Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2010, Article ID 246287, 15 pages
doi:10.1155/2010/246287
Research Article
Weighted Differentiation Composition Operators
from the Mixed-Norm Space to the nth
Weigthed-Type Space on the Unit Disk
Stevo Stević
Mathematical Institute of the Serbian Academy of Sciences and Arts, Knez Mihailova 36/III,
11000 Belgrade, Serbia
Correspondence should be addressed to Stevo Stević, sstevic@ptt.rs
Received 26 March 2010; Accepted 9 May 2010
Academic Editor: Narcisa C. Apreutesei
Copyright q 2010 Stevo Stević. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The boundedness and compactness of the weighted differentiation composition operator from the
mixed-norm space to the nth weighted-type space on the unit disk are characterized.
1. Introduction
Throughout this paper D will denote the open unit disk in the complex plane C, HD
the class of all holomorphic functions on D, and H ∞ H ∞ D the space of all bounded
holomorphic functions on D with the norm f∞ supz∈D |fz|.
The mixed norm space Hp,q,γ Hp,q,γ D, 0 < p, q < ∞, −1 < γ < ∞, consists of all
f ∈ HD such that
q
f Hp,q,γ
1
q
Mp f, r 1 − rγ dr < ∞,
1.1
0
where
Mp f, r 1
2π
1/p
2π p
iθ .
f re dθ
0
1.2
2
Abstract and Applied Analysis
A positive continuous function on D is called weight. Let μz be a weight and n ∈ N0 .
n
The nth weighted-type space on D, denoted by Wμ D, consists of all f ∈ HD such that
n μz
bWn
f
:
sup
z
f
< ∞.
μ D
z∈D
1.3
The space was recently introduced by this author in 1 as an extension of several weightedtype spaces which attracted a lot of attention in last few decades. For instance, when n 0,
the space becomes the weighted-type space Hμ∞ D see, e.g., 2–4, when n 1, the Blochtype space Bμ D see, e.g., 5–7, and for n 2, the Zygmund-type space Zμ D. Some
information on Zygmund-type spaces on D and some operators on them can be found, for
example, in 8–10 and on the unit ball, for example, in 11, 12.
n
f is a seminorm on the nth weighted-type space Wμ D and a
The quantity bWn
μ D
n
norm on Wμ D/Pn−1 , where Pn−1 is the set of all polynomials whose degrees are less than
or equal to n − 1. A natural norm on the nth weighted-type space is introduced as follows:
n−1 j f n f .
f 0 bWn
Wμ D
D
μ
1.4
j0
With this norm the nth weighted-type space becomes a Banach space.
n
n
The little nth weighted-type space, denoted by Wμ,0 D, is a closed subspace of Wμ D
consisting of those f for which
lim μzf n z 0.
|z| → 1
1.5
An analytic self-map ϕ : D → D induces the composition operator Cϕ on HD,
defined by Cϕ fz fϕz for f ∈ HD see, e.g., 8, 13–16.
Let ϕ be an analytic self-map of D, u ∈ HD, and m ∈ N. Then the weighted
m
, is defined on HD by
differentiation composition operator, denoted by Dϕ,u
m
fz uzf m ϕz ,
Dϕ,u
f ∈ HD.
1.6
Recently there has been some interest in studying some particular cases of operator
m
see, e.g., 17–25. For some other products of linear operators on spaces of holomorphic
Dϕ,u
functions see also recent papers 11, 26–32.
m
from Hp,q,γ to
Here we study the boundedness and compactness of the operator Dϕ,u
nth weighted-type spaces, where n ∈ N.
Throughout this paper, constants are denoted by C; they are positive and may differ
from one occurrence to the other. The notation A B means that there is a positive constant
C such that B/C ≤ A ≤ CB.
Abstract and Applied Analysis
3
2. Auxiliary Results
Here we quote some auxiliary results which will be used in the proofs of the main results.
The first lemma can be proved in a standard way see, e.g., in 13, Proposition 3.11 or in 15,
Lemma 3.
Lemma 2.1. Assume that m ∈ N0 , n ∈ N, p, q > 0, γ > −1, ϕ is an analytic self-map of D and
n
n
m
m
u ∈ HD. Then the operator Dϕ,u
: Hp,q,γ → Wμ is compact if and only if Dϕ,u
: Hp,q,γ → Wμ
is bounded and for any bounded sequence fk k∈N in Hp,q,γ which converges to zero uniformly on
n
m
fk → 0 in Wμ as k → ∞.
compact subsets of D, Dϕ,u
The next lemma is known, but we give a proof of it for the benefit of the reader.
Lemma 2.2. Assume that n ∈ N0 , 0 < p, q < ∞, −1 < γ < ∞ and f ∈ Hp,q,γ . Then there is a positive
constant C independent of f such that
f n f z ≤ C Hp,q,γ
1 − |z|2
γ1/q1/pn .
2.1
Proof. By the monotonicity of the integral means, using the well-known asymptotic formula
1
q
Mp f, r 1
q
− r dr f0 γ
1
0
q
Mp f n , r 1 − rγnq dr,
2.2
0
and Theorem 7.2.5 in 33, we have that
q
f Hp,q,γ ≥
1
1|z|/2
q
Mp f n , r 1 − rγnq dr
γ1nq
1 |z| q
≥ CMp f n ,
1 − |z|2
2
γ1nqq/p n q
≥ C 1 − |z|2
f z ,
2.3
from which the result follows.
The following lemma can be found in 34.
Lemma 2.3. For β > −1 and m > 1 β one has
1
0
1β−m
1 − rβ
,
m dr ≤ C 1 − ρ
1 − ρr
0 < ρ < 1.
A proof of the next lemma can be found in 35, Lemma 2.3.
2.4
4
Abstract and Applied Analysis
Lemma 2.4. Assume a > 0 and
1
1
a
a1
aa 1 a 1a 2
Dn a n−2
n−2
aj
aj 1
j0
j0
Then Dn a ···
1
···
an−1
· · · a n − 1a n .
···
n−2
···
aj n−1 j0
2.5
n−1
j1 j!.
The following formula
n
n
f k ϕz
f ◦ϕ
z k1
k1 ,...,kn
n
n!
k1 ! · · · kn ! j1
ϕj z
j!
kj
,
2.6
where the second sum is over all nonnegative integers k1 , k2 , . . . , kn satisfying k k1 k2 · · ·kn and k1 2k2 · · ·nkn n, is attributed to Faà di Bruno 36. By using Bell polynomials
Bn,k x1 , . . . , xn−k1 it can be written as follows:
n
n
f k ϕz Bn,k ϕ
z, ϕ
z, . . . , ϕn−k1 z .
f ◦ϕ
z 2.7
k0
For n ∈ N the last sum can go from k 1 since Bn,0 ϕ
z, ϕ
z, . . . , ϕn1 z 0; however
we will keep the summation since for n 0 the only existing term B0,0 is equal to 1 and we
will use it.
The Leibnitz formula along with 2.6 yields
n
l
n Cln un−l z g k ϕz Bl,k ϕ
z, . . . , ϕl−k1 z .
uzg ϕz
l0
2.8
k0
Hence we have the next result.
Lemma 2.5. Assume that g, u ∈ HD and ϕ is an analytic self-map of D. Then
n
n
n uzg ϕz
g k ϕz
Cln un−l zBl,k ϕ
z, . . . , ϕl−k1 z .
k0
2.9
lk
n
m
3. The Boundedness and Compactness of Dϕ,u
: Hp,q,γ → Wμ
m
This section characterizes the boundedness and compactness of the operator Dϕ,u
: Hp,q,γ →
n
Wμ .
Abstract and Applied Analysis
5
Theorem 3.1. Suppose that m, n ∈ N, 0 < p, q < ∞, −1 < γ < ∞, ϕ is an analytic self-map of the
n
m
: Hp,q,γ → Wμ is bounded if and
unit disk, u ∈ HD, and μ is a weight. Then the operator Dϕ,u
only if for each k ∈ {0, 1, . . . , n}
μz nlk Cln un−l zBl,k ϕ
z, . . . , ϕl−k1 z Ik : sup
< ∞.
2 γ1/q1/pmk
z∈D
1 − ϕz
3.1
n
m
Moreover if Dϕ,u
: Hp,q,γ → Wμ is bounded, then the following asymptotic relation holds
m Dϕ,u n
Hp,q,γ → Wμ /Pn−1
n
Ik .
3.2
k0
n
m
Proof. First assume that Dϕ,u
: Hp,q,γ → Wμ is bounded; then there exists a constant C such
that
m Dϕ,u f n
Wμ
≤ Cf Hp,q,γ
3.3
for all f ∈ Hp,q,γ .
For a fixed w ∈ D, t ≥ γ 1/q, and constants c1 , . . . , cn1 , set
gw z n1
j1
cj
m−1 gw,j z,
3.4
j 1, . . . , n 1.
3.5
j t 1/p l
l0
where
gw,j z 1 − |w|2
jt−γ1/q
1 − wz
1/pjt
,
By 33, Theorem 1.4.10, we get
Mp gw,j , r ≤ C
1 − |w|2
jt−γ1/q
1 − r|w|jt
,
j 1, . . . , n 1.
3.6
6
Abstract and Applied Analysis
Applying Lemma 2.3, we have that
gw,j q
Hp,q,γ
1
q
Mp gw,j , r 1 − rγ dr
0
≤C
qjt−γ1
1 1 − |w|2
0
qjt
1 − r|w|
1 − rγ dr
3.7
≤ C.
Therefore gw ∈ Hp,q,γ , and moreover supw∈D gw Hp,q,γ < ∞.
Now we show that for each s ∈ {m, m 1, . . . , m n}, there are constants c1 , c2 , . . . , cn1 ,
such that
s
gw w 1 − |w|2
ws
sγ1/q1/p ,
t
gw w 0,
t ∈ {m, . . . , m n} \ {s}.
3.8
By differentiating function gw , for each s ∈ {m, . . . , m n}, 3.8 becomes
c1 c2 · · · cn1 0,
t p−1 m 1 c1 t p−1 m 2 c2 · · · t p−1 m n 1 cn1 0,
..
.
s−m s−m t p−1 m j c1 · · · t p−1 m n j cn1 1,
j1
3.9
j1
..
.
n n t p−1 m j c1 · · · t p−1 m n j cn1 0.
j1
j1
Applying Lemma 2.4 with a t 1/p m 1 > 0 and where n → n 1, we see that
the determinant of system 3.9 is different from zero, as claimed.
By gw,k , k ∈ {0, 1, . . . , n}, denote the corresponding family of functions which satisfy
3.8 with s m k. Then, for each fixed k ∈ {0, 1, . . . , n}, inequality 3.3 along with 2.9
and 3.8 implies that for each ϕw /0
km μwϕw nlk Cln un−l wBl,k ϕ
w, . . . , ϕl−k1 w 2 γ1/q1/pkm
1 − ϕw
m ≤ CsupDϕ,u
gϕw,k w∈D
n
Wμ
m ≤ CDϕ,u
n
Hp,q,γ → Wμ
.
3.10
Abstract and Applied Analysis
7
From 3.10 it follows that for each k ∈ {0, 1, . . . , n},
μz nlk Cln un−l zBl,k ϕ
z, . . . , ϕl−k1 z m
≤ CDϕ,u
Hp,q,γ → Wn
.
sup
γ1/q1/pkm
μ
2
|ϕz|>1/2
1 − ϕz
3.11
Let
hk z zk ,
k m, . . . , n m.
3.12
for each k ∈ N.
3.13
Then clearly
hk Hp,q,γ ≤ 1,
By formula 2.9 applied to the function fz hm z we get
m
Dϕ,u
hm
n
m z hm
ϕz
n
Cln un−l zBl,0 ϕ
z, . . . , ϕl1 z
l0
n
m! Cln un−l zBl,0 ϕ
z, . . . , ϕl1 z ,
3.14
l0
n
m
which along with the boundedness of the operator Dϕ,u
: Hp,q,γ → Wμ and 3.13 implies
that
n
m m m n n−l
l1
.
m!sup μz Cl u
zBl,0 ϕ z, . . . , ϕ
z ≤ Dϕ,u
z n ≤ Dϕ,u
n
l0
Wμ
Hp,q,γ → Wμ
z∈D
3.15
Now assume that we have proved that for j ∈ {0, 1, . . . , k − 1} and a k ≤ n
n n n−l
m .
supμz Cl u
zBl,j ϕ
z, . . . , ϕl−j1 z ≤ CDϕ,u
n
Hp,q,γ → Wμ
lj
z∈D
3.16
Applying 2.9 to the function fz hmk z, k ∈ {0, 1, . . . , n}, and noticing that
s
hmk z ≡ 0 for s > m k, we get
m
Dϕ,u
hmk
n
z k
n
mj hmk ϕz
Cln un−l zBl,j ϕ
z, . . . , ϕl−j1 z
j0
lj
n
k
k−j Cln un−l zBl,j ϕ
z, . . . , ϕl−j1 z .
m k · · · k − j 1 ϕz
j0
lj
3.17
8
Abstract and Applied Analysis
n
m
From 3.17, the boundedness of the operator Dϕ,u
: Hp,q,γ → Wμ , the fact
that ϕ∞ ≤ 1, the triangle inequality, noticing that m k! is the coefficient at
n
n n−l
zBl,k ϕ
z, . . . , ϕl−k1 z, and finally using hypothesis 3.16 we get
lk Cl u
n
m n n−l
l−k1
.
sup μz Cl u
zBl,k ϕ z, . . . , ϕ
z ≤ CDϕ,u
n
lk
Hp,q,γ → Wμ
z∈B
3.18
Hence by induction, 3.18 holds for each k ∈ {0, 1, . . . , n}.
From 3.18, for each fixed k ∈ {0, 1, . . . , n}
μz nlk Cln un−l zBl,k ϕ
z, . . . , ϕl−k1 z sup
2 γ1/q1/pkm
|ϕz|≤1/2
1 − ϕz
n
m n n−l
l−k1
≤ C sup μz Cl u
.
zBl,k ϕ z, . . . , ϕ
z ≤ CDϕ,u
n
lk
Hp,q,γ → Wμ
z∈B
3.19
Inequalities 3.11 and 3.19 imply
n
m Ik ≤ CDϕ,u
n
Hp,q,γ → Wμ
k0
.
3.20
Now assume that 3.1 holds. Then for any f ∈ Hp,q,γ , by 2.9 and Lemma 2.2 we have
n
n
m n mk
n
n−l
l−k1
Cl u
μz Dϕ,u f
ϕz
z μz f
zBl,k ϕ z, . . . , ϕ
z k0
lk
n n
mk n n−l
l−k1
≤ μz f
ϕz Cl u
zBl,k ϕ z, . . . , ϕ
z k0
lk
3.21
l−k1
n μz n Cn un−l zB
z l,k ϕ z, . . . , ϕ
lk l
≤ C f Hp,q,γ
3.22
2 γ1/q1/pkm
k0
1 − ϕz
n
μz nlk Cln un−l zBl,k ϕ
z, . . . , ϕl−k1 z ≤ C f Hp,q,γ sup
2 γ1/q1/pkm
k0 z∈D
1 − ϕz
3.23
Abstract and Applied Analysis
9
We also have that for each s ∈ {1, . . . , n − 1}
s
s
m s mk
s s−l
l−k1
Dϕ,u f
Cl u
ϕ0
0 f
0Bl,k ϕ 0, . . . , ϕ
0 k0
lk
l−k1
s s Cs us−l 0B
0 l,k ϕ 0, . . . , ϕ
lk
l
≤ Cf Hp,q,γ
,
2 γ1/q1/pmk
k0
1 − ϕ0
3.24
f Hp,q,γ
2 γ1/q1/pm .
1 − ϕ0
m
Dϕ,u f 0 |u0|f m ϕ0 ≤ C|u0| n
m
Using 3.23, 3.24, and 3.1 it follows that the operator Dϕ,u
: Hp,q,γ → Wμ is bounded.
From 3.23 and 3.20 the asymptotic relation 3.2 follows.
Theorem 3.2. Suppose that m, n ∈ N, 0 < p, q < ∞, −1 < γ < ∞, ϕ is an analytic self-map of the
n
m
unit disk, u ∈ HD, and μ is a weight. Then the operator Dϕ,u
: Hp,q,γ → Wμ,0 is bounded if and
n
m
only if Dϕ,u
: Hp,q,γ → Wμ is bounded and for each k ∈ {0, 1, . . . , n}
n
n n−l
l−k1
lim μz Cl u
zBl,k ϕ z, . . . , ϕ
z 0.
lk
|z| → 1
n
3.25
n
m
m
Proof. The boundedness of Dϕ,u
: Hp,q,γ → Wμ,0 clearly implies that Dϕ,u
: Hp,q,γ → Wμ is
m
hm ∈
bounded. Applying 2.9 to the function fz hm z and using the assumption Dϕ,u
n
Wμ,0 it follows that
n
n m
n
n−l
l1
μz Dϕ,u hm
z m!μz Cl u
zBl,0 ϕ z, . . . , ϕ
z −→ 0,
3.26
l0
as |z| → 1, which is 3.25 for k 0.
Assume that we have proved the following inequalities:
n n n−l
lim μz Cl u
zBl,j ϕ
z, . . . , ϕl−j1 z 0,
|z| → 1
lj
3.27
for j ∈ {0, 1, . . . , k − 1} and a k ≤ n.
Applying formula 2.9 to the function fz hmk z, k ∈ {0, 1, . . . , n}, we get 3.17.
From 3.17, by using the boundedness of function ϕ, the triangle inequality, noticing that the
coefficient at nlk Cln un−l zBl,k ϕ
z, . . . , ϕl−k1 z is independent of z, and finally using
10
Abstract and Applied Analysis
hypothesis 3.27, we easily obtain
n
n n−l
l−k1
lim μz Cl u
zBl,k ϕ z, . . . , ϕ
z 0.
lk
|z| → 1
3.28
Hence by induction we get that 3.25 holds for each k ∈ {0, 1, . . . , n}.
n
m
Now assume that Dϕ,u
: Hp,q,γ → Wμ is bounded and 3.25 holds for each k ∈
{0, 1, . . . , n}. For each polynomial p we have
n
n
m n k
n n−l
l−k1
Cl u
μz Dϕ,u p
ϕz
z μz p
zBl,k ϕ z, . . . , ϕ
z k0
lk
n n
k n n−l
l−k1
≤
zBl,k ϕ z, . . . , ϕ
z −→ 0,
p μz Cl u
∞
k0
lk
3.29
as |z| → 1.
n
m
From 3.29 we have that, for each polynomial p, Dϕ,u
p ∈ Wμ,0 . The set of all
polynomials is dense in Hp,q,γ , so we have that for each f ∈ Hp,q,γ , there is a sequence
of polynomials pk k∈N such that f − pk Hp,q,γ → 0 as k → ∞. Thus the boundedness of
n
m
: Hp,q,γ → Wμ implies
Dϕ,u
m
m
pk Dϕ,u f − Dϕ,u
n
Wμ
m ≤ Dϕ,u
n
Hp,q,γ → Wμ
f − pk −→ 0,
Hp,q,γ
as k −→ ∞.
n
3.30
n
m
m
Hence Dϕ,u
Hp,q,γ ⊆ Wμ,0 , from which the boundedness of Dϕ,u
: Hp,q,γ → Wμ,0 follows,
completing the proof of the theorem.
Theorem 3.3. Suppose that m, n ∈ N, 0 < p, q < ∞, −1 < γ < ∞, ϕ is an analytic self-map of the
n
m
unit disk, u ∈ HD, and μ is a weight. Then the operator Dϕ,u
: Hp,q,γ → Wμ is compact if and
n
m
only if Dϕ,u
: Hp,q,γ → Wμ is bounded and for each k ∈ {0, 1, . . . , n}
μz nlk Cln un−l zBl,k ϕ
z, . . . , ϕl−k1 z 0.
lim
2 γ1/q1/pkm
|ϕz| → 1
1 − ϕz
n
3.31
m
Proof. First assume that Dϕ,u
: Hp,q,γ → Wμ is bounded and 3.31 holds. By Theorem 3.1 we
have that for each k ∈ {0, 1, . . . , n}, 3.1 holds.
Abstract and Applied Analysis
11
Let fi i∈N be a sequence in Hp,q,γ such that supi∈N fi Hp,q,γ ≤ L and fi converges to 0
uniformly on compact subsets of D as i → ∞. By the assumption, for any ε > 0, there is a
δ ∈ 0, 1, such that for each k ∈ {0, 1, . . . , n} and δ < |ϕz| < 1
μz nlk Cln un−l zBl,k ϕ
z, . . . , ϕl−k1 z < ε.
2 γ1/q1/pkm
1 − ϕz
3.32
We have
m Dϕ,u fi n
Wμ
n−1 m n m j supμz Dϕ,u fi
z Dϕ,u fi
0
z∈D
j0
n
n
mk supμz fi
Cln un−l zBl,k ϕ
z, . . . , ϕl−k1 z ϕz
z∈D
k0
lk
j
j
n−1 j j−l
mk l−k1
fi
Cl u
ϕ0
0Bl,k ϕ 0, . . . , ϕ
0 j0 k0
lk
⎛
⎞
n n
mk n n−l
l−k1
⎝
⎠
sup sup
≤
μz fi
ϕz Cl u
zBl,k ϕ z, . . . , ϕ
z k0
lk
|ϕz|≤δ |ϕz|>δ
j
j
n−1 j j−l
mk l−k1
fi
Cl u
ϕ0
0Bl,k ϕ 0, . . . , ϕ
0 J1 J2 J3 .
j0 k0
lk
3.33
Now we estimate J1 , J2 , and J3 :
n n
mk n n−l
l−k1
J1 sup μz fi
ϕz Cl u
zBl,k ϕ z, . . . , ϕ
z |ϕz|≤δ
k0
lk
n
mk
≤
sup fi
w sup μz Cln un−l zBl,k ϕ
z, . . . , ϕl−k1 z |ϕz|≤δ
k0 |w|≤δ
lk
n
μz nlk Cln un−l zBl,k ϕ
z, . . . , ϕl−k1 z mk
≤
sup fi
wsup
2 γ1/q1/pmk
z∈D
k0 |w|≤δ
1 − ϕz
n
n
mk
sup fi
wIk −→ 0,
k0 |w|≤δ
3.34
as i −→ ∞,
where in 3.34 we have used the fact that from fi → 0 uniformly on compact subsets of D as
s
i → ∞ it follows that for each s ∈ N, fi → 0 uniformly on compact subsets of D as i → ∞.
12
Abstract and Applied Analysis
The fact that
j
j
n−1 j j−l
mk l−k1
J3 Cl u
ϕ0
0Bl,k ϕ 0, . . . , ϕ
0 −→ 0,
fi
j0 k0
lk
3.35
as i → ∞, is proved similarly; so we omit it.
By Lemma 2.2 and 3.32 we have that
n
μz nlk Cln un−l zBl,k ϕ
z, . . . , ϕl−k1 z J2 ≤ Cfi Hp,q,γ
sup
< Cεn 1L.
2 γ1/q1/pkm
k0 |ϕz|>δ
1 − ϕz
3.36
From 3.34, 3.35, and 3.36 we obtain
m lim Dϕ,u
fi i→∞
n
Wμ
0.
3.37
From this and applying Lemma 2.1 the implication follows.
n
n
m
m
: Hp,q,γ → Wμ is compact; then clearly Dϕ,u
: Hp,q,γ → Wμ is
Now assume that Dϕ,u
bounded. Let zi i∈N be a sequence in D such that |ϕzi | → 1 as i → ∞. If such a sequence
does not exist, then the conditions in 3.31 automatically hold.
Let gw,k , k ∈ {0, 1, . . . , n} be as in Theorem 3.1. Then the sequences gϕzi ,k i∈N are
m
: Hp,q,γ →
bounded and gϕzi ,k → 0 uniformly on compact subsets of D as i → ∞. Since Dϕ,u
n
Wμ is compact, we have that for each k ∈ {0, 1, . . . , n}
m
lim Dϕ,u
gϕzi ,k i→∞
n
Wμ
0.
3.38
On the other hand, from 3.10 we obtain
m
Dϕ,u gϕzi ,k n
Wμ
km Cμzi ϕzi nlk Cln un−l zi Bl,k ϕ
zi , . . . , ϕl−k1 zi ≥
, 3.39
2 γ1/q1/pkm
1 − ϕzi which along with |ϕzi | → 1 as i → ∞ and 3.38 implies that
μzi nlk Cln un−l zi Bl,k ϕ
zi , . . . , ϕl−k1 zi ,
lim
2 γ1/q1/pkm
i→∞
1 − ϕzi for each k ∈ {0, 1, . . . , n}, from which 3.31 holds in this case.
3.40
Abstract and Applied Analysis
13
n
m
4. The Compactness of the Operator Dϕ,u
: Hp,q,γ → Wμ,0
n
m
The compactness of Dϕ,u
: Hp,q,γ → Wμ,0 is characterized here. The proof of the next lemma
is similar to the proof of the corresponding result in 14.
Lemma 4.1. Suppose that n ∈ N0 and μ is a radial weight such that lim|z| → 1 μz 0. A closed set
n
K in Wμ,0 is compact if and only if it is bounded and satisfies
lim sup μzf n z 0.
|z| → 1 f∈K
4.1
Theorem 4.2. Suppose that m, n ∈ N, 0 < p, q < ∞, −1 < γ < ∞, ϕ is an analytic self-map of
the unit disk, u ∈ HD and μ is a radial weight such that lim|z| → 1 μz 0. Then the operator
n
m
: Hp,q,γ → Wμ,0 is compact if and only if for each k ∈ {0, 1, . . . , n}
Dϕ,u
μz nlk Cln un−l zBl,k ϕ
z, . . . , ϕl−k1 z 0.
lim
2 γ1/q1/pkm
|z| → 1
1 − ϕz
4.2
n
m
Proof. First assume that Dϕ,u
: Hp,q,γ → Wμ,0 is compact. Then it is bounded and since the
test functions in 3.12 belong to Hp,q,γ D, we have that 3.25 holds. Beside this the operator
n
m
: Hp,q,γ → Wμ is compact too, so that 3.31 holds. Hence, if ϕ∞ < 1, from 3.25 for
Dϕ,u
each k ∈ {0, 1, . . . , n} we get
μz nlk Cln un−l zBl,k ϕ
z, . . . , ϕl−k1 z 2 γ1/q1/pkm
1 − ϕz
μz nlk Cln un−l zBl,k ϕ
z, . . . , ϕl−k1 z ≤
−→ 0,
2 γ1/q1/pkm
1 − ϕ∞
4.3
as |z| → 1, hence we obtain 4.2 in this case.
Now assume ϕ∞ 1. Let ϕzi i∈N be a sequence such that |ϕzi | → 1 as i → ∞.
Then from 3.31 we have that for every ε > 0, there is an r ∈ 0, 1 such that for each k ∈
{0, 1, . . . , n}
μz nlk Cln un−l zBl,k ϕ
z, . . . , ϕl−k1 z <ε
2 γ1/q1/pkm
1 − ϕz
4.4
when r < |ϕz| < 1, and from 3.25 there exists a σ ∈ 0, 1 such that for σ < |z| < 1
n
γ1/q1/pkm
n n−l
l−k1
μz Cl u
.
zBl,k ϕ z, . . . , ϕ
z < ε 1 − r 2
lk
4.5
14
Abstract and Applied Analysis
Therefore, when σ < |z| < 1 and r < |ϕz| < 1, we have that
μz nlk Cln un−l zBl,k ϕ
z, . . . , ϕl−k1 z < ε.
2 γ1/q1/pkm
1 − ϕz
4.6
On the other hand, if |ϕz| ≤ r and σ < |z| < 1, from 4.5 we obtain
μz nlk Cln un−l zBl,k ϕ
z, . . . , ϕl−k1 z 2 γ1/q1/pkm
1 − ϕz
<
μz nlk Cln un−l zBl,k ϕ
z, . . . , ϕl−k1 z 1 − r 2 γ1/q1/pkm
4.7
< ε.
Combining the last two inequalities we obtain 4.2, as desired.
Now assume that 4.2 holds. Taking the supremum in 3.22 over f in the unit ball of
Hp,q,γ , then letting |z| → 1 is such obtained inequality and using 4.2 we get
lim
sup
|z| → 1 f
Hp,q,γ ≤1
m n μz Dϕ,u f
z 0.
4.8
n
m
Hence by Lemma 4.1 the compactness of the operator Dϕ,u
: Hp,q,γ → Wμ,0 follows.
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