Compact Weighted Composition Operators and Multiplication Operators between Hardy Spaces

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


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 by 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 ϕ f u f • ϕ .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, 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 B 1 into H q B 1 1 < p ≤ q < ∞ in terms of the pull-back measure.Here, B 1 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 B 1 → H q B 1 has been studied by Cucković 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.The one variable case of the first estimate for uC ϕ e in above theorem may be found in the work 14 by Cucković 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 lim sup t→0 sup ζ∈∂B N 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.

Carleson-type measures
For each u ∈ H q B N , we can define a finite positive Borel measure μ ϕ,u on B N by 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

2.4
Then there exists a constant K > 0 such that 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 that where Mf is the admissible maximal function of f which is defined by for ζ ∈ ∂B N .By 2.6 , we have Since f ∈ H p B N , it follows from 17, Theorem 5.6.5 that By 2.8 and 2.9 , we have

2.10
This completes the proof.
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 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.
for some constant C > 0.Then, there exists a constant K > 0 such that 15 Here, the notation f * denotes the function defined on 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 .
Proof.Suppose that E, F ⊂ ∂B N and ϕ * E ⊂ F with σ E > 0 and σ F 0. Put μ μ ϕ,u | ∂B N .As in the case of composition operators, it is well known that the boundedness of uC ϕ : 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 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.
a.e.σ on ∂B N .Here the notation f * is used as in Lemma 2.3.Proof (cf.[19,Lemma 1.6]).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 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 that 0

3.1
Proof of the lower estimates.For each w ∈ B N , we define the function f w on B N by

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 By the definition of f w , we also see that Combining this with 3.3 , we get Since this holds for every compact operator K, it follows that uC ϕ q e,H p →H q ≥ C lim sup e,H p →H q , 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 expansion For the homogeneous expansion of f and any integer n ≥ 1, let and , 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.
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.
This completes the proof of the proposition.
Proof.Let K w be the reproducing kernel for H 2 B N and let C f be the Cauchy-Szeg ö projection.Then, the orthogonality of monomials ζ α implies that H ölder's inequality and the expansion of K z w give

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. 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 set

3.20
By the notation 3.18 , for given ε > 0, we can choose an Since the function 3.27 for all integers n ≥ 1. Condition 3.24 and Lemma 2.3 implies that

3.28
On the other hand, by Lemma 3.5, we have The boundedness of uC ϕ implies that u ∈ H q B N and the convergence of the series

3.30
So we obtain

3.32
Since Corollary 3.4 implies that sup n≥1 R n < ∞, and ε > 0 was arbitrary, we conclude that

Multiplication operators between Hardy spaces
In this section, we consider the compact multiplication operator M u between Hardy spaces.As a consequence of Theorem 3.1, we obtain the following results.
Corollary 4.1.Suppose that 1 < p ≤ q < ∞.For the bounded multiplication operator M u : H p B N → H q B N , the following inequality holds: Theorem 4.2.Suppose that 1 < p ≤ q < ∞.Then M u : H p B N → H q B N is compact if and only if u 0 in B N .
Proof.If u ≡ 0, then M u is compact.Thus, we only prove that the compactness of M u implies u ≡ 0. The boundedness of M u implies that u ∈ H q B N .Hence, the Poisson representation for u gives that where 1/q 1/q 1.By the assumption 1 < p ≤ q < ∞, we see that s ≡ 1 − 1 p q q p − 1 p q − 1 ≤ pq − p p q − 1 1, 4.5 and so we have Since u ∈ H q B N , this implies that u has a K-limit 0 on a set of positive σ-measure in ∂B N .Hence 17, page 83, Theorem 5.5.9 shows that u ≡ 0. This completes the proof.

where
P w, ζ is the Poisson kernel.H ölder's inequality shows that |u w | ≤ ∂B N u * ζ P w, ζ dσ ζ
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 have 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 ∈ L 1 ∂B N .By 2.12 , we have 0 ∂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 have Furthermore, we put w 1 − t ζ for each t, 0 < t < 1 and ζ ∈ 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 that for every integer n ≥ 1.By Proposition 3.2, we see , t , the inequality 3.21 implies that < t ≤ t 2 .Let μ 1 and μ 2 be the restrictions of μ ϕ,u to B N \ 1 − t 1 B N and B N \ 1 − t 2 B N , respectively.We claim that μ j j 1, 2 also satisfies the Carleson measure condition ∂B N and t > 0. By 3.22 or 3.23 , these conditions are true for all t, 0 < t ≤ t j .Hence, we assume that t > t j .For a finite cover {Q w k , t j /3 }, wherew k ∈ Q ζ, t of the set Q ζ, t {z ∈ ∂B N : |1 − z, ζ | ≤ t},the covering property implies that there exists a disjoint subcollection Γ of {Q w k , t j /3 } so that depends only on p, q, and the dimension N. Now, we take a function f ∈ H p B N with f H p ≤ 1.By Lemma 2.6, we have H p →H q ∼lim sup Furthermore, M u : H p B N → H q B N iscompact if and only ifBy using Corollary 4.1, we can completely characterize the compactness of a multiplication operator M u from H p B N into H q B N .