Download Composition followed by differentiation between weighted Bergman-Nevanlinna spaces

Mathematica Aeterna, Vol. 2, 2012, no. 5, 379 - 388
Composition followed by differentiation between
weighted Bergman-Nevanlinna spaces
Ambika Bhat
School of Mathematics,
Shri Mata Vaishno Devi University,
Kakryal, Katra-182320, J & K, India
[email protected]
Zaheer Abbas
,
Department of Applied Mathematics,
Baba Gulam Shah Bad Shah University,
Rajouri, J & K, India.
[email protected]
Ajay K. Sharma
School of Mathematics,
Shri Mata Vaishno Devi University,
Kakryal, Katra-182320, J & K, India
aksju− [email protected]
Abstract
In this paper, we characterize boundedness of Cϕ D acting on weighted
Bergman-Nevanlinna spaces, where Cϕ is the composition operator and
D is the differentiation operator. We also provide a necessary condition
and a sufficient condition for Cϕ D to be compact on weighted BergmanNevanlinna spaces.
Mathematics Subject Classification: Primary 47B33, 46E10, Secondary 30D55.
Keywords: composition operator, differentiation operator, weighted Bergman
Nevanlinna space.
380
1
A. Bhat, Z. Abbas and A. K. Sharma
Introduction
Let D be the open unit disk in the complex plane C, H(D) be the algebra
of all functions holomorphic on D and λ ∈ (−1, ∞) be a real number. Let
1
1
dA(z) = dxdy = rdrdθ be the normalized area measure on D. For each
π
π
λ ∈ (−1, ∞), we set dνλ (z) = (λ + 1)(1 − |z|2 )λ dA(z), z ∈ D. Then dνλ is
a probability measure on D. The weighted Bergman Nevanlinna space A0λ (D)
consists of all f ∈ H(D) such that
||f ||A0λ (D) =
Z
log+ |f (z)|dνλ (z) < ∞,
log x
0
D
where
log+ x =
if x ≥ 1
if x < 1.
In fact, ||f ||A0λ (D) fails to be a norm, but (f, g) → ||f − g||A0λ (D) defines a
translation invariant metric on A0λ (D) and this turns A0λ(D) into a complete
metric space. The space A0λ (D) appears in the limit as p → 0 of the weighted
Bergman space
Apλ (D) = f ∈ H(D) : ||f ||Apλ (D) =
in the sense of
Z
D
|f (z)|p dνλ (z)
1/p
<∞ ,
tp − 1
= log+ t, 0 < t < ∞.
p→0
p
lim
The Bergman-Nevanlinna space A0λ (D) contains all the Bergman spaces Apλ (D)
for all p > 0. Obviously, the inequality
log+ (x) ≤ log(1 + x) ≤ 1 + log+ (x);
x≥0
implies that f ∈ A0λ (D) if and only if
||f ||A0λ(D) ≍
Z
D
log(1 + |f (z)|)dνλ (z) < ∞,
where X ≍ Y means that there is a positive constant C such that C −1 X ≤
Y ≤ CX. See [3] for more about weighted Bergman spaces and weighted
Bergman-Nevanlinna spaces. By the subharmonicity of log(1 + |f (z)|), we
have
||f ||A0λ (D)
, z∈D
(1.1)
log(1 + |f (z)|) ≤ Cλ
(1 − |z|2 )λ+2
for all f ∈ A0λ (D). In particular, (1.1) tells us that if fn → f in A0λ (D), then
fn → f locally uniformly. Here locally uniform convergence means the uniform
381
Composition followed by differentiation
convergence on every compact subset of D.
Let ϕ be a holomorphic self-map D of itself. The composition operator
Cϕ is defined as follows Cϕ (f )(z) = f (ϕ(z)) for all f ∈ H(D). Let D be
the differentiation operator. We know that on a general space of holomorphic functions, the differentiation operator D is typically unbounded. On the
other hand, the composition operator Cϕ is bounded on most of the spaces
of holomorphic functions (see [1] and [6] for details), though the product is
possibly still unbounded there. Hibschweiler and Portnoy [4] defined DCϕ and
Cϕ D and investigated boundedness and compactness of the operators DCϕ
and Cϕ D between weighted Bergman spaces. S. Ohno [5] discussed boundedness and compactness of Cϕ D between Hardy spaces. Recently, there are
some papers that deal with these operators from a particular domain space of
holomorphic functions into another space (see for example, [4],[5] and [7]-[18].
In this paper, we characterize boundedness of Cϕ D : A0λ (D) → A0λ (D). We
also provide a necessary condition and a sufficient condition for Cϕ D to be
compact on weighted Bergman-Nevanlinna spaces.
2
Preliminary Notes
Denote by D(z, r) the pseudohyperbolic disk whose pseudohyperbolic centre
is z and whose pseudo hyperbolic radius is r, that is:
D(z, r) = ω ∈ D :
For z, ω ∈ D with
ρ(z, ω) = (z − ω) <r .
(1 − zω)
(z − ω) < r; 0 < r < 1,
(1 − zω)
we have
(1 − |z|2 )
(1 − |z|2 )
≍
≍ 1 and νλ (D(z, r)) ≍ (1 − |z|2 )(λ+2) .
|1 − zω|
(1 − |ω|2)
See [1] for more information on pseudohyperbolic disks. The next two lemmas
can also be found in [1] (see [2] also).
Lemma 2.1 Let 0 < r < 1. Then there is a sequence {an } in D and a
positive integer M such that
(i) ∪∞
n=1 D(an , r) = D;
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A. Bhat, Z. Abbas and A. K. Sharma
(ii) Each z ∈ D is in atmost M of the
pseudohyperbolic disks D(a1 , 2r), D(a2, 2r), D(a3, 2r) · · · ;
(iii) If n 6= m, then ρ(an , am ) ≥ r/2.
Lemma 2.2 Let λ ∈ (−1, ∞) and β > 0, then there exists a constant
C = C(λ, β) such that
dνλ (ω)
≍ 1, z ∈ D.
D |1 − zω|2+λ+β
Definition 2.3 A positive Borel measure µ on D is called an λ-Carleson
measure if and only if
µ(D(z, r))
< ∞,
sup
2 λ
z∈D (1 − |z| )
and it is called a vanishing λ-Carleson measure if
µ(D(z, r))
lim
= 0.
|z|→1 (1 − |z|2 )λ
(1 − |z|2 )β
Z
The next lemma is proved in [2].
Lemma 2.4 Let λ ∈ (−1, ∞) and 0 < r < 1, then there exists a constant
C = C(λ, r) such that the following inequality holds:
log(1 + |f ′ (z)|) ≤ C
log(1 + |f (ω)|)
dνλ (ω).
(1 − |ω|)λ+3
Z
D(z,r)
Lemma 2.5 Let λ ∈ (−1, ∞) and 0 < r < 1, be fixed. If µ is (λ + 3)Carleson measure on D, then there exists a constant C = C(λ, r) such that
the following inequality holds:
Z
Z
log(1 + |f ′(ω)|)dµ(ω) ≤ C
D
log(1 + |f (ω)|)dµ(ω).
D
Proof. Let 0 < r < 1, be fixed . Pick a sequence {an } in D satisfying
the conditions of Lemma 2.1. For f ∈ A0λ (D), we have
Z
′
log(1+|f (ω)|)dµ(ω) ≤
D
≤
∞ Z
X
n=1 D(an ,r)
∞
X
≤
log(1+|f ′ (ω)|)dµ(ω)
µ(D(an , r))
log(1 + |f ′(ω)|)
sup
ω∈D(an ,r)
n=1
∞
X
µ(D(an , r)) Z
log(1 + |f (ω)|)dµ(ω).
λ+3 D(a ,2r)
n
n=1 (1 − |a|)
Now µ is (λ + 3) -Carleson measure on D, so we have
Z
D
′
log(1 + |f (ω)|)dµ(ω) ≤ C
∞ Z
X
n=1 D(an ,2r)
= CM
Z
D
log(1 + |f (ω)|)dµ(ω)
log(1 + |f (ω)|)dµ(ω).
383
Composition followed by differentiation
3
Boundedness and compactness of CϕD on
A0λ(D)
In this section, we characterize boundedness of Cϕ D : A0λ (D) → A0λ (D). We
also provide a necessary condition and a sufficient condition for Cϕ D to be
compact on A0λ (D).
Theorem 3.1 Let ϕ be a holomorphic self-map of D. Then the following
are equivalent:
1. Cϕ D : A0λ (D) → A0λ (D) is bounded.
2. The pull-back measure νλ ◦ ϕ−1 is a (λ + 3)-Carleson measure on D.
Proof. Suppose that Cϕ D : A0λ(D) → A0λ (D) is bounded. Consider the
function
(1 − |z|2 )λ+4
fz (ω) =
, z ∈ D.
(1 − zω)λ+3
By Lemma 2.2, we have
||fz ||A0λ (D) ≤ ||fz ||A1λ (D) ≍ (1 − |z|)λ+3
for all z ∈ D. Also
fz′ (ω) = (λ + 3)z
(1 − |z|2 )λ+4
.
(1 − zω)λ+4
Therefore ,
|fz′ (ω)| ≤ |z|(λ + 3)
(1 − |z|2 )λ+4
,
(1 − zω)λ+4
and so we have |fz′ (ω)| ≤ C for some constant C = C(λ). Thus log(1 +
|fz′ (ω)|) ≍ |fz′ (ω)| for all z, ω ∈ D. In addition, we have
(1 − |z|2 )
(1 − |z|2 )
≍
≍1
|1 − zω|
(1 − |ω|2)
for ω ∈ D(z, r). Thus |fz′ (ω)| ≍ |z| for ω ∈ D(z, r). Since Cϕ D : A0λ (D) →
A0λ (D) is bounded, there exists C > 0 such that
||Cϕ Dfz ||A0λ (D) ≤ C||fz ||A0λ (D) ≍ (1 − |z|2 )λ+3 .
That is,
(1 − |z|2 )λ+3 ≍ ||Cϕ Dfz ||A0λ (D) ≍
≥C
Z
D
|fz′ (ω)|d(νλ
Z
D
log(1 + |fz′ (ϕ(z)|))dνλ (ω)
−1
◦ ϕ )(ω) ≥ C
Z
D(z,r)
|fz′ (ω)|d(νλ ◦ ϕ−1 )(ω)
384
A. Bhat, Z. Abbas and A. K. Sharma
≍ |z|νλ ◦ ϕ−1 D(z, r),
for all z ∈ D. Consequently,
(νλ ◦ ϕ−1 )D(z, r)
< ∞.
(1 − |z|2 )λ+3
z∈D
sup
Hence νλ ◦ ϕ−1 is an (λ + 3) measure on D. Conversely, suppose that νλ ◦ ϕ−1
is an (λ + 3) measure on D. Then by Lemma 2.4, we have for f ∈ A0λ(D),
||Cϕ Dfz ||A0λ (D) =
Z
log(1 + |f ′(ϕ(ω))|)dνλ(ω)
=
Z
log(1 + |f ′(ϕ(ω))|)d(νλ ◦ ϕ−1 )(ω)
D
D
≤C
Z
D
log(1 + |f (ϕ(ω))|)dνλ(ω) ≍ ||f ||A0λ(D) .
Lemma 3.2 Let ϕ be a holomorphic map of D such that ϕ(D) ⊂ D. Then
Cϕ D : A0λ (D) → A0λ (D) is compact if and only if for every sequence {fn } which
is bounded in A0λ (D) and converges to zero uniformly on compact subsets of D
as n → ∞, we have ||Cϕ Dfn ||A0λ (D) → 0.
Proof follows on the same lines as the proof of proposition 3.11 in [1]. We omit
the details.
We now present a sufficient condition for the compactness of of Cϕ D : A0λ(D) →
A0λ (D).
Theorem 3.3 Let ϕ be a holomorphic map of D such that ϕ(D) ⊂ D.
Then Cϕ D : A0λ (D) → A0λ (D) is compact if the pull-back measure νλ ◦ ϕ−1 is
a vanishing (λ + 3)-Carleson measure on D.
Proof. Suppose that νλ ◦ ϕ−1 is a vanishing (λ + 3)-Carleson measure on D.
Then
(νλ ◦ ϕ−1 )D(a, r)
→ 0 as |a| → 1.
(1 − |a|2 )λ+3
Suppose that {fm } is a bounded sequence in A0λ (D) that converges to zero
uniformly on compact subsets of D. Let {an } be a sequence as in Lemma 2.1
such that |a1 | < |a2 | < |a3 | · · · . Then for each ǫ > 0 we have
(νλ ◦ ϕ−1 )(D(an , r)) < ǫ(1 − |an |2 )λ+3
for all an ∈ D such that |an | > r. Thus
||Cϕ Dfm ||A0λ (D) ≍
Z
D
′
log(1 + |fm
(ϕ(z))|)dνλ (z)
385
Composition followed by differentiation
=
Z
=
Z
D
′
log(1 + |fm
(z)|)d(νλ ◦ ϕ−1 )(z)
|z|≤r0
+
Z
′
log(1 + |fm
(z)|)d(νλ ◦ ϕ−1 )(z)
|z|>r0
′
log(1 + |fm
(z)|)d(νλ ◦ ϕ−1 )(z).
Since {fm } is a bounded sequence in A0λ (D) that converges to zero uniformly
on compact subsets of D,
lim
Z
m→∞ |z|≤r0
′
log(1 + |fm
(z)|)d(νλ ◦ ϕ−1 )(z) = 0,
whereas the second term in the above expression is bounded by
∞ Z
X
′
log(1 + |fm
(z)|)d(νλ ◦ ϕ−1 )(z)
n=k+1 D(an ,r)
≤
∞
X
(νλ ◦ ϕ−1 )(D(an , r))
sup
z∈D(an ,r)
n=k+1
∞
X
(νλ ◦ ϕ−1 )(D(an , r))
≤
(1 − |an |2 )λ+3
n=k+1
≤ ǫCM
Z
D
Z
′
log(1 + |fm
(z)|)
D(an ,2r)
log(1 + |fm (z)|)dνλ (z)
log(1 + |fm (z)|)dνλ (z) = ǫCM||fm ||A0λ .
Since ǫ > 0 is arbitrary, we have ||Cϕ Dfm ||A0λ (D) → 0 as m → ∞. Hence
Cϕ D : A0λ (D) → A0λ (D) is compact.
Finally, we provide a necessary condition for compactness of Cϕ D : A0λ (D) →
A0λ (D).
Theorem 3.4 Let ϕ be a holomorphic map of D such that ϕ(D) ⊂ D.
Then if Cϕ D : A0λ (D) → A0λ (D) is bounded, then νλ ◦ ϕ−1 is a vanishing
(λ + 2)-Carleson measure on D.
Proof. Let {an } be a sequence in D such that |an | → 1 as n → ∞.
Consider the family of functions
(1 − |an |)λ+3
(1 − |an |2 )λ+2
fn (z) =
exp
.
2|an |(λ + 2)
(1 − an z)2(λ+2)
Clearly, fn → 0 uniformly on compact subsets of D as n → ∞. Also
||fn ||A0λ (D) ≤ 1 +
Z
D
log
+
(1 − |an |2 )λ+2 exp
dνλ (z)
2|an |(λ + 2)
(1 − an z)2(λ+2) (1 − |an |)λ+3
h
386
A. Bhat, Z. Abbas and A. K. Sharma
≤1+C
Z
D
(1 − |an |2 )λ+2
dνλ (z) ≤ 1 + C.
|1 − an z|2(λ+2)
Moreover,
fn′ (z)
and so
|fn′ (z)|
(1 − |an |2 )2λ+5
(1 − |an |2 )λ+2
,
a
exp
=
n
|an |(1 − an z)2λ+5
(1 − an z)2(λ+2)
(1 − |an |2 )2λ+5
(1 − |an |2 )λ+2
=
exp
Re
|1 − an z|2λ+5
(1 − an z)2(λ+2)
Now
Re
(1 − |an |2 )λ+2
(1 − an z)2(λ+2)
≍
.
1
,
(1 − |an |)λ+2
whenever z ∈ D(an , r). Thus
log(1 + |fn′ (z)|) ≥ log+ |fn′ (z)| ≥
C
(1 − |an |2 )λ+2
if z ∈ D(an , r). Therefore,
C
(νλ ◦ ϕ−1 )(D(an , r))
(1 − |an |2 )λ+2
≤
Z
D(an ,r)
log+ |fn′ (z)|d(νλ ◦ ϕ−1 )(z) ≤ ||Cϕ Dfn ||A0λ (D) .
But compactness of Cϕ D forces ||Cϕ Dfn ||A0λ (D) to tend to zero as |an | → 1.
Thus
(νλ ◦ ϕ−1 )(D(an , r))
lim
= 0,
|an |→1
(1 − |an |2 )λ+2
and so νλ ◦ ϕ−1 is a vanishing (λ + 2)-Carleson measure on D.
ACKNOWLEDGEMENTS. The work of first author and third author
is a part of the research project sponsored by (NBHM)/DAE, India (Grant
No. 48/4/2009/R&D-II/426).
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Received: August, 2011