This paper finds some lower and upper bounds for the essential norm of the weighted composition operator from
α-Bloch spaces to the weighted-type space Hμ∞ on the unit ball for the case α≥1.

1. Introduction

Let 𝔹={z∈ℂn:|z|<1} be the open unit ball in ℂn, H(𝔹) be the class of all holomorphic functions on
the unit ball, and let H∞(𝔹) be the class of all bounded holomorphic
functions on 𝔹 with the norm∥f∥∞=supz∈𝔹|f(z)|.Let z=(z1,…,zn) and w=(w1,…,wn) be points in ℂn and 〈z,w〉=∑k=1nzkw¯k.
For a holomorphic function f,
we denote∇f=(∂f∂z1,…,∂f∂zn).

A positive continuous function ϕ on the interval [0,1) is called normal (see [1]) if there is δ∈[0,1) and a and b, 0<a<b such thatϕ(r)(1−r)aisdecreasingon[δ,1)andlimr→1ϕ(r)(1−r)a=0,ϕ(r)(1−r)bisincreasingon[δ,1)andlimr→1ϕ(r)(1−r)b=∞.From now on, if we say that a
function ϕ:𝔹→[0,∞) is normal, we will also assume that it is
radial, that is, ϕ(z)=ϕ(|z|),z∈𝔹.

The weighted space Hμ∞=Hμ∞(𝔹) consists of all f∈H(𝔹) such thatsupz∈𝔹μ(z)|f(z)|<∞,where μ is normal on the interval [0,1).
For μ(z)=(1−|z|2)β, β>0, we obtain the weighted space Hβ∞=Hβ∞(𝔹) (see, e.g., [2, 3]).

The α-Bloch space ℬα=ℬα(𝔹),α>0, is the space of all f∈H(𝔹) such thatbα(f)=supz∈𝔹(1−|z|2)α|∇f(z)|<∞.With the norm∥f∥ℬα=|f(0)|+bα(f),the space ℬα is a Banach space ([4–6]).

The little α-Bloch space ℬ0α=ℬ0α(𝔹) is the subspace of ℬα consisting of all f∈H(𝔹) such thatlim|z|→1(1−|z|2)α|∇f(z)|=0.

Let u∈H(𝔹) and φ be a holomorphic self-map of the unit ball. Weighted
composition operator on H(𝔹),
induced by u and φ is defined by(uCφf)(z)=u(z)f(φ(z)),z∈𝔹.This operator can be regarded as
a generalization of a multiplication operator and a composition operator. It is
interesting to provide a function theoretic characterization when u and φ induce a bounded or compact weighted
composition operator between some spaces of holomorphic functions on 𝔹.
(For some classical results in the topic see, e.g., [5]. For some recent results on
this and related operators, see, e.g., [2–4, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25] and the references therein.)

In [18], Ohno has characterized the boundedness and
compactness of weighted composition operators between H∞ and the Bloch space ℬ on the unit disk. In the setting of the unit
polydisk 𝔻n,
we have given some necessary and sufficient conditions for a weighted
composition operator to be bounded or compact from H∞(𝔻n) to the Bloch space ℬ(𝔻n) in [12] (see, also [21]). Corresponding results for the case of the unit ball
are given in [14].
Among other results, in [14],
we have given some necessary and sufficient conditions for the compactness of
the operator uCφ:ℬα(𝔹)→H∞(𝔹), which we incorporate in the following theorem.

Theorem 1.

Let φ=(φ1,…,φn) be a holomorphic self-map of 𝔹 and u∈H(𝔹).

If α>1, then the following statements are equivalent:

uCφ:ℬ0α→H∞ is a compact operator,

uCφ:ℬα→H∞ is a compact operator,

u∈H∞, andlim|φ(z)|→1|u(z)|(1−|φ(z)|2)α−1=0.

If α=1, then the following statements are equivalent:

uCφ:ℬ0→H∞ is a compact operator,

uCφ:ℬ→H∞ is a compact operator,

u∈H∞, and lim|φ(z)|→1|u(z)|ln21−|φ(z)|2=0.

We would also like to point out that if α≥1,
then the boundedness of uCφ:ℬα→Hμ∞ and uCφ:ℬ0α→Hμ∞ are equivalent (see [22] for the case α=1, and the proof of Theorem 3 in [14]).

The essential norm of an operator is its distance in
the operator norm from the compact operators. More precisely, assume that X1 and X2 are Banach spaces and A:X1→X2 is a bounded linear operator, then the
essential norm of A,
denoted by ∥A∥e,X1→X2,
is defined as follows:∥A∥e,X1→X2=inf{∥A+L∥X1→X2:L:X1→X2,Liscompact},where ∥⋅∥X1→X2 denotes the
operator norm. If X1=X2,
it is simply denoted by ∥⋅∥e (see, e.g., [5, page 132]). If A:X1→X2 is an unbounded linear operator, then clearly ∥A∥e,X1→X2=∞.

Since the set of all compact operators is a closed
subset of the set of bounded operators, it follows that an operator A is compact if and only if ∥A∥e,X1→X2=0.

Motivated by Theorem A, in this paper, we find some
lower and upper bounds for the essential norm of the weighted composition
operator uCφ:ℬα(𝔹)(orℬ0α(𝔹))→Hμ∞(𝔹),
when α≥1.

Throughout this paper, constants are denoted by C,
they are positive and may differ from one occurrence to another. The notation a≼b means that there is a positive constant C such that a≤Cb.
If both a≼b and b≼a hold, then we say that a≍b.

2. Auxiliary Results

In this section, we quote several auxiliary results
which we need in the proofs of the main results in this paper. The following
lemma should be folklore.

Lemma 2.1.

Let f∈ℬα(𝔹),0<α<∞. Then,
|f(z)|≤{C∥f∥ℬα,α∈(0,1),|f(0)|+b1(f)12ln1+|z|1−|z|,α=1,C∥f∥ℬα(1−|z|2)α−1,α>1,
for some C>0 independent of f.

The proof of the lemma for the case α≠1 can be found, for example, in [26]. The formulation of the
corresponding estimate in [26], for the case α=1,
is slightly different. In this case, Lemma 2.1 follows from the following
estimate:|f(z)−f(0)|=|∫01〈∇f(tz),z¯〉|≤b1(f)∫01|z|dt1−|z|2t2=b1(f)12ln1+|z|1−|z|.

The next lemma can be proved in a standard way (see, e.g., the proofs of the corresponding results in [5, 27–29]).

Lemma 2.2.

Assume α>0, g∈H(𝔹), μ is normal, and φ is an analytic self-map of 𝔹. Then, uCφ:ℬα(orℬ0α)→Hμ∞ is compact if and only if uCφ:ℬα(orℬ0α)→Hμ∞ is bounded and for any bounded sequence (fk)k∈ℕ in ℬα(orℬ0α) converging to zero uniformly on compacts of 𝔹 as k→∞, one has∥uCφfk∥Hμ∞→0 as k→∞.

Lemma 2.3.

Let
fw(z)=(1−|w|2)ε(1−〈z,w〉)α+ε−1,w∈𝔹,where α>1 and ε∈(0,1]. Then,∥fw∥ℬα=(1−|w|2)ε+(α+ε−1)(2α)α|w|(α2+|w|2ε2−|w|2α2+ε)ε(α+ε)ε(α2+|w|2ε2−|w|2α2+α)α.

Proof.

We have
(1−|z|2)α|∇fw(z)|=(α+ε−1)(1−|z|2)α(1−|w|2)ε|w||1−〈z,w〉|α+ε,
from which it easily follows
that
bα(fw)≤(α+ε−1)2α+ε.
Set
gs(x)=(α+ε−1)s(1−s2)ε(1−x2)α(1−sx)α+ε,x∈[0,1],s∈[0,1).
Then,
g′s(x)=(α+ε−1)s(1−s2)ε(1−x2)α−1s(α−ε)x2−2αx+s(α+ε)(1−sx)α+ε+1.
Hence, the points
xM,m=α±α2+s2ε2−s2α2s(α−ε)
are stationary for the function gs(x). Since xM>1, it follows that gs(x) attains its maximum on the interval [0,1] at the point
xm=α−α2+s2ε2−s2α2s(α−ε)=s(α+ε)α+α2+s2ε2−s2α2∈(0,1).
By some long but elementary calculations, it follows that
gs(xm)=(α+ε−1)(2α)αs(α2+s2ε2−s2α2+ε)ε(α+ε)ε(α2+s2ε2−s2α2+α)α.From this and since fw(0)=(1−|w|2)ε, (2.4) follows.

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In this section, we prove the main results in this
paper. Before we formulate and prove these results, we prove another auxiliary
result.

Lemma 3.1.

Assume α∈(0,∞), u∈H(𝔹),μ is normal, φ is a holomorphic self-map of 𝔹 such that ∥φ∥∞<1, and the operator uCφ:ℬα(orℬ0α)→Hμ∞ is bounded. Then, uCφ:ℬα(orℬ0α)→Hμ∞ is compact.

Proof.

First note that since uCφ:ℬα(orℬ0α)→Hμ∞ is bounded and f0(z)≡1∈ℬ0α⊂ℬα, it follows that uCφ(f0)=u∈Hμ∞. Now, assume that (fk)k∈ℕ is a bounded sequence in ℬα(orℬ0α) converging to zero on compacts of 𝔹 as k→∞. Then, we have
∥uCφ(fk)∥Hμ∞≤∥u∥Hμ∞supw∈φ(𝔹)|fk(w)|→0,
as k→∞, since φ(𝔹) is contained in the ball |w|≤∥φ∥∞ which is a compact subset of 𝔹, according to the assumption, ∥φ∥∞<1. Hence, by Lemma 2.2
, the operator uCφ:ℬα(orℬ0α)→Hμ∞ is compact.

Theorem 3.2.

Assume α>1, μ is normal, u∈H(𝔹), φ=(φ1,…,φn) is a holomorphic self-map of 𝔹, and uCφ:ℬα→Hμ∞ is bounded. Then,
2−αα−1limsup|φ(z)|→1μ(z)|u(z)|(1−|φ(z)|2)α−1≤∥uCφ∥e,ℬ0α→Hμ∞≤∥uCφ∥e,ℬα→Hμ∞≤Climsup|φ(z)|→1μ(z)|u(z)|(1−|φ(z)|2)α−1,
for some positive constant C.

Proof.

Since uCφ:ℬα→Hμ∞ is bounded, recall that u∈Hμ∞. If ∥φ∥∞<1, then, from Lemma 3.1, it follows that uCφ:ℬα→Hμ∞ is compact which is equivalent with ∥uCφ∥e,ℬα→Hμ∞=0, and, consequently, ∥uCφ∥e,ℬ0α→Hμ∞=0. On the other hand, it is clear that in this
case the condition |φ(z)|→1 is vacuous, so that inequalities in (3.2) are
vacuously satisfied.

Hence, assume ∥φ∥∞=1. Let (zk)k∈ℕ be a sequence in 𝔹 such that limk→∞|φ(zk)|=1 and ε∈(0,1) be fixed. Set
fkε(z)=(1−|φ(zk)|2)ε(1−〈z,φ(zk)〉)α+ε−1,k∈ℕ. By Lemma 2.3, it follows that supk∈ℕ∥fkε∥ℬα<∞, moreover, it is easy to see that fkε∈ℬ0α for every k∈ℕ and fkε→0 uniformly on compacts of 𝔹 as k→∞. Then, by [6, Theorem 7.5], it follows that fkε converges to zero weakly as k→∞. Hence, for every compact operator L:ℬ0α→Hμ∞, we have ∥Lfkε∥Hμ∞→0 as k→∞.

We have
∥fkε∥ℬα∥uCφ+L∥ℬ0α→Hμ∞≥∥(uCφ+L)(fkε)∥Hμ∞≥∥uCφfkε∥Hμ∞−∥Lfkε∥Hμ∞
for every compact operator L:ℬ0α→Hμ∞.

Letting k→∞ in (3.4) and using the definition of fkε, we obtain
C(ε,α)∥uCφ+L∥ℬ0α→Hμ∞=limsupk→∞∥uCφfkε∥Hμ∞≥limsupk→∞μ(zk)|u(zk)|(1−|φ(zk)|2)α−1,where C(ε,α) is the quantity in (2.12).

Taking in (3.5) the infimum over the set of all compact
operators L:ℬ0α→Hμ∞, then letting ε→0+ in such obtained inequality, and using (2.13), we obtain
2α(α−1)∥uCφ∥e,ℬ0α→Hμ∞≥limsupk→∞μ(zk)|u(zk)|(1−|φ(zk)|2)α−1,
from which the first inequality in (3.2) follows.

Since the second inequality in (3.2) is obvious, we only
have to prove the third one. By Lemma 3.1, we have that for each fixed ρ∈(0,1) the operator uCρφ:ℬα→Hμ∞ is compact.

Let δ∈(0,1) be fixed, and let (ρm)m∈ℕ be a sequence of positive numbers which
increasingly converges to 1, then for each fixed m∈ℕ, we have
∥uCφ∥e,ℬα→Hμ∞≤∥uCφ−uCρmφ∥ℬα→Hμ∞=sup∥f∥ℬα≤1∥(uCφ−uCρmφ)(f)∥Hμ∞=supz∈𝔹sup∥f∥ℬα≤1μ(z)|u(z)(f(φ(z))−f(ρmφ(z)))|≤sup|φ(z)|≤δsup∥f∥ℬα≤1μ(z)|u(z)||f(φ(z))−f(ρmφ(z))|+sup|φ(z)|>δsup∥f∥ℬα≤1μ(z)|u(z)||f(φ(z))−f(ρmφ(z))|.

By the mean-value theorem, we have
sup|φ(z)|≤δsup∥f∥ℬα≤1μ(z)|u(z)(f(φ(z))−f(ρmφ(z)))|≤(1−ρm)sup|φ(z)|≤δsup∥f∥ℬα≤1μ(z)|u(z)||φ(z)|sup|w|≤δ|∇f(w)|≤(1−ρm)δ(1−δ2)α∥u∥Hμ∞sup∥f∥ℬα≤1sup|w|≤δ(1−|w|2)α|∇f(w)|≤(1−ρm)δ(1−δ2)α∥u∥Hμ∞→0,
as m→∞.

Moreover, by Lemma 2.1 (case α>1), and known inequality
∥fr∥ℬα≤∥f∥ℬα,
where fr(z)=f(rz), r∈[0,1), we have
μ(z)|u(z)(f(φ(z))−f(ρmφ(z)))|≤C∥f−fρm∥ℬαμ(z)|u(z)|(1−|φ(z)|2)α−1≤2C∥f∥ℬαμ(z)|u(z)|(1−|φ(z)|2)α−1
for some positive constant C. Replacing (3.10) in (3.7), letting in such
obtained inequality m→∞, employing (3.8), and then letting δ→1, the third inequality in (3.2) follows, finishing the proof of the theorem.

Corollary 3.3.

Assume α>1, μ is normal, u∈H(𝔹), φ=(φ1,…,φn) is a holomorphic self-map of 𝔹, and the operator uCφ:ℬα(orℬ0α)→Hμ∞ is bounded. Then, uCφ:ℬα(orℬ0α)→Hμ∞ is compact if and only if
limsup|φ(z)|→1μ(z)|u(z)|(1−|φ(z)|2)α−1=0.

Theorem 3.4.

Assume u∈H(𝔹),μ is normal, φ=(φ1,…,φn) is a holomorphic self-map of 𝔹, and uCφ:ℬ→Hμ∞ is bounded. Then,
Climsup|φ(z)|→1μ(z)|u(z)|ln1+|φ(z)|1−|φ(z)|≤∥uCφ∥e,ℬ0→Hμ∞≤∥uCφ∥e,ℬ→Hμ∞≤limsup|φ(z)|→1μ(z)|u(z)|ln1+|φ(z)|1−|φ(z)|
for a positive constant C.

Proof.

Clearly, u=(uCφ)(1)∈Hμ∞. If ∥φ∥∞<1, then the result follows as in Theorem 3.2.

Hence, assume ∥φ∥∞=1.
We use the following family of test functionshw(z)=(ln(1+|w|)21−〈z,w〉)2(ln1+|w|1−|w|)−1,w∈𝔹.

We have
supz∈𝔹(1−|z|2)|∇hw(z)|=supz∈𝔹2(1−|z|2)|w||1−〈z,w〉||ln(1+|w|)21−〈z,w〉|(ln1+|w|1−|w|)−1≤4|w|(2ln1+|w|1−|w|+2π)(ln1+|w|1−|w|)−1≤4(2+2πln3),
when |w|≥1/2.

From this and since |hw(0)|≤4(ln2)2/ln3, for |w|≥1/2, we have that
∥hw∥ℬ≤4(2+2πln3+(ln2)2ln3)=C0.

Assume (φ(zk))k∈ℕ is a sequence in 𝔹 such that |φ(zk)|→1 as k→∞. Note that (hφ(zk))k∈ℕ is a bounded sequence in ℬ (moreover in ℬ0) converging to zero uniformly on compacts of 𝔹. Then, by [6, Theorem 7.5], it follows that hφ(zk) converges to zero weakly as k→∞. Hence, for every compact operator L:ℬ0→Hμ∞, we have
limk→∞∥Lhφ(zk)∥Hμ∞=0.

On the other hand, for every compact operator L:ℬ0→Hμ∞,
we have∥hφ(zk)∥ℬ∥uCφ+L∥ℬ0→Hμ∞≥∥(uCφ+L)hφ(zk)∥Hμ∞≥∥uCφhφ(zk)∥Hμ∞−∥Lhφ(zk)∥Hμ∞.
Using (3.15), letting k→∞ in (3.17), and applying (3.16), it follows that

Taking in (3.18) the infimum over the set of all compact
operators L:ℬ0→Hμ∞, we obtain
∥uCφ∥e,ℬ0→Hμ∞≥1C0limsupk→∞μ(zk)|u(zk)|ln1+|φ(zk)|1−|φ(zk)|,
from which the first inequality in (3.12) follows.

As in Theorem 3.2, we need only to prove the third inequality in (3.12).

Recall that for each ρ∈(0,1), the operator uCρφ:ℬ→Hμ∞ is compact. Let the sequence (ρm)m∈ℕ be as in Theorem 3.2. Note that inequality (3.7)
and relationship (3.8) also hold for α=1. Hence, we should only estimate the quantity
Imδ=sup|φ(z)|>δsup∥f∥ℬ≤1μ(z)|u(z)||f(φ(z))−f(ρmφ(z))|.
On the other hand, by Lemma 2.1 (case α=1) applied to the function f−fρm, which belongs to the Bloch space for each m∈ℕ, and inequality (3.9) with α=1, we have
μ(z)|u(z)(f(φ(z))−f(ρmφ(z)))|≤12∥f−fρm∥ℬμ(z)|u(z)|ln1+|φ(z)|1−|φ(z)|≤∥f∥ℬμ(z)|u(z)|ln1+|φ(z)|1−|φ(z)|.

From (3.21), by letting m→∞ in (3.7) and using (3.8) (with α=1), and letting δ→1 in such obtained inequality, we obtain∥uCφ∥e,ℬ→Hμ∞≤limsup|φ(z)|→1μ(z)|u(z)|ln1+|φ(z)|1−|φ(z)|,
as desired.

Corollary 3.5.

Assume u∈H(𝔹),μ is normal, φ=(φ1,…,φn) is a holomorphic self-map of 𝔹, and the operator uCφ:ℬ(orℬ0α)→Hμ∞ is bounded. Then, uCφ:ℬ(orℬ0α)→Hμ∞ is compact if and only if
limsup|φ(z)|→1μ(z)|u(z)|ln1+|φ(z)|1−|φ(z)|=0.

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