ISSN 2291-8639
Volume 12, Number 2 (2016), 87-97 http://www.etamaths.com
PROPERTIES OF WEIGHTED COMPOSITION OPERATORS ON SOME WEIGHTED HOLOMORPHIC FUNCTION CLASSES IN THE UNIT BALL
A. E. SHAMMAKY AND M. A. BAKHIT∗
Abstract. In this paper, we introduceNK-type spaces of holomorphic functions in the unit ball of
Cn by the help of a non-decreasing functionK: (0,∞)→[0,∞).Several important properties of these spaces in the unit ball are provided. The results are applied to characterize boundedness and compactness of weighted composition operatorsWu,φfromNK(B) spaces into Beurling-type classes. We also find the essential norm estimates forWu,φfromNK(B) spaces into Beurling-type classes.
1. Introduction
Through this paper, Bis the unit ball of then-dimensional complex Euclidean space Cn,S is the boundary ofB.We denote the class of all holomorphic functions, with the compact-open topology on the unit ballBbyH(B). For anyz= (z1, z2, . . . , zn), w= (w1, w2, . . . , wn)∈Cn,the inner product is defined byhz, wi=z1w1+. . .+znwn, and write|z|=
p
hz, zi.
Two quantities Af and Bf, both depending on a function f ∈ H(B), are said to be equivalent, written asAf ≈Bf,if there exists a finite positive constant M not depending onf, such that
1
MBf ≤Af≤M Bf
for everyf ∈ H(B).If the quantitiesAf andBf,are equivalent, then in particular we haveAf <∞if
and only ifBf <∞.As usual, the letterM will denote a positive constant, possibly different on each
occurrence.
Given a pointa∈B,we can associate wit it the following automorphism Φa(z)∈Aut(B) :
Φa(z) =
a−Pa(z)−SaQa(z)
1− hz, ai , z∈B,
(1.1)
whereSa=
p
1− |a|2, P
a(z) is the orthogonal projection ofCn on a subspace [a] generated bya,that is
Pa(z) =
(
0, ifa= 0;
ahz,ai
|a|2 , ifa6= 0,
andQa =I−Pa the projection on orthogonal complement [a] (see, for example,[8] or [10]).
The map Φa has the following properties that Φa(0) =a, Φa(a) = 0, Φa= Φ−1a and
1− hΦa(z),Φa(w)i=
(1− |a|2)(1− hz, wi)
(1− hz, ai)(1− ha, wi), (1.2)
wherezandware arbitrary points inB. In particular,
1− |Φa(z)|2=
(1− |a|2)(1− |z|2) |1− hz, ai|2 .
(1.3)
2010Mathematics Subject Classification. Primary 30D45, 47B33; Secondary 30B10. Key words and phrases. weighted composition operators;NK spaces; Beurling-type spaces.
c
2016 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.
LetV be the Lebesgue volume measure onCn,normalized so thatV(B)≡1 andσbe the normalized surface measure onS,so thatσ(B)≡1.Let
dτ(z) = dV(z) (1− |z|2)n+1,
which is M¨obius invariant, that is for any ψ∈Aut(B), f ∈L1(
B),we have Z
B
f(z)dτ(z) = Z
B
f ◦ψ(z)dτ(z).
For a∈B,the M¨obius invariant Green function in Bdenoted byG(z, a) =g(Φa(z)) whereg(z) is
defined by:
g(z) =n+ 1 2n
Z 1
|z|
(1−t2)n−1t1−2ndt.
(1.4)
LetH∞(B) denote the Banach space of bounded functions inH(B) with the norm
kfk∞= sup z∈B
|f(z)|.
Forα > 0, the Beurling-type space (sometimes also called the Bers-type space) Hα∞(B) in the unit ballBconsists of those functionsf ∈ H(B) for which
kfkH∞
α(B)= sup z∈B
|f(z)|(1− |z|)α<∞.
The Bergman spaceA2(
B) consists of those functionsf ∈ H(B) for which
kfk2 A2(B)=
Z
B
|f(z)|2dV(z)<∞.
Let K : (0,∞) → [0,∞) be a right-continuous, non-decreasing function and is not equal to zero identically. TheNK(B) space consists of all functionsf ∈ H(B) such that
kfk2
NK(B)= sup a∈B
Z
B
|f(z)|2K(G(z, a))dτ(z)<∞.
Moreover,f ∈ H(B) is said to belong toNK,0(B) if
lim
|a|→1
Z
B
|f(z)|2K(G(z, a))dτ(z) = 0.
Clearly, if K(t) =tp,then N
K(B) =Np(B); since G(z, a)≈(1− |ϕa(z)|2) (see [6]). For K(t) = 1
it gives the Bergman spaceA2(
B). IfNK(B) consists of just the constant functions, we say that it is trivial. Several important properties of theNK(B) spaces in the unit disk in the complex plane have been characterized in [1], [4] and [9]. We assume from now that allK: (0,∞)→[0,∞) to appear in this paper are right-continuous, non-decreasing function and not equal to zero identically.
Given u ∈ H(B) and φ a holomorphic self-map of B. The weighted composition operator Wu,φ : H(B)→ H(B) is defined by
Wu,φ(f)(z) =u(z)(f◦φ)(z), z∈B.
It is obvious thatWu,φcan be regarded as a generalization of the multiplication operatorMuf =u·f
and composition operatorCφf =f◦φ.Weighted composition operators are a general class of operators
and appear naturally in the study of isometries on most of the function spaces. Operators of this kind also appear in many branches of analysis; the theory of dynamical systems, semigroups, the theory of operator algebras, the theory of solubility of equations with deviating argument and so on. The behavior of those operators is studied extensively on various spaces of holomorphic functions (see for example [2], [5], [6] and [9]).
Recall that the pseudohyperbolic metric in the ball is defined as:
ρ(z, w) =|Φw(z)|, z∈B. (1.5)
It is true metric (see [3]). Also it is easy to verify, in particular, thatρ(0, w) =|w|andρ(Φa(z), w) =|z|.
Lemma 1.1. Forz, w∈B, ifρ(z, w)≤ 12. Then
(1.6) 1
6 ≤
1− |z|2
1− |w|2 ≤6.
The following proposition was proved as part of Lemma 2.3 in [6], and hence, we omit the details.
Proposition 1.1. Forz, w∈B, ifρ(z, w)≤ 14. Then
|f(z)−f(w)| ≤2√n|f(Φa(z))|ρ(z, w).
Recall that a linear operator T :X → Y is said to be bounded if there exists a constantM > 0 such thatkT(f)kY ≤MkfkX for all maps f ∈X. Moreover,T : X →Y is said to be compact if it
takes bounded sets inX to sets in Y which have compact closure. For a Complex Banach spaces X
andY of H(B), T :X →Y is compact (respectively weakly compact) if it maps the closed unit ball ofX onto a relatively compact (respectively relatively weakly compact) set inY.
In this paper, we introduce NK(B) spaces, in terms of the right continuous and non-decreasing functionK : (0,∞) →[0,∞) on the unit ball B. We prove that NK(B) contained in Beurling-type spaceHα∞(B), α= n+12 .A sufficient and necessary condition forNK(B) non-trivial is given. We discus
the nesting property of NK(B). We obtain the complete characterizations of the boundedness and compactness of weigted composition operators fromNK(B) spaces into Beurling-type classes. We also find the essential norm estimates for these operators. Our results contain the results in the unit disk as particular cases (for example [4], [6] and [9]).
2. NK(B) spaces in the unit ball
The following results play an important role in the proof of our main result. They also have their own interest.
Proposition 2.1. LetK: (0,∞)→[0,∞)be non-decreasing function. ThenNK(B)⊂H∞n+1 2
(B).
Proof. For a∈B, let B1
2 ={z ∈B: |z|<
1
2}. Without loss of generality, assume that K( 3
4)>0.If
f ∈ NK(B),then
kfk2
NK(B)≥K(3/4)
Z
B1 2
|f(z)|2dτ(z).
By By the subharmonicity of|f(z)|2 and hence by ([7], Theorem 2.1.4), we have
|f(0)|2≤ 1
V(B1 2)
Z
B1 2
|f(z)|2dτ(z) = 4nZ B1
2
|f(z)|2dτ(z).
Thus
|f(0)|2≤ 4n
K(3/4)kfk
2
NK(B), f ∈ NK(B).
For every fixedz∈B,we put
(2.1) F(w) =(1− |z|
2)n+12 |f(Φ a(w))|
(1− hw, zi)n+1 , w∈B,
which is clearlyF ∈ H(B). We can prove thatkFk2
NK(B)≤ kfk 2
NK(B),and soF ∈ NK(B).Then, we have
|f(a)|2(1− |a|2)n+1=|F(0)|2≤ 4 n
K(3/4)kfk
2 NK(B),
for allz∈B,which implies that:
kfk2H∞ n+1
2
(B)= sup a∈B
|f(a)|2(1− |a|2)n+1≤ 4 n
K(3/4)kfk
2 NK(B).
That is,NK(B)⊂H∞n+1 2
Proposition 2.2. Forz, w∈Band f ∈ NK(B), we have
(2.2) |f(z)−f(w)| ≤MkfkNK(B)max{(1− |z|2)− n+1
2 ,(1− |w|2)−
n+1
2 }ρ(z, w).
Here,M = 6
n+1
2 22(n+1)√n(3+2√3)
K(3/4) .
Proof. We consider two cases:
Case 1: ρ(z, w)≥1 4.
Since|f(z)−f(w)| ≤ |f(z)|+|f(w)|,by Proposition 2.1, we have
min{(1− |z|2)n+12 ,(1− |w|2)n+12 }|f(z)−f(w)|
≤ (1− |z|2)n+12 |f(z)|+ (1− |w|2)n+12 |f(w)|
≤ 2kfkH∞ n+1
2
(B)≤
22n+1
K(3/4)kfk
2 NK(B)
≤ 2
2n+3ρ(z, w)
K(3/4) kfk
2 NK(B).
Which implies that
|f(z)−f(w)| ≤ 2 2n+3
K(3/4)kfkNK(B)max{(1− |z|
2)−n+1
2 ,(1− |w|2)−
n+1
2 }ρ(z, w).
Case 2: ρ(z, w)>14.
Take and fixw∈B,fromρ(Φa(z), w) =|z|it follows that ifz∈B1
2,(B=B∪S),thenρ(Φa(z), w)<
1 2.
In this case, by Proposition 2.1 and Lemma 1.1, we have
|f(Φw(z))| ≤
kfkH∞ n+1
2
(B)
(1− |Φw(z)|2) n+1
2
≤ 2
2n+1
K(3/4)
kfk2 NK(B)
(1− |Φw(z)|2) n+1
2
= 2
2n+1
K(3/4)
kfk2 NK(B)
(1− |w|2)n+12
1− |w|2
1− |Φw(z)|2
n+12
≤ 6 n+1
2 22n+1 K(3/4)
kfk2 NK(B)
(1− |w|2)n+12 .
By Proposition 1.1, we have
|f(z)−f(w)| ≤ 6 n+1
2 22(n+1)
√
n(3 + 2√3)
K(3/4)
kfk2 NK(B)
(1− |w|2)n+12 ρ(z, w).
Combining the results of the two cases yields
|f(z)−f(w)| ≤MkfkNK(B)max{(1− |z|2)− n+1
2 ,(1− |w|2)−
n+1
2 }ρ(z, w),
whereM = 6
n+1
2 22(n+1)√n(3+2√3)
K(3/4) .
Lemma 2.1. Fora∈B,0< δ <1 andf ∈ NK(B), we have
(2.3) |f(a)−f(δa)| ≤ M
(1− |a|2)n+12 kfk 2 NK(B).
Consequently, for any0< r <1, we have
(2.4) sup
|a|≤r
|f(a)−f(δa)| ≤ M
(1−r2)n+12 kfk 2 NK(B).
Proof. Proposition 2.2 shows that
|f(a)−f(δa)| ≤MkfkNK(B)max{(1− |a|2)− n+1
2 ,(1− |δa|2)−
n+1
2 }ρ(a, δa).
The well-known formula
1− |ρ(a, δa)|2= 1− |Φ
a(δa)|2=
(1− |a|2)(1− |δa|2) |1− ha, δai|2 ,
together with simple calculations gives
ρ(a, δa) = (1− |a|
2)|a|
1−δ|a|2 ≤1.
On the other hand, (1− |δa|2)−n+1
2 ≤(1− |a|2)−
n+1
2 . The inequalities in (2.3) now follow.
If|a| ≤r,then 1−δ|a|2≥1−r2. Taking supremum of (2.3) inayields (2.4).
Theorem 2.1. If
(2.5)
Z 1
0
r2n−1
(1−r2)n+1K(g(r))dr <∞,
thenNK(B)contains all polynomials; otherwise, NK(B) contains only constant functions.
Proof. First assume that (2.5) holds. Let f(z) be a polynomial i.e. (there exists aM >0 such that
|f(z)|2≤M for allz∈
B. Then,
Z
B
|f(z)|2K(G(z, a))dτ(z) = Z
B
|f(Φa(z))|2K(g(z))
dV(z) (1− |z|2)n+1
= 2n
Z 1
0
r2n−1
(1−r2)n+1K(g(r))dr
Z
S
|f◦ϕa(rζ)|2dσ(ζ)
≤ 2nM
Z 1
0
r2n−1
(1−r2)n+1K(g(r))dr.
Sinceais arbitrary, it follows that
kfkNK(B)≤2nM
Z 1
0
r2n−1
(1−r2)n+1K(g(r))dr <∞.
Thus,f ∈ NK(B) and the first half of the theorem is proved.
Now, we assume that the integral in (2.5) is divergent. Let α = (α1,· · · , αn) is an n-tuple of
non-negative integers,|α|=α1+α2+· · ·+αn≥1,f(z) =zα. Then, we have |f(rζ)|2=r2|α||ζα|2,
and
Z
S
|(rζ)α|2dσ(rζ)≥r2|α| (n−1)!α!
(n−1 +|α|)!≥M r
2|α|.
Thus,
kfkNK(B)≥
nM
22|α|−1
Z 1
1 2
r2n−1
(1−r2)n+1K(g(r))dr.
(2.6)
There existsa∈Bsuch thatf(a)6= 0,by the subharmonicity of|f ◦Φa(rζ)|2,
kfkNK(B)≥
3n
2 |f(a)|
2
Z 12
0
r2n−1
(1−r2)n+1K(g(r))dr.
(2.7)
Combining (2.6) and (2.7), we see that (2.5) implies thatkfkNK(B)=∞.It is proved thatf /∈ NK(B) and, sinceαis arbitrary, any non-constant polynomial is not contained in NK(B).We conclude that
Lemma 2.2. Forw∈Bwe define the probe function in NK(B)as
hw(z) =
(1− |w|2)n+1
(1− hz, wi)32(n+1) .
Suppose that condition (2.5) is satisfied. Thenhw∈ NK(B)andkhwkNK(B)≤1.
Proof. Trivially hw∈ NK(B).It is also easy to see that
khwk2NK(B)= sup a∈B
Z
B
(1− |w|2)n+1
(1− hz, wi)32(n+1)
2
K(G(z, a))dτ(z)≤1,
this by a change of variables and since condition (2.5) is satisfied.
Such hw is a normalized reproducing kernel function in the Bergman spaceA2(B). Also note that
hw(w) =
1 1− |w|2
n+12
, ∀w∈B.
In Section 4, we will discuss the estimation for the lower bounded of kWu,φke We will make use of
weakly convergent sequences in the Bergman spaceA2(B).The following lemma plays an important role.
Lemma 2.3. Suppose{fm}m≥1∈A2(B)is a sequence that converges weakly to zero in A2(B). Then
{fm}m≥1 converges weakly to zero in NK(B)as well.
Proof. Let Γ∈ NK(B) be a bounded liner functional onNK(B).By the fact thatkfkNK(B)≤ kfkA2(B),
then
kΓkA2(B) = sup
f∈A2(B)
|Γ(f)| kfkA2(B)
≤ sup
f∈A2(B)
|Γ(f)| kfkNK(B)
≤ sup
f∈NK(B)
|Γ(f)| kfkNK(B)
=kΓkN K(B),
which implies Γ is also a bounded linear functional on A2(B). Since fm → 0 weakly in A2(B), we conclude that Γ(fm)→0.Therefore,fm→0 weakly inNK(B) as well.
Corollary 2.1. Let {wm}m∈N ⊂Band |wm| →1 asm → ∞, then{hwm} converges weakly to zero
inNK(B).
Proof. It is well known that hwm →0 weakly inA2(B) as m→ ∞.Indeed, for any f ∈A2(B),using the reproducing property, we have
hf, hwmi= (1− |wm|
2)n+12 f(w m),
which converges to zero as m → ∞, because the set of polynomials is dense in A2(
B) (see [10], Proposition 2.6). The conclusion of the corollary follows immediately from Lemma 2.3.
3. Weighted composition operators from NK(B) intoHα∞(B)
In this section, we will consider the operator Wu,φ:NK(B)→Hα∞(B).
Theorem 3.1. Let φ : B → B be a holomorphic mapping and u ∈ H(B). For 0 < α < ∞, then
Wu,φ:Hα∞(B)→ NK(B) is a bounded operator if and only if
(3.1) sup
a∈B
|u(z)|(1− |z|2)α
(1− |φ(z)|2)n+12 <∞.
Proof. First assume that condition (3.1) holds, by Proposition 1.1, we have
kWu,φ(f)kH∞
α(B) = sup z∈B
|u(z)||f(φ(z))|(1− |z|2)α
≤ kfkH∞ n+1
2
sup
z∈B
|u(z)|(1− |z|2)α
This implies thatWu,φ:Hα∞(B)→ NK(B) is a bounded operator.
Conversely, assume thatWu,φ:NK(B)→Hα∞(B) is bounded, thenkWu,φ(f)kH∞
α(B)≤ kfkNK(B). Let
hw be the test function in Lemma 2.2 withw=φ(z),then we get
hφ(z)(φ(z)) =
1
1− |φ(z)|2
n+12 .
Hence, there exist a positive constantM such that:
M ≥ khwkNK(B)≥ kWu,φ(hw)kH∞ α(B)≥
|u(z)|(1− |z|2)α
(1− |φ(z)|2)n+12 .
This completes the proof of the theorem.
Using the standard arguments similar to those outlined in proposition 3.11 of [2], we have the following lemma:
Lemma 3.1. Let φ : B → B be a holomorphic mapping and u ∈ H(B). For 0 < α < ∞, then
Wu,φ:Hα∞(B)→ NK(B) is compact if and only if
lim
m→∞kWu,φ(fm)kNK(B)= 0,
for every bounded sequence{fm} ⊂ NK(B) which converges to 0 uniformly on any compact subsets ofBasm→ ∞.
Theorem 3.2. Let φ : B → B be a holomorphic mapping and u ∈ H(B). For 0 < α < ∞, then
Wu,φ:NK(B)→Hα∞(B) is compact if and only if
(3.2) lim
r→1− sup |φ(z)|>r
|u(z)|(1− |z|2)α
(1− |φ(z)|2)n+12 = 0.
Proof. First assume that Wu,φ : NK(B) → Hα∞(B) is compact, then it is bounded. By Theorem
theorem 3.1, we have
L= sup
a∈B
|u(z)|(1− |z|2)α
(1− |φ(z)|2)n+12 <∞,
Note that lim
r→1−L(r) always exists, where:
L(r) = sup
|φ(z)|>r
|u(z)|(1− |z|2)α
(1− |φ(z)|2)n+12 .
Now, we show that (3.4) holds.
Assume on the contrary that there existsε0>0 such that
lim
r→1− sup |φ(z)|>r
|u(z)|(1− |z|2)α
(1− |φ(z)|2)n+12 =ε0.
There exists anr0 ∈(0,1) such that r0 < r <1, we haveL(r)> ε20.Then, by the standard diagonal
process, we can construct a sequence {zm} ⊂ B such that |φ(z)| → 1 asm → ∞, and also for each
m∈N,
|u(z)|(1− |z|2)α
(1− |φ(z)|2)n+12 ≥
ε0
4 .
Clearly, we can assume thatwn =φ(zm) tends tow0 ∈∂Bas m→ ∞. Lethwm =
(1−|wm|2)n+1
(1−hzm,wmi)
3 2(n+1)
be the function in Lemma 2.2 withwn =φ(zm).Thenhwn→hw0 with respect to the compact-open
topology. Definefm=hwm−hw0. Then kfmkNK(B) ≤1 and fm→0 uniformly on compact subsets
ofB. Thus,fm◦φ→0 inHα∞(B) by assumption. But, formbig enough,
kWu,φ(fn)kH∞ α(B)≥
|u(zm)|(1− |zm|2)α
(1− |φ(zm)|2) n+1
2
≥ε0
4,
which is a contradiction.
to zero uniformly on every compact subset ofmathbbB, then for all ε >0 there existsδ∈(0,1) and
mε< msuch that forδ < r <1,we have
kWu,φ(fn)kH∞
α(B) ≤ sup |φ(z)|>r
|u(z)||fm(φ(z))|(1− |z|2)α
+ sup
|φ(z)|≤r
|fm(φ(z))|(1− |z|2)α≤ε.
From this kWu,φ(fn)kH∞
α(B) →0 as m→ ∞, it follows that Wu,φ : NK(B)→ H ∞
α (B) is a compact
operator. This completes the proof of the theorem. As a corollary of Theorems 3.1 and 3.2, we have:
Corollary 3.1. Let φ:B→Bbe a holomorphic mapping and0< α <∞.Then composition operator
Cφ:NK(B)→Hα∞(B)
• is bounded if and only if
(3.3) sup
z∈B
(1− |z|2)α
(1− |φ(z)|2)n+12 <∞;
• is compact if and only if
(3.4) lim
r→1− sup |φ(z)|>r
(1− |z|2)α
(1− |φ(z)|2)n+12
= 0.
4. Essential norms of weighted composition operators from NK(B) intoHα∞(B)
In this section, we study the essential norm of weighted composition operator Wu,φ : NK(B) →
Hα∞(B). Let us denote byC:=C(NK(B), Hα∞(B)) the set of all compact operators acting fromNK(B) intoHα∞(B). Then the essential norm ofWu,φ is defined as follows:
(4.1) kWu,φke= inf
O∈C{kWu,φ− Ok}.
Obviously, the essential norm of a compact operator is zero.
Note that by using the standard argument, it can be shown that a composition operator Cφ : NK(B)→ NK(B) is compact if and only if for any bounded sequence {fm} ⊂ NK(B) converging to zero uniformly on every compact subset ofB,the sequence{kfm◦φk}converges to zero as m→ ∞.
Lemma 4.1. Supposeφ:B→Bis a holomorphic mapping such thatkφk∞<1, andu∈ H(B). For 0< α <∞, thenWu,φ:Hα∞(B)→ NK(B)is compact if and only if
lim
m→∞kWu,φ(fm)kNK(B)= 0,
Proof. Letr=kφk∞ and take an arbitraryf ∈ NK(B).Then we have
kWu,φ(f)kNK(B)≤ kf◦φk∞kukNK(B)≤
sup
{z:|z|≤r} |f(z)|
kukNK(B)<∞.
This show thatWu,φ mapsNK(B) into itself.
Now suppose that{fm}is a bounded sequence inNK(B) that converges to zero uniformly on every compact subset ofB.Applying the above estimate withf =fm,we have
kWu,φ(fm)kNK(B)≤
sup
{z:|z|≤r} |fm(z)|
kukNK(B)<∞.
Since the setz:|z| ≤ris compact, the right-hand side of the last quantity converges to 0 asm→ ∞,
hence so does the sequence{kWu,φ(fm)kNK(B)}.This means that Wu,φis compact.
Theorem 4.1. Letφ:B→Bbe a holomorphic mapping andu∈ H(B). For0< α <∞,suppose that
Wu,φ:NK(B)→Hα∞(B) is a bounded operator. Then
(4.2) kWu,φke≤M lim
r→1− sup |φ(z)|>r
|u(z)|(1− |z|2)α
(1− |φ(z)|2)n+12 ,
whereM = 6
n+1
2 22(n+1)√n(3+2√3)
K(3/4) is the constant from Proposition 2.2.
Proof. SinceWu,φis bounded, we see thatu∈Hα∞(B), and Theorems 3.1, 3.2 shows that
lim
r→1− sup |φ(z)|>r
|u(z)|(1− |z|2)α
(1− |φ(z)|2)n+12
exists and is a real number.
First, we prove that for anyr∈[0,1),
(4.3) kWu,φke≤M sup
|φ(z)|>r
|u(z)|(1− |z|2)α
(1− |φ(z)|2)n+12 .
For eachk∈N,set φk(z) = k+1kz for allz∈B.By Lemma 4.1, Cφk is compact onNK(B), and hence,
O=Wu,φ◦Cφk ∈ C is compact acting fromNK(B) intoHα∞(B).Then for anyk∈N,we have
kWu,φke = inf
O∈C{kWu,φ− Ok} ≤ kWu,φ−Wu,φ◦Cφkke
= sup
kfkNK(B)≤1
k(Wu,φ−Wu,φ◦Cφk)(f)kHα∞(B),
which implies that
(4.4) kWu,φke≤ inf
k∈N
sup
kfkNK(B)≤1
k(Wu,φ−Wu,φ◦Cφk)(f)kHα∞(B)
.
Forf ∈ NK(B),we estimate
k(Wu,φ−Wu,φ◦Cφk)(f)kH∞ α(B)
= sup
z∈B
|u(z)|
f(φ(z))−f
kφ(z)
k+ 1
(1− |z|2)α
≤ sup
|φ(z)|>r
|u(z)|
f(φ(z))−f
kφ(z)
k+ 1
(1− |z|2)α
+ sup
|φ(z)|≤r
|u(z)|
f(φ(z))−f
kφ(z)
k+ 1
(1− |z|2)α
.
On the one hand, by Lemma 2.1, equation (2.3), we have
sup
|φ(z)|>r
|u(z)|
f(φ(z))−f
kφ(z)
k+ 1
(1− |z|2)α
≤ sup |φ(z)|>r
f(φ(z))−f
kφ(z)
k+ 1 sup z∈B
|u(z)|(1− |z|2)α
≤
sup
|φ(z)|>r
M|u(z)|(1− |z|2)α
(1− |φ(z)|2)n+12
kfkNK(B).
On the one hand, by Lemma 2.1, equation (2.4), we have
sup
|φ(z)|≤r
|u(z)|
f(φ(z))−f
kφ(z)
k+ 1
(1− |z|2)α
≤
M rkuk
H∞ α(B)
(k+ 1)(1−r2)α
Therefore, ifkfkNK(B)≤1,then
k(Wu,φ−Wu,φ◦Cφk)(f)kH∞ α(B)
≤ sup
z∈B
|u(z)|
f(φ(z))−f
kφ(z)
k+ 1
(1− |z|2)α
≤ M
sup
|φ(z)|>r
|u(z)|(1− |z|2)α
(1− |φ(z)|2)n+12
+ M rkukHα∞(B)
(k+ 1)(1−r2)α
.
It then follows that
inf
k∈N
sup
kfkNK(B)≤1
k(Wu,φ−Wu,φ◦Cφk)(f)kH∞ α(B)
≤ M inf
k∈N
sup
|φ(z)|>r
|u(z)|(1− |z|2)α
(1− |φ(z)|2)n+12
+ M rkukH∞α(B)
(k+ 1)(1−r2)α
≤ M
sup
|φ(z)|>r
|u(z)|(1− |z|2)α
(1− |φ(z)|2)n+12
.
(4.5)
Combining (4.4) and (4.5), we obtain (4.3).
Now lettingr→1 in (4.3), we arrive at the desired inequality
(4.6) kWu,φke≤M lim
r→1
sup
|φ(z)|>r
|u(z)|(1− |z|2)α
(1− |φ(z)|2)n+12
.
This completes the proof of the theorem.
We now discuss the estimation for the lower bound of the essential norm ofWu,φ:NK(B)→Hα∞(B).
Theorem 4.2. Letφ:B→Bbe a holomorphic mapping andu∈ H(B). For0< α <∞,suppose that
Wu,φ:NK(B)→Hα∞(B) is a bounded operator. Then
(4.7) kWu,φke≥ lim
r→1− sup |φ(z)|>r
|u(z)|(1− |z|2)α
(1− |φ(z)|2)n+12 .
Proof. The case kφk∞<1 is obvious since the right hand side is zero. Now assume thatkφk∞= 1.
For anyr∈(0; 1),the setSr:={z∈B:|φ(z)|> r is not empty. For eachz∈B,consider the probe functionhwin Lemma 2.2 withw=φ(z). Then for any compact operatorO ∈ C we have
kWu,φ− Ok = sup kfkNK(B)≤1
k(Wu,φ− O)(f)kHα∞(B)
≥ k(Wu,φ− O)(hφ(z))kH∞ α(B) ≥ kWu,φ(hφ(z))kH∞
α(B)− kO(hφ(z))kHα∞(B)
≥ |u(z)|(1− |z| 2)α
(1− |φ(z)|2)n+12 − kO(hφ(z))kHα∞(B),
which is equivalent to
(4.8) kWu,φ− Ok+kO(hφ(z))kH∞ α(B)≥
|u(z)|(1− |z|2)α
(1− |φ(z)|2)n+12 .
Taking the supremum onzover the setSr on both sides of (4.8) yields
kWu,φ− Ok+ sup z∈Sr
kO(hφ(z))kH∞
α(B)≥ sup z∈Sr
|u(z)|(1− |z|2)α
(1− |φ(z)|2)n+12 .
which is
(4.9) kWu,φ− Ok+ sup
|φ(z)|>r
kO(hφ(z))kH∞
α(B)≥ sup |φ(z)|>r
|u(z)|(1− |z|2)α
(1− |φ(z)|2)n+12 .
Denote H(r) = sup
|φ(z)|>r
kO(hφ(z))kH∞
α(B). Since H(r) decreases as r increases, limr→1H(r) exists. We
lim
r→1H(r) =L >0.Then there is a sequence{zm} ⊂Bsatisfying|φ(zm)| →1 asm→ ∞,and for each
m∈N,
(4.10) kO(hφ(z))kH∞
α(B)≥
1 2L.
By Corollary 3.1,{hφ(zm)}converges weakly to zero inNK(B).SinceOis compact, we have{kO(hφ(z))kH∞ α(B)}
converges to zero asm→ ∞,which contradicts (4.10). Therefore,
lim
r→1− sup |φ(z)|>r
kO(hφ(z))kH∞ α(B)= 0.
Lettingr→1− on both sides of (4.9), we conclude that for any compact operatorO ∈ C,
kWu,φ− Ok ≥ lim r→1− sup
|φ(z)|>r
|u(z)|(1− |z|2)α
(1− |φ(z)|2)n+12 .
From this, it follows that
kWu,φke= inf
O∈CkWu,φ− Ok ≥r→1lim− sup |φ(z)|>r
|u(z)|(1− |z|2)α
(1− |φ(z)|2)n+12 .
This completes the proof of the theorem.
In conclusion, combining Theorems 4.1 and 4.2, we obtain a full description of the essential norm ofWu,φ.
Theorem 4.3. Letφ:B→Bbe a holomorphic mapping andu∈ H(B). For0< α <∞,suppose that
Wu,φ:NK(B)→Hα∞(B) is a bounded operator. Then
(4.11) kWu,φke≈ lim
r→1− sup |φ(z)|>r
|u(z)|(1− |z|2)α
(1− |φ(z)|2)n+12 .
Acknowledgement. The authors would like to express their thanks to the Deanship of Scien-tific Research at Jazan University Saudi Arabia for funding the work through research project no. 207/SABIC 2/36.
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Department of Mathematics, Faculty of Science, Jazan university, Jazan, Saudi Arabia ∗
