Compact Weighted Composition Operators and Multiplication Operators between Hardy Spaces

Abstract

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

1. Introduction

Let 𝑁 be a fixed integer. Let 𝐵_𝑁 denote the unit ball of ℂ^𝑁 and let 𝐻(𝐵_𝑁) denote the space of all holomorphic functions in 𝐵_𝑁. For each 𝑝,1≤𝑝<∞, the Hardy space   𝐻^𝑝(𝐵_𝑁) is defined by𝐻^𝑝𝐵^𝑁={𝑓∈𝐻𝐵^𝑁∶sup_0<𝑟<1∫_𝜕𝐵𝑁|||𝑓(𝑟𝜁)|||^𝑝‖‖𝑓‖‖𝑑𝜎(𝜁)<∞},_𝑝=∫_𝜕𝐵𝑁|||𝑓_∗(𝜁)|||^𝑝𝑑𝜎(𝜁)^1/𝑝,(1.1)where 𝑑𝜎 is the normalized Lebesgue measure on the boundary 𝜕𝐵_𝑁 of 𝐵_𝑁.

For a given holomorphic self-map 𝜑 of 𝐵_𝑁 and holomorphic function 𝑢 in 𝐵_𝑁, the weighted composition operator 𝑢𝐶_𝜑 is defined by 𝑢𝐶_𝜑𝑓=𝑢(𝑓0𝑥025𝑒6𝜑). In particular, if 𝑢 is the constant function 1, then 𝑢𝐶_𝜑 becomes the composition operator 𝐶_𝜑. In the special case that 𝜑 is the identity mapping of 𝐵_𝑁, 𝑢𝐶_𝜑 is called the multiplication operator and is denoted by 𝑀_𝑢.

Let 𝑋 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,‖‖𝑇‖‖_{𝑒,𝑋→𝑌}=inf{‖‖𝑇−𝒦‖‖∶𝒦iscompactfrom𝑋into𝑌},(1.2) 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 𝐵_𝑁. 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 𝑢𝐶_𝜑 from 𝐻^𝑝(𝐵₁) into 𝐻^𝑞(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 𝑢𝐶_𝜑. The essential norm of 𝑢𝐶_𝜑∶𝐻^𝑝(𝐵₁)→𝐻^𝑞(𝐵₁) has been studied by ^⌣Cu^⌣cković and Zhao [13, 14]. In the higher-dimensional case, Ueki [15] characterized the boundedness and compactness of 𝑢𝐶_𝜑∶𝐻^𝑝(𝐵_𝑁)→𝐻^𝑞(𝐵_𝑁), in terms of the pull-back measure and the integral operator. The purpose of this paper is to estimate the essential norm of 𝑢𝐶_𝜑∶𝐻^𝑝(𝐵_𝑁)→𝐻^𝑞(𝐵_𝑁). The following theorem is our main result.

Main 1 Theorem. Let 1<𝑝≤𝑞<∞. If 𝑢𝐶_𝜑is a bounded weighted composition operator from 𝐻^𝑝(𝐵_𝑁)into 𝐻^𝑞(𝐵_𝑁), then‖‖𝑢𝐶_𝜑‖‖^{𝑞𝑒,𝐻}_{𝑝→𝐻𝑞}∼l||imsup_𝑤||_→1⁻_𝜕𝐵𝑁|||𝑢_∗(𝜁)|||^𝑞{1−||𝑤||²|||1−𝜑_∗(𝜁),𝑤|||²∼limsup_𝑡→0+}^{𝑞𝑁/𝑝}𝑑𝜎(𝜁)sup_{𝜁∈𝜕𝐵𝑁}𝜇_𝜑,𝑢𝑆(𝜁,𝑡)𝑡^{𝑞𝑁/𝑝},(1.3)where 𝜇_𝜑,𝑢 is the pull-back measure induced by 𝜑 and 𝑢, 𝑆(𝜁,𝑡) is the Carleson set of 𝐵_𝑁, 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 ‖𝑢𝐶_𝜑‖_𝑒 in above theorem may be found in the work [14] by ^⌣Cu^⌣cković and Zhao. In the case 𝑝=𝑞=2 and 𝑢=1, Choe [1] and Luo [16] showed that the essential norm ‖𝐶_𝜑‖_𝑒 is comparable to the value limsup_𝑡→0⁺sup_{𝜁∈𝜕𝐵𝑁}(𝜇_𝜑(𝑆(𝜁,𝑡))/𝑡^𝑁).

We give the proof of main theorem in Section 3. 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 symbol 𝐶 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 each 𝑢∈𝐻^𝑞(𝐵_𝑁), we can define a finite positive Borel measure 𝜇_𝜑,𝑢 on 𝐵_𝑁 by𝜇_𝜑,𝑢(𝐸)=_{𝜑∗−1(𝐸)}|||𝑢_∗|||^𝑞𝑑𝜎(∀Borelsets𝐸of𝐵_𝑁),(2.1)where 𝜑^∗ denotes the radial limit map of the mapping 𝜑 considered as a map of 𝜕𝐵_𝑁→𝐵_𝑁. A change-of-variable formula from measure theory shows that_𝐵𝑁𝑔𝑑𝜇_𝜑,𝑢=_𝜕𝐵𝑁|||𝑢_∗|||^𝑞𝑔0𝑥025𝑒6𝜑^∗𝑑𝜎,(2.2)for each nonnegative measurable function 𝑔 on 𝐵_𝑁. This type of pull-back measure played an important role in past studies of composition operators on Hardy spaces of 𝐵_𝑁.

For each 𝜁∈𝜕𝐵_𝑁 and 𝑡>0, let𝐵(𝜁,𝑡)𝑆=(𝜁𝑆,(𝑡𝜁),=𝑡){∩𝑧𝐵∈_𝑁𝐵_𝑁∶|||1−⟨𝑧,𝜁⟩|||<𝑡},,𝑄(𝜁,𝑡)=𝑆(𝜁,𝑡)∩𝜕𝐵_𝑁.(2.3)It is well known that 𝜎(𝑄(𝜁,𝑡)) is comparable to 𝑡^𝑁 ([17, page 67]).

The proof of the following lemma is essentially the same as that of Power's theorem in [18].

Lemma 2.1. Let 1≤𝛼<∞. Suppose that 𝜇 is a positive Borel measure on 𝐵_𝑁 and that there exists a constant 𝐶>0 such that 𝜇𝐵(𝜁,𝑡)≤𝐶𝑡^𝛼𝑁𝜁∈𝜕𝐵^𝑁,𝑡>0.(2.4)Then there exists a constant 𝐾>0 such that _𝐵𝑁||𝑓||^𝑝𝛼𝑑𝜇^1/𝑝𝛼≤𝐾‖‖𝑓‖‖_𝐻𝑝𝑓∈𝐻^𝑝𝐵^𝑁.(2.5)

Proof. Fix 𝑓∈𝐻^𝑝(𝐵_𝑁) 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 that𝜇{𝑧∈𝐵_𝑁∶|||𝑓(𝑧)|||≥𝑠}≤𝐶𝜎{𝜁∈𝜕𝐵_𝑁∶𝑀𝑓(𝜁)≥𝑠}^𝛼,(2.6)where 𝑀𝑓 is the admissible maximal function of 𝑓 which is defined by𝑀𝑓(𝜁)=sup{|||𝑓(𝑧)|||∶𝑧∈ℂ^𝑛,|||1−⟨𝑧,𝜁⟩|||<1−||𝑧||²},(2.7)for 𝜁∈𝜕𝐵_𝑁. By (2.6), we have_𝐵𝑁||𝑓||^𝑝𝛼𝑑𝜇=𝑝𝛼^∞0𝜇{||𝑓||>𝑠}𝑠^𝑝𝛼−1𝑑𝑠≤𝐶𝑝𝛼^∞0𝜎{𝑀𝑓≥𝑠}^𝛼𝑠^𝑝𝛼−1𝑑𝑠.(2.8)Since 𝑓∈𝐻^𝑝(𝐵_𝑁), it follows from [17, Theorem 5.6.5] that𝜎{𝑀𝑓≥𝑠}^𝛼−1𝑠^{𝑝𝛼−𝑝}≤_𝜕𝐵𝑁{𝑀𝑓(𝜁)}^𝑝𝑑𝜎(𝜁)^𝛼−1≤𝐶‖‖𝑓‖‖^{𝐻𝑝(𝛼−1)}_𝑝.(2.9)By (2.8) and (2.9), we have_𝐵𝑁||𝑓||^𝑝𝛼𝑑𝜇≤≤𝐶𝐶‖‖‖‖𝑓𝑓‖‖‖‖^{𝐻𝑝(𝛼−1)}_{𝑝𝐻𝑝(𝛼−1)𝑝}𝑝^∞0𝜎{𝑀𝑓≥𝑠}𝑠^𝑝−1𝑑𝑠_𝜕𝐵𝑁{𝑀𝑓(𝜁)}^𝑝𝑑𝜎(𝜁)≤𝐶‖‖𝑓‖‖^𝐻𝑝𝛼_𝑝.(2.10)This completes the proof.

Lemma 2.2. Let 1≤𝛼<∞. Suppose that 𝜇 is a positive Borel measure on 𝜕𝐵_𝑁 such that 𝜇𝑄(𝜁,𝑡)≤𝐶𝑡^𝛼𝑁𝜁∈𝜕𝐵^𝑁,𝑡>0,(2.11)for some constant 𝐶>0.
(a) If 𝛼=1, then there exist a 𝑔∈𝐿^∞(𝜕𝐵_𝑁) 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) If 𝛼>1, then 𝜇≡0 for all Borel sets of 𝜕𝐵_𝑁.

Proof. Part (a) is completely analogous to [19, page 238, Lemma 1.3]. So we only prove part (b). Combining 𝜎(𝑄(𝜁,𝑡))∼𝑡^𝑁 with (2.11), we have𝜇𝑄(𝜁,𝑡)𝜎𝑄(𝜁,𝑡)≤𝐶𝑡^{𝑁(𝛼−1)}(2.12)for all 𝜁∈𝜕𝐵_𝑁 and 𝑡>0. Hence we see that the maximal function 𝑀𝜇 of the positive measure 𝜇 satisfies 𝑀𝜇(𝜁)<∞ for all 𝜁∈𝜕𝐵_𝑁. By [17, page 70, Theorem 5.2.7], we obtain 𝑑𝜇=𝑔𝑑𝜎 for some 𝑔∈𝐿¹(𝜕𝐵_𝑁). By (2.12), we have0≤1𝜎𝑄(𝜁,𝑡)^{𝑄(𝜁,𝑡)}𝑔𝑑𝜎=𝜇𝑄(𝜁,𝑡)𝜎𝑄(𝜁,𝑡)≤𝐶𝑡^{𝑁(𝛼−1)}(2.13)for all 𝜁∈𝜕𝐵_𝑁 and 𝑡>0. Letting 𝑡→0⁺, we see that 𝑔=0 a.e. on 𝜕𝐵_𝑁, and so 𝜇≡0. This completes the proof of part (b).

Combining Lemma 2.1 with Lemma 2.2 and using the same argument as in [19, page 239], we obtain the following lemma.

Lemma 2.3. Let 1<𝑝≤𝑞<∞. Suppose that 𝜇 is a positive Borel measure on 𝐵_𝑁 such that 𝜇𝑆(𝜁,𝑡)≤𝐶𝑡^{𝑞𝑁/𝑝}𝜁∈𝜕𝐵^𝑁,𝑡>0,(2.14)for some constant 𝐶>0. Then, there exists a constant 𝐾>0 such that _𝐵𝑁|||𝑓_∗|||^𝑞𝑑𝜇^{1/𝑞≤𝐾}‖‖_𝑓‖‖_𝐻𝑝,(2.15)for all 𝑓∈𝐻^𝑝(𝐵_𝑁). Here, the notation 𝑓^∗ denotes the function defined on 𝐵_𝑁 by 𝑓^∗=𝑓 in 𝐵_𝑁 and 𝑓^∗=lim_𝑟→1⁻𝑓_𝑟 a.e. [𝜎] on 𝜕𝐵_𝑁.

Remark 2.4. In Lemma 2.3 (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 𝑢𝐶_𝜑.

Proposition 2.5. Let 1<𝑝≤𝑞<∞. Suppose that 𝜑∶𝐵_𝑁→𝐵_𝑁 is a holomorphic map and 𝑢∈𝐻^𝑞(𝐵_𝑁)⧵{0} such that 𝑢𝐶_𝜑∶𝐻^𝑝(𝐵_𝑁)→𝐻^𝑞(𝐵_𝑁) is bounded. Then 𝜑^∗ cannot carry a set of positive 𝜎-measure in 𝜕𝐵_𝑁 into a set of 𝜎-measure 0 in 𝜕𝐵_𝑁.

Proof. Suppose that 𝐸,𝐹⊂𝜕𝐵_𝑁 and 𝜑^∗(𝐸)⊂𝐹 with 𝜎(𝐸)>0 and 𝜎(𝐹)=0. Put 𝜇=𝜇_𝜑,𝑢|_𝜕𝐵𝑁. As in the case of composition operators, it is well known that the boundedness of 𝑢𝐶_𝜑∶𝐻^𝑝(𝐵_𝑁)→𝐻^𝑞(𝐵_𝑁) implies𝜇𝑆(𝜁,𝑡)≤𝐶𝑡^{𝑞𝑁/𝑝}𝜁∈𝜕𝐵^𝑁,𝑡>0,(2.16)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 have0≥𝜇𝜑^∗(𝐸)≡_{𝜑∗−1(𝜑∗(𝐸))}|||𝑢_∗|||^𝑞𝑑𝜎≥_𝐸|||𝑢_∗|||^𝑞𝑑𝜎.(2.17)That is, 𝑢^∗=0 a.e. on 𝐸. Hence [17, page 83, Theorem 5.5.9] gives that 𝑢≡0 in 𝐵_𝑁. This contradicts 𝑢≢0.

Lemma 2.6. Let 1<𝑝≤𝑞<∞ and 𝑓∈𝐻^𝑝(𝐵_𝑁). Suppose that 𝜑∶𝐵_𝑁→𝐵_𝑁 is a holomorphic map and 𝑢∈𝐻^𝑞(𝐵_𝑁)⧵{0} such that 𝑢𝐶_𝜑∶𝐻^𝑝(𝐵_𝑁)→𝐻^𝑞(𝐵_𝑁) is bounded. Then 𝑢^∗(𝑓0𝑥025𝑒6𝜑)^∗=𝑢^∗(𝑓^∗0𝑥025𝑒6𝜑^∗) a.e. [𝜎] on 𝜕𝐵_𝑁. Here the notation 𝑓^∗ is used as in Lemma 2.3.

Proof. Since 𝜑^∗ cannot carry a set of positive measure in 𝜕𝐵_𝑁 into a set of measure 0 in 𝜕𝐵_𝑁 (by Proposition 2.5) and since the radial limit of 𝜑, 𝑓 and 𝜓 exist on a set of full measure in 𝜕𝐵_𝑁, we have lim_𝑟→1⁻𝑢^∗(𝑓_𝑟0𝑥025𝑒6𝜑^∗)=𝑢^∗(𝑓^∗0𝑥025𝑒6𝜑^∗) a.e. [𝜎] on 𝜕𝐵_𝑁.
On the other hand, since 𝑓_𝑟 is in the ball algebra 𝐴(𝐵_𝑁) and 𝑓_𝑟→𝑓 as 𝑟→1⁻ in 𝐻^𝑝(𝐵_𝑁), the boundedness of 𝑢𝐶_𝜑 shows that0≤_𝜕𝐵𝑁|||𝑢_∗≤liminf_𝑟→1−(𝜁)(𝑓0𝑥025𝑒6𝜑)^∗(𝜁)−𝑢^∗(𝜁)𝑓^∗0𝑥025𝑒6𝜑^∗(𝜁)|||^𝑞𝑑𝜎(𝜁)_𝜕𝐵𝑁|||𝑢_∗=liminf_𝑟→1−(𝜁)(𝑓0𝑥025𝑒6𝜑)^∗(𝜁)−𝑢^∗(𝜁)𝑓_𝑟0𝑥025𝑒6𝜑^∗(𝜁)|||^𝑞𝑑𝜎(𝜁)‖‖𝑢𝐶_𝜑𝑓−𝑢𝐶_𝜑𝑓_𝑟‖‖^𝑞𝐻_𝑞=0.(2.18)This implies that 𝑢^∗(𝑓0𝑥025𝑒6𝜑)^∗=𝑢^∗(𝑓^∗0𝑥025𝑒6𝜑^∗) a.e. [𝜎] on 𝜕𝐵_𝑁.

3. Weighted Composition Operators between Hardy Spaces

Theorem 3.1. Let 1<𝑝≤𝑞<∞. If 𝑢𝐶_𝜑 is a bounded weighted composition operator from 𝐻^𝑝(𝐵_𝑁) into 𝐻^𝑞(𝐵_𝑁), then ‖‖𝑢𝐶_𝜑‖‖^{𝑞𝑒,𝐻}_{𝑝→𝐻𝑞}∼l||imsup_𝑤||_→1⁻_𝜕𝐵𝑁|||𝑢_∗(𝜁)|||^𝑞{1−||𝑤||²|||1−𝜑_∗(𝜁),𝑤|||²∼limsup_𝑡→0+}^{𝑞𝑁/𝑝}𝑑𝜎(𝜁)sup_{𝜁∈𝜕𝐵𝑁}𝜇_𝜑,𝑢𝑆(𝜁,𝑡)𝑡^{𝑞𝑁/𝑝}.(3.1)

Proof of the lower estimates. For each 𝑤∈𝐵_𝑁, we define the function 𝑓_𝑤 on 𝐵_𝑁 by𝑓_𝑤(𝑧)={1−||𝑤||²1−⟨𝑧,𝑤⟩²}^𝑁/𝑝.(3.2)Then the functions {𝑓_𝑤∶𝑤∈𝐵_𝑁} belong to the ball algebra 𝐴(𝐵_𝑁) and form a bounded sequence of 𝐻^𝑝(𝐵_𝑁). Take a compact operator 𝒦∶𝐻^𝑝(𝐵_𝑁)→𝐻^𝑞(𝐵_𝑁) arbitrarily. Since the bounded sequence {𝑓_𝑤} converges to 0 uniformly on compact subsets of 𝐵_𝑁 as |𝑤|→1⁻, we have ‖𝒦𝑓_𝑤‖_𝐻^𝑞→0 as |𝑤|→1⁻. Thus we obtain‖‖𝑢𝐶_𝜑−𝒦‖‖_{𝐻𝑝→𝐻𝑞}≥𝐶l||imsup_𝑤||_→1⁻‖‖𝑢𝐶_𝜑−𝒦𝑓_𝑤‖‖_𝐻𝑞≥𝐶l||imsup_𝑤||_→1⁻‖‖𝑢𝐶_𝜑𝑓_𝑤‖‖_𝐻𝑞.(3.3)By the definition of 𝑓_𝑤, we also see that‖‖𝑢𝐶_𝜑𝑓_𝑤‖‖^𝑞𝐻_𝑞=_𝜕𝐵𝑁|||𝑢_∗(𝜁)|||^𝑞{1−||𝑤||²|||1−𝜑_∗(𝜁),𝑤|||²}^{𝑞𝑁/𝑝}𝑑𝜎(𝜁).(3.4)Combining this with (3.3), we get‖‖𝑢𝐶_𝜑−𝒦‖‖^𝑞𝐻_{𝑝→𝐻𝑞}≥𝐶l||imsup_𝑤||_→1⁻_𝜕𝐵𝑁|||𝑢_∗(𝜁)|||^𝑞{1−||𝑤||²|||1−𝜑_∗(𝜁),𝑤|||²}^{𝑞𝑁/𝑝}𝑑𝜎(𝜁).(3.5)Since this holds for every compact operator 𝒦, it follows that‖‖𝑢𝐶_𝜑‖‖^{𝑞𝑒,𝐻}_{𝑝→𝐻𝑞}≥𝐶l||imsup_𝑤||_→1⁻_𝜕𝐵𝑁|||𝑢_∗(𝜁)|||^𝑞{1−||𝑤||²|||1−𝜑_∗(𝜁),𝑤|||²}^{𝑞𝑁/𝑝}𝑑𝜎(𝜁).(3.6)
Furthermore, we put 𝑤=(1−𝑡)𝜁 for each 𝑡,0<𝑡<1 and 𝜁∈𝜕𝐵_𝑁 in the definition of 𝑓_𝑤. Since we see that |𝑓_{(1−𝑡)𝜁}(𝑧)|≥𝐶𝑡^{−𝑞𝑁/𝑝} for all 𝑧∈𝑆(𝜁,𝑡), we have𝐶sup_{𝜁∈𝜕𝐵𝑁}𝜇_𝜑,𝑢𝑆(𝜁,𝑡)𝑡^{𝑞𝑁/𝑝}≤sup_{𝜁∈𝜕𝐵𝑁}_{𝑆(𝜁,𝑡)}|||𝑓_{(1−𝑡)𝜁}|||^𝑞𝑑𝜇_𝜑,𝑢≤sup_{𝜁∈𝜕𝐵𝑁}‖‖𝑢𝐶_𝜑𝑓_{(1−𝑡)𝜁}‖‖^𝑞𝐻_𝑞.(3.7)Letting 𝑡→0⁺, we get𝐶limsup_𝑡→0+sup_{𝜁∈𝜕𝐵𝑁}𝜇_𝜑,𝑢𝑆(𝜁,𝑡)𝑡^{𝑞𝑁/𝑝}≤limsup_𝑡→0+sup_{𝜁∈𝜕𝐵𝑁}‖‖𝑢𝐶_𝜑𝑓_{(1−𝑡)𝜁}‖‖^𝑞𝐻_𝑞.(3.8)Combining this with (3.6), we obtain𝐶limsup_𝑡→0+sup_{𝜁∈𝜕𝐵𝑁}𝜇_𝜑,𝑢𝑆(𝜁,𝑡)𝑡^{𝑞𝑁/𝑝}≤‖‖𝑢𝐶_𝜑‖‖^{𝑞𝑒,𝐻}_{𝑝→𝐻𝑞},(3.9)completing the proof of the lower estimates.

To prove the upper estimates, we need some technical results about the polynomial approximation of 𝑓∈𝐻^𝑝(𝐵_𝑁). Recall that a holomorphic function 𝑓 in 𝐵_𝑁 has the homogeneous expansion𝑓(𝑧)=^∞_𝑘=0||_𝛾||_=𝑘𝑐(𝛾)𝑧^𝛾,(3.10)where 𝛾=(𝛾₁,…,𝛾_𝑁) is a multi-index, |𝛾|=𝛾₁+⋯+𝛾_𝑁, and 𝑧^𝛾=𝑧_1𝛾¹⋯𝑧_𝑁𝛾^𝑁. For the homogeneous expansion of 𝑓 and any integer 𝑛≥1, let𝑅_𝑛𝑓(𝑧)=^∞_𝑘=𝑛||_𝛾||_=𝑘𝑐(𝛾)𝑧^𝛾,(3.11)and 𝐾_𝑛=𝐼−𝑅_𝑛, where 𝐼𝑓=𝑓 is the identity operator.

Proposition 3.2. Suppose that 𝑋 is a Banach space of holomorphic functions in 𝐵_𝑁 with the property that the polynomials are dense in 𝑋. Then ‖𝐾_𝑛𝑓−𝑓‖_𝑋→0 as 𝑛→∞ if and only if sup{‖𝐾_𝑛‖∶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. If 1<𝑝<∞, then ‖𝐾_𝑛𝑓−𝑓‖_𝐻^𝑝→0 as 𝑛→∞ for each 𝑓∈𝐻^𝑝(𝐵_𝑁).

Proof. For each 𝑓∈𝐻^𝑝(𝐵_𝑁) and 𝑟,0<𝑟<1, the slice function (𝑓_𝑟)_𝜁(𝜁∈𝜕𝐵_𝑁) of 𝑓_𝑟 is in the disc algebra 𝐴(𝔻). Here, 𝑓_𝑟 denotes the dilated function of 𝑓, that is 𝑓_𝑟(𝑧)=𝑓(𝑟𝑧). Hence [20, Corollary 3 and Proposition 1] implies that there is a positive constant 𝐶_𝑝 such that12𝜋^𝜋−𝜋|||𝐾_𝑛𝑓^𝑟_𝜁𝑒^𝑖𝜃|||^𝑝𝑑𝜃≤𝐶_𝑝12𝜋^𝜋−𝜋|||𝑓_𝑟_𝜁𝑒^𝑖𝜃|||^𝑝𝑑𝜃,(3.12)for every integer 𝑛≥1. Since 𝐾_𝑛(𝑓_𝑟)_𝜁(𝑒^𝑖𝜃)=𝐾_𝑛𝑓(𝑟𝑒^𝑖𝜃𝜁), integration by slices (see [17, page 15, Proposition 1.4.7.]) shows_𝜕𝐵𝑁|||𝐾_𝑛𝑓(𝑟𝜁)|||^𝑝𝑑𝜎(𝜁)≤𝐶_𝑝_𝜕𝐵𝑁|||𝑓(𝑟𝜁)|||^𝑝𝑑𝜎(𝜁),(3.13)that is, ‖𝐾_𝑛‖≤𝐶^𝑝1/𝑝 for every integer 𝑛≥1. By Proposition 3.2, we see ‖𝐾_𝑛𝑓−𝑓‖_𝐻^𝑝→0 as 𝑛→∞. This completes the proof of the proposition.

Corollary 3.4. If 1<𝑝<∞, then 𝑅_𝑛 converges to 0 pointwise in 𝐻^𝑝(𝐵_𝑁) as 𝑛→∞. Moreover, sup{‖𝑅_𝑛‖∶𝑛≥1}<∞.

Proof. Since 𝑅_𝑛𝑓=𝑓−𝐾_𝑛𝑓, Proposition 3.3 shows that ‖𝑅_𝑛𝑓‖_𝑝→0 as 𝑛→∞. Furthermore, the principle of uniform boundedness implies that sup_𝑛≥1‖𝑅_𝑛‖<∞.

Lemma 3.5. Let 1<𝑝<∞. For each 𝑓∈𝐻^𝑝(𝐵_𝑁) and 𝑛≥1, |||𝑅_𝑛𝑓(𝑧)|||≤‖‖𝑓‖‖_𝐻𝑝^∞_𝑘=𝑛Γ(𝑁+𝑘)𝑘!Γ(𝑁)||𝑧||^𝑘.(3.14)

Proof. Let 𝐾_𝑤 be the reproducing kernel for 𝐻²(𝐵_𝑁) and let 𝐶[𝑓] be the Cauchy-Szegö projection. Then, the orthogonality of monomials 𝜁^𝛼 implies that𝑅_𝑛𝑓(𝑧)=𝐶𝑅_𝑛𝑓(𝑧)=_𝜕𝐵𝑁𝑅_𝑛𝑓(𝜁)𝐾_𝑧(𝜁)𝑑𝜎(𝜁)=_𝜕𝐵𝑁𝑓(𝜁)𝑅_𝑛𝐾_𝑧(𝜁)𝑑𝜎(𝜁).(3.15)Hölder's inequality and the expansion of 𝐾_𝑧(𝑤) give|||𝑅_𝑛𝑓(𝑧)|||≤_𝜕𝐵𝑁|||𝑓(𝜁)||||||𝑅_𝑛≤{_𝜕𝐵𝑁𝐾_𝑧(𝜁)|||𝑑𝜎(𝜁)|||𝑓(𝜁)|||^𝑝𝑑𝜎(𝜁)}^1/𝑝{_𝜕𝐵𝑁|||𝑅_𝑛𝐾_𝑧(𝜁)|||^𝑞𝑑𝜎(𝜁)}^1/𝑞≤‖‖𝑓‖‖_𝐻𝑝^∞_𝑘=𝑛Γ(𝑁+𝑘)𝑘!Γ(𝑁)||𝑧||^𝑘.(3.16)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. Let 1<𝑝≤𝑞<∞. If 𝑢𝐶_𝜑 is bounded from 𝐻^𝑝(𝐵_𝑁) into 𝐻^𝑞(𝐵_𝑁), then ‖‖𝑢𝐶_𝜑‖‖_{𝑒,𝐻𝑝→𝐻𝑞}≤liminf_𝑛→∞‖‖𝑢𝐶_𝜑𝑅_𝑛‖‖_{𝐻𝑝→𝐻𝑞}.(3.17)

Let us prove the upper estimates for the essential norm of 𝑢𝐶_𝜑.

Proof of the upper estimates. For the sake of convenience, we set𝑀₁=l||imsup_𝑤||_→1⁻∫_𝜕𝐵𝑁|||𝑢_∗(𝜁)|||^𝑞{1−||𝑤||²|||1−𝜑_∗(𝜁),𝑤|||²}^{𝑞𝑁/𝑝}𝑀𝑑𝜎(𝜁),(3.18)₂=limsup_𝑡→0+sup_{𝜁∈𝜕𝐵𝑁}𝜇_𝜑,𝑢𝑆(𝜁,𝑡)𝑡^{𝑞𝑁/𝑝}𝐷(𝜁,𝑡)={𝑧∈,(3.19)𝐵_𝑁∶||𝑧||>1−𝑡,𝑧||𝑧||∈𝑄(𝜁,𝑡)}.(3.20)𝜀>0𝑅₁,0<𝑅₁<1
By the notation (3.18), for given _𝜕𝐵𝑁|||𝑢_∗(𝜁)|||^𝑞{1−||𝑤||²|||1−𝜑_∗(𝜁),𝑤|||²}^{𝑞𝑁/𝑝}𝑑𝜎(𝜁)<𝑀₁+𝜀,(3.21), we can choose an 𝑤∈𝐵_𝑁 such that|𝑤|≥𝑅₁for 𝜁∈𝜕𝐵_𝑁 with 𝑡,0<𝑡≤1−𝑅₁≡𝑡₁. For each 𝑤₁=(1−𝑡)𝜁 and 𝑓_𝑤1(𝑧)={(1−|𝑤₁|²)/(1−⟨𝑧,𝑤₁⟩)²}^𝑁/𝑝, we put |𝑓_𝑤1(𝑧)|^𝑝>4^−𝑁𝑡^−𝑁. Since the function 𝑧∈𝑆(𝜁,𝑡) satisfies 𝜇_𝜑,𝑢𝑆(𝜁,𝑡)𝑡^{𝑞𝑁/𝑝}<𝐶_{𝑆(𝜁,𝑡)}|||𝑓_𝑤1(𝑧)|||^𝑞𝑑𝜇_𝜑,𝑢(𝑧)<𝐶(𝑀₁+𝜀)(3.22) for all 𝜁∈𝜕𝐵_𝑁, the inequality (3.21) implies that𝑡,0<𝑡≤𝑡₁for all 𝑡₂,0<𝑡₂<1 and all sup_{𝜁∈𝜕𝐵𝑁}𝜇_𝜑,𝑢𝑆(𝜁,𝑡)𝑡^{𝑞𝑁/𝑝}<𝑀₂+𝜀(3.23).
By the notation (3.19), we can also choose a 𝑡,0<𝑡≤𝑡₂, so that𝜇₁for all 𝜇₂. Let 𝜇_𝜑,𝑢 and 𝐵_𝑁⧵(1−𝑡₁)𝐵_𝑁 be the restrictions of 𝐵_𝑁⧵(1−𝑡₂)𝐵_𝑁 to 𝜇_𝑗(𝑗=1,2) and 𝜇_𝑗𝑆(𝜁,𝑡)≤𝐶𝑀^𝑗+𝜀𝑡^{𝑞𝑁/𝑝}(3.24), respectively. We claim that 𝜁∈𝜕𝐵_𝑁 also satisfies the Carleson measure condition𝑡>0for all 𝑡,0<𝑡≤𝑡_𝑗 and 𝑡>𝑡_𝑗. By (3.22) or (3.23), these conditions are true for all {𝑄(𝑤_𝑘,𝑡_𝑗/3)}. Hence, we assume that 𝑤_𝑘∈𝑄(𝜁,𝑡). For a finite cover 𝑄(𝜁,𝑡)={𝑧∈𝜕𝐵_𝑁∶|1−⟨𝑧,𝜁⟩|≤𝑡}, where Γ of the set {𝑄(𝑤_𝑘,𝑡_𝑗/3)}, the covering property implies that there exists a disjoint subcollection 𝑄(𝜁,𝑡)⊂_Γ𝑄𝑤_𝑘,𝑡_𝑗.(3.25) of card(Γ)≤𝐶(𝑡/𝑡_𝑗)^𝑁 so that𝜇_𝑗𝑆(𝜁,𝑡)≤≤𝜇^𝑗𝐷(𝜁,𝑡)≤_Γ𝜇_𝑗𝐷𝑤^𝑘,𝑡_𝑗_Γ𝜇_𝑗𝑆𝑤^𝑘,2𝑡_𝑗≤𝐶𝑡𝑡_𝑗^𝑁𝑀^𝑗+𝜀𝑡_{𝑗𝑞𝑁/𝑝}=𝐶𝑀_𝑗+𝜀𝑡^𝑁𝑡_𝑗^{𝑞/𝑝−1}^𝑁≤𝐶𝑀_𝑗+𝜀𝑡^{𝑞𝑁/𝑝},(3.26)Furthermore, we obtain 𝐶. By the notation (3.20), we have𝑝,𝑞where the constant 𝑁 depends only on 𝑓∈𝐻^𝑝(𝐵_𝑁), and the dimension ‖𝑓‖_𝐻^𝑝≤1.
Now, we take a function ‖‖𝑢𝐶_𝜑𝑅_𝑛𝑓‖‖^𝑞𝐻_𝑞=_𝜕𝐵𝑁|||𝑢_∗=𝑅^𝑛𝑓^∗0𝑥025𝑒6𝜑^∗|||^𝑞𝑑𝜎_𝐵𝑁|||𝑅_𝑛𝑓^∗|||^𝑞𝑑𝜇_𝜑,𝑢=_𝐵𝑁|||𝑅_𝑛𝑓^∗|||^𝑞𝑑𝜇_𝑗+_{(1−𝑡𝑗)𝐵𝑁}|||𝑅_𝑛𝑓|||^𝑞𝑑𝜇_𝜑,𝑢(3.27) with 𝑛≥1. By Lemma 2.6, we have_𝐵𝑁|||𝑅_𝑛𝑓^∗|||^𝑞𝑑𝜇_𝑗≤𝐶𝑀_𝑗+𝜀‖‖𝑅_𝑛𝑓‖‖^𝑞𝐻_𝑝≤𝐶sup_𝑛≥1‖‖𝑅_𝑛‖‖^𝑞𝑀^𝑗+𝜀.(3.28)for all integers _{(1−𝑡𝑗)𝐵𝑁}|||𝑅_𝑛𝑓|||^𝑞𝑑𝜇_𝜑,𝑢≤‖‖𝑓‖‖^𝑞𝐻_𝑝{^∞_𝑘=𝑛Γ(𝑁+𝑘)𝑘!Γ(𝑁)|||1−𝑡^𝑗|||^𝑘}^𝑞‖‖𝑢‖‖^𝑞𝐻_𝑞.(3.29). Condition (3.24) and Lemma 2.3 implies that𝑢𝐶_𝜑On the other hand, by Lemma 3.5, we have𝑢∈𝐻^𝑞(𝐵_𝑁)The boundedness of ∑(Γ(𝑁+𝑘)/𝑘!Γ(𝑁))|1−𝑡_𝑗|^𝑘 implies that ^∞_𝑘=𝑛Γ(𝑁+𝑘)𝑘!Γ(𝑁)|||1−𝑡^𝑗|||^𝑘⟶0as𝑛⟶∞.(3.30) and the convergence of the series _{(1−𝑡𝑗)𝐵𝑁}|||𝑅_𝑛𝑓|||^𝑞𝑑𝜇_𝑢,𝜑⟶0as𝑛⟶∞.(3.31) implies that‖‖𝑢𝐶_𝜑‖‖^{𝑞𝑒,𝐻}_{𝑝→𝐻𝑞}≤liminf_𝑛→∞‖‖𝑢𝐶_𝜑𝑅_𝑛‖‖^𝑞𝐻_{𝑝→𝐻𝑞}≤𝐶sup_𝑛≥1‖‖𝑅_𝑛‖‖^𝑞𝑀^𝑗+𝜀.(3.32)So we obtainsup_𝑛≥1‖𝑅_𝑛‖<∞Combining (3.27), (3.28), and (3.31) with Lemma 3.6, we have𝜀>0Since Corollary 3.4 implies that ‖‖𝑢𝐶_𝜑‖‖^{𝑞𝑒,𝐻}_{𝑝→𝐻𝑞}≤⎧⎪⎪⎪⎨⎪⎪⎪⎩𝐶l||imsup^𝑤||_→1⁻_𝜕𝐵𝑁|||𝑢_∗(𝜁)|||^𝑞{1−||𝑤||²|||1−𝜑_∗(𝜁),𝑤|||²𝐶limsup_𝑡→0+}^{𝑞𝑁/𝑝}𝑑𝜎(𝜁),sup_{𝜁∈𝜕𝐵𝑁}𝜇_𝜑,𝑢𝑆(𝜁,𝑡)𝑡^{𝑞𝑁/𝑝},(3.33), and 1<𝑝≤𝑞<∞ was arbitrary, we conclude that𝑢𝐶_𝜑∶𝐻^𝑝(𝐵_𝑁)→𝐻^𝑞(𝐵_𝑁)which were to be proved.

Corollary 3.7 (see [15]). Suppose that 𝑢𝐶_𝜑∶𝐻^𝑝(𝐵_𝑁)→𝐻^𝑞(𝐵_𝑁). For the bounded weighted composition operator 𝑢, the following conditions are equivalent:
(a)𝜑 is compact;(b)||lim_𝑤||_→1⁻_𝜕𝐵𝑁|||𝑢_∗(𝜁)|||^𝑞{1−||𝑤||²|||1−𝜑_∗(𝜁),𝑤|||²}^{𝑞𝑁/𝑝}𝑑𝜎(𝜁)=0;(3.34) and 𝑢 satisfy𝜑(c)lim_𝑡→0+sup_{𝜁∈𝜕𝐵𝑁}𝜇_𝜑,𝑢𝑆(𝜁,𝑡)𝑡^{𝑞𝑁/𝑝}=0.(3.35) and 𝑀_𝑢 satisfy1<𝑝≤𝑞<∞

4. Multiplication Operators between Hardy Spaces

In this section, we consider the compact multiplication operator 𝑀_𝑢∶𝐻^𝑝(𝐵_𝑁)→𝐻^𝑞(𝐵_𝑁) between Hardy spaces. As a consequence of Theorem 3.1, we obtain the following results.

Corollary 4.1. Suppose that ‖‖𝑀_𝑢‖‖^{𝑞𝑒,𝐻}_{𝑝→𝐻𝑞}∼l||imsup_𝑤||_→1⁻_𝜕𝐵𝑁|||𝑢_∗(𝜁)|||^𝑞{1−||𝑤||²|||1−⟨𝜁,𝑤⟩|||²}^{𝑞𝑁/𝑝}𝑑𝜎(𝜁).(4.1). For the bounded multiplication operator 𝑀_𝑢∶𝐻^𝑝(𝐵_𝑁)→𝐻^𝑞(𝐵_𝑁), the following inequality holds: ||lim_𝑤||_→1⁻_𝜕𝐵𝑁|||𝑢_∗(𝜁)|||^𝑞{1−||𝑤||²|||1−⟨𝜁,𝑤⟩|||²}^{𝑞𝑁/𝑝}𝑑𝜎(𝜁)=0.(4.2)Furthermore, 𝑀_𝑢 is compact if and only if 𝐻^𝑝(𝐵_𝑁)

By using Corollary 4.1, we can completely characterize the compactness of a multiplication operator 𝐻^𝑞(𝐵_𝑁) from 1<𝑝≤𝑞<∞ into 𝑀_𝑢∶𝐻^𝑝(𝐵_𝑁)→𝐻^𝑞(𝐵_𝑁).

Theorem 4.2. Suppose that 𝑢=0. Then 𝐵_𝑁 is compact if and only if 𝑢≡0 in 𝑀_𝑢.

Proof. If 𝑀_𝑢, then 𝑢≡0 is compact. Thus, we only prove that the compactness of 𝑀_𝑢 implies 𝑢∈𝐻^𝑞(𝐵_𝑁). The boundedness of 𝑢 implies that 𝑢(𝑤)=_𝜕𝐵𝑁𝑢^∗(𝜁)𝑃(𝑤,𝜁)𝑑𝜎(𝜁)𝑤∈𝐵_𝑁,(4.3). Hence, the Poisson representation for 𝑃(𝑤,𝜁) gives that||𝑢(𝑤)||≤≤{_{𝜕𝐵𝑁𝜕𝐵𝑁}|||𝑢_∗(𝜁)|||𝑃(𝑤,𝜁)𝑑𝜎(𝜁)|||𝑢_∗(𝜁)|||^𝑞𝑃(𝑤,𝜁)^𝑞/𝑝𝑑𝜎(𝜁)}^1/𝑞{_𝜕𝐵𝑁𝑃(𝑤,𝜁)^1−1/𝑝^𝑞′𝑑𝜎(𝜁)}^1/𝑞′,(4.4)where 1/𝑞+1/𝑞^′=1 is the Poisson kernel. Hölder's inequality shows that1<𝑝≤𝑞<∞where 𝑠≡1−1𝑝𝑞^′=𝑞(𝑝−1)𝑝(𝑞−1)≤𝑝𝑝(𝑞𝑞−−1𝑝)=1,(4.5). By the assumption _𝜕𝐵𝑁𝑃(𝑤,𝜁)^1−1/𝑝^𝑞′𝑑𝜎(𝜁)≤{_𝜕𝐵𝑁{𝑃(𝑤,𝜁)^𝑠}^1/𝑠𝑑𝜎(𝜁)}^𝑠=1.(4.6), we see thatlim_|𝑤|→1⁻|𝑢(𝑤)|=0and so we have𝑢∈𝐻^𝑞(𝐵_𝑁)Inequality (4.4) and Corollary 4.1 give that 𝑢. Since 𝐾, this implies that 0 has a 𝜎-limit 𝜕𝐵_𝑁 on a set of positive 𝑢≡0-measure in 𝐻^∞. Hence [17, page 83, Theorem 5.5.9] shows that 𝛼. This completes the proof.

Acknowledgment

The authors would like to thank the referee for the careful reading of the first version of this paper and for the several suggestions made for improvement.