝑀_𝑞𝑡,ℛ^𝑚+1𝑓𝑑𝑡≲sup_|𝑧|<1/2𝑓(𝑧)+¹₀(1−𝑡)^𝑚𝑀_𝑞𝑡,ℛ^𝑚+1𝑓𝑑𝑡. (2.14)
Let 0<𝜌<1. By (2.11), we have ℛ^𝑚+1𝑓(𝑧)≲𝑀_𝑝𝜌,ℛ^𝑚+1𝑓𝜌²−|𝑧|^{2−𝑛/𝑝},|𝑧|<𝜌. (2.15) For 0<𝑟<1, we take 𝜌=(1+𝑟)/2. Then, 𝑀_∞𝑟,ℛ^𝑚+1𝑓≲𝑀_𝑝𝜌,ℛ^𝑚+1𝑓(1−𝑟)^−𝑛/𝑝. (2.16) For 𝑞≥𝑝, we have 𝑀_𝑞𝑟,ℛ^𝑚+1𝑓=_𝕊ℛ^𝑚+1𝑓(𝑟𝜁)^𝑝ℛ^𝑚+1𝑓(𝑟𝜁)^𝑞−𝑝𝑑𝜎(𝜁)^1/𝑞≲𝑀_∞𝑟,ℛ^𝑚+1𝑓^{1−𝑝/𝑞}𝑀_𝑝𝑟,ℛ^𝑚+1𝑓^𝑝/𝑞≲𝑀_𝑝𝜌,ℛ^𝑚+1𝑓^{1−𝑝/𝑞}(1−𝑟)^{−𝑛/𝑝(1−𝑝/𝑞)}𝑀_𝑝𝑟,ℛ^𝑚+1𝑓^𝑝/𝑞≲𝑓_Λ^𝑝_𝑠(1−𝑟)^{𝑛(1/𝑞−1/𝑝)−1+𝛼}. (2.17)
By (2.14) and (2.17), we have 𝑀_𝑞𝑟,𝑓≲𝑓_Λ^𝑝_𝑠¹₀1(1−𝑡)^{𝑛(1/𝑝−1/𝑞)+1−𝛼−𝑚}𝑑𝑡<∞, (2.18) since 𝑛(1/𝑝−1/𝑞)+1−𝛼−𝑚<𝑠+1−𝛼−𝑚=1. Thus, we get the result.
(ii) If 𝑠=𝑛/𝑝 in (2.18), then, for any 𝑝<𝑞<∞, we have 𝑀_𝑞𝑟,𝑓≲𝑓_Λ^𝑝_𝑛/𝑝^𝑟₀1(1−𝑡)^{1−𝑛/𝑞}≲𝑓𝑑𝑡_{Λ𝑝𝑛/𝑝}. (2.19)

Remark 2.4. The obvious question: Is Λ^𝑝_𝑛/𝑝 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 for 𝐵𝑀𝑂𝐴 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 𝔹^𝑛 (see [19]). Even though we cannot prove that (1−|𝑧|²)|ℛ𝑓(𝑧)|²𝑑𝑉(𝑧) is a Carleson measure for 𝑓∈Λ^𝑝_𝑛/𝑝, 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∈Λ^𝑝_𝑛/𝑝 and 1≤𝑝<∞. By (2.17), we have Λ^𝑝_𝑛/𝑝⊂Λ^𝑞_𝑛/𝑞. This means that 𝑀_𝑞≲1𝑟,ℛ𝑓(1−𝑟)^{1−𝑛/𝑞}. (2.20) Hence, we have 1−|_𝔹𝑛𝑧|^2𝑞−1ℛ𝑓(𝑧)^𝑞1−|𝑎|^2𝑛|1−⟨𝑎,𝑧⟩|^2𝑛(𝑧)≲𝑑𝑉₁₀(1−𝑟)^𝑞−11−|𝑎|^2𝑛(1−𝑟|𝑎|)^2𝑛𝑀^𝑞_𝑞≲|𝑎|𝑟,ℛ𝑓𝑑𝑟1−^2𝑛1₀(1−𝑟)^𝑛−11(1−𝑟|𝑎|)^2𝑛𝑑𝑟≲1. (2.21) Thus, (1−|𝑧|²)^𝑞−1|ℛ𝑓(𝑧)|^𝑞𝑑𝑉(𝑧) is a Carleson measure (see [19]). However, the embedding problem Λ^𝑝_𝑛/𝑝⊂𝐵𝑀𝑂𝐴 is still open.

3. Regular Cases

We need an elementary variant of Hölder's inequality.

Lemma 3.1. Let 0<𝑝,𝑞,𝛿<∞ with 1/𝑝=1/𝑞+1/𝛿. Then, for 𝑓∈𝐿^𝑞 and 𝑔∈𝐿^𝛿, the product 𝑓𝑔 is in 𝐿^𝑝 and 𝑓𝑔_𝐿𝑝≤𝑓_𝐿𝑞𝑔_𝐿𝛿. (3.1)

Theorem 3.2. If 𝑠>𝑛/𝑝, then Λ^𝑝_𝑠 is a multiplicative algebra.

Proof. Let 𝑚 be the greatest integer less than 𝑠 and 𝛼=𝑠−𝑚. Let 𝑓,𝑔∈Λ^𝑝_𝑠. We will prove that 𝑀_𝑝𝑟,ℛ^𝑚+1≲1𝑓𝑔(1−𝑟)^1−𝛼. (3.2) We note that ℛ^𝑚+1𝑓𝑔=^𝑚+1_𝑗=0𝑐_𝑗ℛ^𝑗𝑓ℛ^{𝑚+1−𝑗}𝑔. (3.3) Since 𝑠>𝑛/𝑝, by (iii) of Proposition 2.1, we have Λ^𝑝_𝑠⊂𝐴(𝔹^𝑛)⊂𝐻^∞.
Let 0≤𝑘≤𝑚+1. Then, 𝑀_{(𝑚+1)𝑝/𝑘}𝑟,ℛ^𝑘𝑓≤𝐶𝑓_𝐻^{1−𝑘/(𝑚+1)}_∞𝑀_𝑝𝑟,ℛ^𝑚+1𝑓^{𝑘/(𝑚+1)}, (3.4) 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 𝑀_{(𝑚+1)𝑝/𝑗}𝑟,ℛ^𝑗𝑓≤𝐶𝑓_𝐻^{1−𝑗/(𝑚+1)}_∞𝑀_𝑝𝑟,ℛ^𝑚+1𝑓^{𝑗/(𝑚+1)},𝑀_{(𝑚+1)𝑝/(𝑚+1−𝑗)}𝑟,ℛ^{𝑚+1−𝑗}𝑔≤𝐶𝑔_𝐻^{1−(𝑚+1−𝑗)/(𝑚+1)}_∞𝑀_𝑝𝑟,ℛ^𝑚+1𝑔^{(𝑚+1−𝑗)/(𝑚+1)}. (3.5) Therefore, by a variant of Hölder's inequality, 𝑀_𝑝𝑟,ℛ^𝑚+1𝑓𝑔≲^𝑚+1_𝑗=0𝑀_{(𝑚+1)𝑝/𝑗}𝑟,ℛ^𝑗𝑓𝑀_{(𝑚+1)𝑝/(𝑚+1−𝑗)}𝑟,ℛ^{𝑚+1−𝑗}𝑔≲𝑓_𝐻^{1−𝑗/(𝑚+1)}_∞𝑔_𝐻^{1−(𝑚+1−𝑗)/(𝑚+1)}_∞𝑀_𝑝𝑟,ℛ^𝑚+1𝑓^{𝑗/(𝑚+1)}𝑀_𝑝𝑟,ℛ^𝑚+1𝑔^{(𝑚+1−𝑗)/(𝑚+1)}≲𝑓_𝐻^{1−𝑗/(𝑚+1)}_∞𝑔_𝐻^{1−(𝑚+1−𝑗)/(𝑚+1)}_∞𝑓_Λ^{𝑗/(𝑚+1)}_𝑝𝑠𝑔_Λ^{(𝑚+1−𝑗)/(𝑚+1)}_𝑝𝑠1(1−𝑟)^1−𝛼≲𝑓_Λ^𝑝_𝑠𝑔_Λ^𝑝_𝑠1(1−𝑟)^1−𝛼. (3.6) That is, 𝑓𝑔_Λ^𝑝_𝑠≲𝑓_Λ^𝑝_𝑠𝑔_Λ^𝑝_𝑠. (3.7)

4. Nonregular Cases

A function 𝜑 is a multiplier for Λ^𝑝_𝑠 if the multiplication operator 𝑀_𝜑𝑓=𝜑𝑓 is continuous from Λ^𝑝_𝑠 to itself. The space of those multipliers will be denoted by ℳ(Λ^𝑝_𝑠).

Let 𝜑∈ℳ(Λ^𝑝_𝑠). Since 1∈Λ^𝑝_𝑠, we have 𝜑=𝜑⋅1∈Λ^𝑝_𝑠. Thus, ℳΛ^𝑝_𝑠⊂Λ^𝑝_𝑠. (4.1) Hence, ℳΛ^𝑝_𝑠⊂𝐻𝔹^𝑛. (4.2)

The following is a special case of Lemma 5.1 in [7].

Lemma 4.1 (see [7]). Let 𝑠>0. Then, 𝜑∈ℳ(Λ^𝑝_𝑠) is bounded pointwise by its multiplier norm.

Corollary 4.2. Let 𝑠>0. Then, we have ℳΛ^𝑝_𝑠⊊𝐻^∞. (4.3)

Proof. Let 𝜑∈ℳ(Λ^𝑝_𝑠). Since 1∈Λ^𝑝_𝑠, we have 𝜑=𝜑⋅1∈Λ^𝑝_𝑠. Thus, ℳΛ^𝑝_𝑠⊂Λ^𝑝_𝑠. (4.4) Hence, ℳΛ^𝑝_𝑠⊂𝐻𝔹^𝑛. (4.5) Thus, by Lemma 4.1, we get ℳ(Λ^𝑝_𝑠)⊂𝐻^∞. By (i) of Proposition 2.1, there is a function 𝜓∈𝐴(𝔹^𝑛)⧵Λ^𝑝_𝑠⊂𝐻^∞⧵Λ^𝑝_𝑠. Since ℳ(Λ^𝑝_𝑠)⊂Λ^𝑝_𝑠, it follows that 𝜓∉ℳ(Λ^𝑝_𝑠). Thus, we get the result.

Theorem 4.3. Let 0<𝑠≤𝑛/𝑝. Then, Λ^𝑝_𝑠 is not a multiplicative algebra.

Proof. If Λ^𝑝_𝑠 is multiplicative, then Λ^𝑝_𝑠=ℳΛ^𝑝_𝑠⊊𝐻^∞. (4.6) By (ii) of Proposition 2.1, this is a contradiction.

5. Some Sufficient Conditions for a Nonregular Case

Lemma 5.1. Let 𝑓∈Λ^𝑝_𝑠. Let 𝑚 be the integer part of 𝑠 and 𝛼=𝑠−𝑚. Let 0<𝑟<1. Then, for 𝑞>𝑝, we have 𝑀_𝑞𝑟,ℛ^𝑚+1𝑓≲𝑓_Λ^𝑝_𝑠(1−𝑟)^{𝑛(1/𝑞−1/𝑝)−1+𝛼}. (5.1)

Proof. We proved this inequality at (2.17) in the proof of (i) of Proposition 2.3.

Lemma 5.2. Let 0≤𝑙<𝑘 and 1/2<𝑟<1. Then, we have 𝑀_𝑝𝑟,ℛ^𝑙𝑓≲sup_|𝑧|<1/2𝑓(𝑧)+^𝑟₀(𝑟−𝑡)^{𝑘−𝑙−1}𝑀_𝑝𝑡,ℛ^𝑘𝑓𝑑𝑡. (5.2)

Proof. We proved this inequality at (2.14) in the proof of (i) of Proposition 2.3.

Theorem 5.3. Let 1≤𝑝<𝑞<∞. Then, Λ^𝑞_𝑛/𝑝⊂ℳ(Λ^𝑝_𝑛/𝑝).

Proof. Let 𝜑∈Λ^𝑞_𝑛/𝑝 and 𝑓∈Λ^𝑝_𝑛/𝑝. Let 𝑚 be the integer part of 𝑛/𝑝 and 𝛼=𝑛/𝑝−𝑚. It is enough to prove that 𝑀_𝑝𝑟,ℛ^𝑚+1≲1𝜑𝑓(1−𝑟)^1−𝛼,0<𝑟<1. (5.3)
We note that ℛ^𝑚+1𝜑𝑓=^𝑚+1_𝑗=0𝑐_𝑗ℛ^𝑗𝜑ℛ^{𝑚+1−𝑗}𝑓. (5.4) Thus, 𝑀_𝑝𝑟,ℛ^𝑚+1𝜑𝑓≲^𝑚+1_𝑗=0𝑀_𝑝𝑟,ℛ^𝑗𝜑ℛ^{𝑚+1−𝑗}𝑓. (5.5)
Let 𝑚=0. Then, 𝑀_𝑝𝑟,ℛ𝜑𝑓≤𝑀_𝑝𝑟,ℛ𝜑𝑓+𝑀_𝑝𝑟,𝜑ℛ𝑓. (5.6) By Proposition 2.1, we have 𝜑∈Λ^𝑞_𝑛/𝑝⊂Λ_{𝑛/𝑝−𝑛/𝑞}⊂𝐻^∞,𝑓∈Λ^𝑝_𝑛/𝑝⊂_𝛿>0𝐻^𝛿. (5.7) Choose 𝛿>1 with 1/𝑝=1/𝑞+1/𝛿. Then, 𝑀_𝑝𝑟,ℛ𝜑𝑓≤𝑀_𝑞𝑟,ℛ𝜑𝑀_𝛿𝑟,𝑓≲𝑀_𝑞𝑟,ℛ𝜑𝑓_𝐻𝛿≲𝑀_𝑞𝑟,ℛ𝜑𝑓_Λ^𝑝_𝑛/𝑝≲𝜑_Λ^𝑞_𝑛/𝑝𝑓_Λ^𝑝_𝑛/𝑝1(1−𝑟)^1−𝛼. (5.8) Also, 𝑀_𝑝𝑟,𝜑ℛ𝑓≲𝜑_𝐻∞𝑀_𝑝≲𝜑𝑟,ℛ𝑓_𝐻∞𝑓_Λ^𝑝_𝑛/𝑝1(1−𝑟)^1−𝛼. (5.9) Thus, 𝜑𝑓_Λ^𝑝_𝑛/𝑝≲𝜑_Λ^𝑞_𝑛/𝑝𝑓_Λ^𝑝_𝑛/𝑝. (5.10)
Now, let 𝑚≥1.
If 𝑗=0, then, since 𝜑∈𝐻^∞ and 𝑓∈Λ^𝑝_𝑛/𝑝, 𝑀_𝑝𝑟,𝜑ℛ^𝑚+1𝑓≲𝜑_𝐻∞𝑀_𝑝𝑟,ℛ^𝑚+1𝑓≲𝜑_𝐻∞𝑓_Λ^𝑝_𝑛/𝑝1(1−𝑟)^1−𝛼. (5.11)
If 𝑗=𝑚+1, we choose 𝛿>𝑝 such that 1𝑝=1𝑞+1𝛿. (5.12) Then, 𝑀_𝑝𝑟,ℛ^𝑚+1𝜑𝑓≤𝑀_𝑞𝑟,ℛ^𝑚+1𝜑𝑀_𝛿𝑟,𝑓. (5.13) By (ii) of Proposition 2.3, Λ^𝑝_𝑛/𝑝⊂⋂_𝛿>0𝐻^𝛿. By a variant of Hölder's inequality, we have 𝑀_𝑝𝑟,ℛ^𝑚+1𝜑𝑓≲𝑀_𝑞𝑟,ℛ^𝑚+1𝜑𝑓_𝐻𝛿≲𝑀_𝑞𝑟,ℛ^𝑚+1𝜑𝑓_Λ^𝑝_𝑛/𝑝≲𝜑_Λ^𝑞_𝑛/𝑝𝑓_Λ^𝑝_𝑛/𝑝1(1−𝑟)^1−𝛼. (5.14)
Now, we consider the case 1≤𝑗≤𝑚. Since Λ^𝑞_𝑛/𝑝 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 𝜇_𝑗>0 such that 𝑛𝜇_𝑗𝑛=𝑗−𝑝−𝑛𝑞. (5.15) Then, 𝑛1𝑞−1𝜇_𝑗=𝑛𝑝−𝑗>0. (5.16) Since 𝑝<𝑞<𝜇_𝑗, we can choose 𝛿_𝑗>0 such that 1𝑝=1𝜇_𝑗+1𝛿_𝑗,𝑗=1,2,…,𝑚. (5.17)
By a variant of Hölder's inequality, we have 𝑀_𝑝𝑟,ℛ^𝑗𝜑ℛ^{𝑚+1−𝑗}𝑓≤𝑀_𝜇𝑗𝑟,ℛ^𝑗𝜑𝑀_𝛿𝑗𝑟,ℛ^{𝑚+1−𝑗}𝑓. (5.18) By Lemmas 5.2 and 5.1, 𝑀_𝜇𝑗𝑟,ℛ^𝑗𝜑≲sup_|𝑧|<1/2𝜑(𝑧)+^𝑟₀(𝑟−𝑡)^𝑚−𝑗𝑀_𝜇𝑗𝑡,ℛ^𝑚+1𝜑≲𝜑𝑑𝑡_{Λ𝑞𝑠𝑟0}(𝑟−𝑡)^𝑚−𝑗(1−𝑡)^{𝑛(1/𝑞−1/𝜇𝑗)+1−𝛼}𝑑𝑡. (5.19) Here, ^𝑟₀(𝑟−𝑡)^𝑚−𝑗(1−𝑡)^{𝑛(1/𝑞−1/𝜇𝑗)+1−𝛼}𝑑𝑡≲^𝑟₀111−𝑡𝑑𝑡≲log.1−𝑟 (5.20) Moreover, by Lemmas 5.2 and 5.1 again, we have 𝑀_𝛿𝑗𝑟,ℛ^{𝑚+1−𝑗}𝑓≲sup_|𝑧|<1/2𝑓(𝑧)+^𝑟₀(𝑟−𝑡)^𝑗−1𝑀_𝛿𝑗𝑡,ℛ^𝑚+1𝑓≲𝑓𝑑𝑡_{Λ𝑝𝑛/𝑝𝑟0}(𝑟−𝑡)^𝑗−1(1−𝑡)^{𝑛(1/𝑝−1/𝛿𝑗)+1−𝛼}≲𝑓𝑑𝑡_{Λ𝑝𝑛/𝑝}1(1−𝑟)^{1−𝛼−(𝑛/𝑝−𝑛/𝑞)}. (5.21) Thus, it follows that 𝑀_𝑝𝑟,ℛ^𝑗𝜑ℛ^{𝑚+1−𝑗}𝑓≲𝜑_Λ^𝑝_𝑛/𝑝𝑓_Λ^𝑝_𝑛/𝑝1(1−𝑟)^1−𝛼𝑐_𝑟, (5.22) where 𝑐_𝑟1=log()1−𝑟1−𝑟^{𝑛/𝑝−𝑛/𝑞}⟶0as𝑟⟶1. (5.23) Thus, 𝑀_𝑝𝑟,ℛ^𝑗𝜑ℛ^{𝑚+1−𝑗}𝑓≲1(1−𝑟)^1−𝛼. (5.24) This completes the proof.

Acknowledgment

The author wishes to thank the referee for some very helpful suggestions and comments that improved the presentation of the paper. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (NRF-2009-0073976).