RESEARCH PAPER
CHARACTERIZATION OF WEIGHTED ANALYTIC
BESOV SPACES IN TERMS OF OPERATORS
OF FRACTIONAL DIFFERENTIATION
Alexey Karapetyants 1 , Ferdos Kodzoeva
2
Abstract
Let D stand for the unit disc in the complex plane C. Given 0 < p < ∞,
−1 < λ < ∞, the analytic weighted Besov space Bpλ (D) is defined to consist
of analytic in D functions such that
(1 − |z|2 )N p−2 |f (N ) (z)|p dμλ (z) < ∞,
D
where dμλ (z) = (λ + 1)(1 − |z|2 )λ dμ(z), dμ(z) = π1 dxdy, and N is an
arbitrary fixed natural number, satisfying N p > 1 − λ.
We provide a characterization of weighted analytic Besov spaces Bpλ (D),
0 < p < ∞, in terms of certain operators of fractional differentiation Rzα,t
of order t. These operators are defined in terms of construction known
as Hadamard product composition with the function b. The function b
is calculated from the condition that Rzα,t (uniquely) maps the weighted
Bergman kernel function (1 − zw)−2−α to the similar (weight parameter
shifted) kernel function (1 − zw)−2−α−t , t > 0. We also show that Bpλ (D)
can be thought as the image of certain weighted Lebesgue space Lp (D, dνλ )
under the action of the weighted Bergman projection PDα .
MSC 2010 : Primary 30H25; Secondary 47G20
Key Words and Phrases: analytic Besov space, fraction integro-differentiation
c 2014 Diogenes Co., Sofia
pp. 897–906 , DOI: 10.2478/s13540-014-0204-2
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1. Introduction
The Bergman, Hardy, Besov, Lipshitz, Bloch and BMOA spaces are
widely studied in the modern literature among other spaces of analytic
functions of one or several variables. Without claiming for completeness,
we mention the books [11, 6, 15, 16] (see also references therein). A wide
range of problems is very common for these spaces: integral representations
of functions in these spaces, atomic decomposition, duality, interpolation,
characterization in terms of derivatives, including fractional ones. The investigation of these problems deals with classic Bergman type operators,
operators of fractional integro-differentiation, Berezin transform and mean
oscillation technique. These spaces, except for BMOA, are naturally considered as a part of a more general family of analytic Sobolev spaces, but
a specific study of each concrete space is of independent interest.
The analytic Besov spaces on the unit disk in C as well as the socalled Qp -spaces defined without use of derivatives have been extensively
studied within the mainstream of the study of spaces of functions which
are invariant under Mobius transformations of the unit disk. An important
class of analytic Besov spaces, the so-called diagonal Besov space Bp (D),
is introduced and described in Zhu [12] in the case of the unit disk, and
furthermore studied in the case of bounded symmetric domains in [13], [14].
Here we study weighted analytic Besov spaces Bpλ (D), as introduced
in the abstract. As it is stated in Theorem 2.1, the definition of Bpλ (D)
does not depend on N > 1−λ
p . In particular, for 1 − λ < p < ∞ one can
set N = 1. We describe these spaces in terms of the Bergman projection
and certain differential operator Rα,t of fractional order. The operator Rα,t
is defined by its action on weighted Bergman kernel function, as it was
specified in the abstract. This operator Rα,t belongs to the general class of
operators of fractional integro-differentiation of analytic functions, known
as Hadamard’s product composition operators (see [10], Section 22, and
references therein). We find it useful to discuss this issue below for the
sake of completeness. For proving the main results, we follow some ideas
of [15]. We also generalize some results from [7], though using a different
approach.
2. Auxiliaries
Let A2λ (D) stand for the weighted Bergman space on D, which consists of
analytic in D functions that belong to weighted L2 (D, dμλ ), −1 < λ < ∞.
PDλ denotes the corresponding weighted Bergman projection. As usual,
H(D) stands for the space of analytic in D functions equipped with topology
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CHARACTERIZATION OF WEIGHTED ANALYTIC . . .
of uniform convergence on compacts. Set dν(z) =
|z|2 )λ dν(z).
dμ(z)
(1−|z|2 )2 ,
899
dνλ (z) = (λ +
Note that dν(z) is invariant with respect to Moebius
1)(1 −
transformations of D. For further properties we refer to [6], [15].
The proof of the next two results can be derived from analogous theorems in [15], so we omit details here.
Theorem 2.1. Let 0 < p < ∞, −1 < λ < ∞ and f be analytic in D,
then the following are equivalent:
(1) (1 − |z|2 )N f (N ) (z) ∈ Lp (D, dνλ ) for some natural N > 1−λ
p ,
1−λ
2
N
(N
)
p
(2) (1 − |z| ) f (z) ∈ L (D, dνλ ) for all natural N > p .
Theorem 2.2. Let 0 < p < ∞, −1 < λ < ∞. Then the space Bpλ (D)
is complete with respect to the norm
N
−1
p
(m)
p
|f (0)| + (1 − |z|2 )N p−2 |f (N ) (z)|p dμλ (z),
f B λ (D) =
p
m=0
D
where N is an arbitrary natural number N >
dense in Bpλ (D).
1−λ
p .
The set {z n }n=0,1,2,... is
3. On fractional integro-differentiation
We shall discuss some preliminaries on the theory of fractional integrodifferentiation basing ourselves on the book [10] (see also [8]). There exist
several approaches to define fractional order integro-differentiation of functions on the complex plane. We follow [10] to outline the four most general
ones. The first is based on representation of a function via series of exponents (Liouville’s approach) and power series (Hadamard’s approach).
That is, the formulas
D ν eiaz = aν eiaz , D ν (z − z0 )μ =
Γ(1 + μ)
(z − z0 )μ−ν
Γ(1 + μ − ν)
are taken as definitions. The second one, used
by Hardy and Littlewood ([4],
n
[5]), is based on the Weyl calculus: if f (z) = ∞
n=0 fn z , then the fractional
∞
f
integral is I ν f (z) = n=0 (in)n ν z n . This approach may be considered as
Weyl fractional integro-differentiation of a function f (reiϕ ) with respect to
angular variable ϕ. The third approach is based on the direct introduction
of the Riemman-Liouville operator of integro-differentiation in a complex
plane with integration over straight line interval or a curve with z as a final
point. The fourth one uses the Cauchy integral representation formula.
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A. Karapetyants, F. Kodzoeva
Note that working with these definitions in complex plane may require
certain conditions on the domain or need of using certain special classes of
functions (cf. [10] and [8], Ch. 2 and 5).
The operators of integro-differentiation that we use in this paper are
further generalization
approach, called Hadamard’s product com∞of Weyl’s
n
position. If b(z) = n=0 bn z is analytic in D, then the operator
D(b, f )(z) =
∞
bn fn z n
n=0
is known as Hadamard product composition of functions b and f . If bn → ∞
when n → ∞, then this composition generalizes the (fractional) differentiis invertible with inverse
ation. If bn = 0, n = 0, 1, 2, . . ., then the
operator
−1 z n .
b
constructed via the function b∗ (z) = ∞
n=0 n
Now we are in position to specify the operators Rα,t used in our study
(we follow [14], [15]). Let in the further considerations −1 < α < ∞, t ∈ R,
and neither 1 + α nor 1 + α + t be negative integer. We may also write Rzα,t
to outline that the operator acts in the z variable.
A very important construction in our study is the (weighted) Bergman
projection. So, the main idea is to consider the operator which is “nicely”
acting on the weighted Bergman kernel. That is, to consider unique operator Rα,t on H(D) such that
1
1
α,t
.
=
Rz
2+α
(1 − zw)
(1 − zw)2+α+t
Here α is connected to weight parameter and t stands for the integrodifferentiation parameter (fractional order). From the series expansion of
the weighted Bergman kernel, it is easy see that the operator Rα,t on functions z n acts as
Γ(n + 2 + α + t)Γ(2 + α) n
z , n = 0, 1, 2, ...
Rα,t (z n ) =
Γ(2 + α + t)Γ(n + 2 + α)
That is, in our case we have
∞
Γ(2 + α) Γ(n + 2 + α + t) n
z
b(z) = bt (z) =
Γ(2 + α + t) n=0 Γ(n + 2 + α)
Due to Stirling’s formula,
Γ(2 + α)
Γ(n + 2 + α + t)Γ(2 + α)
∼
nt ,
Γ(2 + α + t)Γ(n + 2 + α)
Γ(2 + α + t)
(3.1)
n → ∞,
that is why for t > 0 the operator Rα,t is naturally thought as operator
of fractional differentiation of order t. As it is now clear, this operator is
nothing but the Hadamard product composition of the function b from (3.1)
with f . Note, that a very close fractional differentiation
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CHARACTERIZATION OF WEIGHTED ANALYTIC . . .
f (z) =
∞
fn z →
n
n=0
901
∞
fn
zn,
t
(n
+
1)
n=0
along with some modifications, was studied by Flett in [1], [2], [3], including
the action between weighted Bergman spaces. Related investigations in
various settings were also made by A.A. Pekarsky, V.S. Kiryakova and I.H.
Dimovski, M.M. Djrbashian and A.B. Nersesian, and many others (see [10],
[8] for further information and extensive lists of references).
For t > 0 one can verify that
Rα,t (z n ) =
D
wn
dμα (w),
(1 − zw)2+α+t
f (rw)
dμα (w), f ∈ H(D).
r→1−0 D (1 − zw)2+α+t
The limit above exists for any f ∈ H(D), and for functions f integrable
over unit disc one can set r = 1. The operator Rα,t is invertible on H(D)/C.
The operator Rα,t inverse to Rα,t is defined on z n as
Rα,t f (z) = lim
Rα,t (z n ) =
Γ(2 + α + t)Γ(n + 2 + α) n
z ,
Γ(n + 2 + α + t)Γ(2 + α)
and
Rα,t (z n ) =
1+α+t
1+t
D
(1 − |w|2 )α wn
dμt (w)
(1 − zw)2+α
for t > 0, and is understood in a similar manner when z n is replaced
by f ∈ H(D). The operator Rα,t (for t > 0) is naturally refereed to as
fractional integration of order t.
For the sake of completeness let us provide, following [10], integral
representations for Rα,t , Rα,t involving b, b∗ :
α,t
b(zτ )f (τ )dσ(τ ), Rα,t f (z) =
b∗ (zτ )f (τ )dσ(τ ),
R f (z) =
|τ |=r
|τ |=r
where dσ(τ ) stands for normalized measure of the unit circle ∂D : ∂D dσ(τ )
= 1, and |z| < r < 1.
Finally, in the particular important “unweighed” case α = 0 the function b is easy to calculate and is of the form:
1
1
−1 .
b(z) =
z(1 + α) (1 − z)1+α
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A. Karapetyants, F. Kodzoeva
4. Main results
We start with the theorem which describes the analytic weighted Besov
space Bpλ (D) as the image of Lp (D, dνλ ) under the action of Bergman projection.
Theorem 4.1. Let 1 p < ∞, −1 < α < ∞, −1 < λ < ∞. Then
Bpλ (D) = PDα Lp (D, dνλ ).
α,t is well defined.
P r o o f. Fix t > 1−λ
p , and such that the operator R
Based on Theorem 2.19 from [15], one can show that the operator Rα,t
is bounded and invertible as operator from Bpλ (D) to Appt−2+λ (D). That
is, an analytic in D function f belongs to Bpλ (D) if and only if Rα,t f in
Appt−2+λ (D). Set β = t + α + λ, γ = pt − 2 + λ. Then p(β + 1) > γ + 1 and
Apγ (D) = PDβ Lp (D, dμγ )
due to Theorem 2.11 from [15]. This means that f ∈ Bpλ (D) if and only if
Rα,t f = PDβ g, where g ∈ Lp (D, dμγ ), or otherwise,
g(w)(1 − |w|2 )β dμλ (w)
α,t
.
R f (z) = (β + 1)
(1 − zw)2+α+t
D
The integral in the above equality is converging on compacts in D, and the
operator Rα,t is continuous on H(D). Due to
Rα,t (1 − zw)−2−α−t = (1 − zw)−2−α ,
we have
Rα,t
f (z) = (β + 1)
=
β+1
α+1
D
D
1
g(w)(1 − |w|2 )β dμλ (w)
(1 − zw)2+α+t
g(w)(1 − |w|2 )t dμα (w)
.
(1 − zw)2+α
Note that g ∈ Lp (D, dμγ ) is equivalent to (1 − |w|2 )t g(w) ∈ Lp (D, dνλ ).
Hence f ∈ Bpλ (D) is equivalent to the statement that f belongs PDα Lp (D, dνλ ).
2
In the next theorem we characterize the functions from Bpλ (D) in terms
of differential operator of integer order Rα,N . This result generalizes Theorem 3.3 from [7], where the case 1 < p < ∞, α = 0 was treated. Here we
use different approach, following the lines of [15].
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CHARACTERIZATION OF WEIGHTED ANALYTIC . . .
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Theorem 4.2. Let 0 < p < ∞, −1 < λ < ∞, and 1+α be not negative
integer, N be natural number such that N > 1−λ
p , and the function f be
λ
analytic in D. Then f ∈ Bp (D) if and only if FN (z) = (1 − |z|2 )N Rα,N f (z)
belongs to Lp (D, dνλ ).
P r o o f. If f ∈ Bpλ (D), then f (N ) ∈ AppN −2+λ (D). Hence f (m) belongs
to AppN −2+λ (D) for any m N . In that case due to Proposition 1.15 from
[15], we have Rα,N f ∈ AppN −2+λ (D), that is, FN ∈ Lp (D, dνλ ).
To show that FN ∈ Lp (D, dνλ ) implies f ∈ Bpλ (D), we prove that there
exists C > 0 such that
f (N ) Ap
pN−2+λ (D)
CRα,N f Ap
pN−2+λ (D)
.
Let β = α + K, where K is natural number, such that β >
f is analytic in D, then
f (w)
dμβ (w), z ∈ D.
f (z) =
(1 − zw)2+β
λ−1
p
− 1. Since
D
The operators
Rα,t ,
Rα,t are continuous in H(D), hence
Rα,N f (w)
dμβ (w), z ∈ D.
f (z) = Rα,N
(1 − zw)2+β
D
According to Lemma 2.18 from [15], there exists a polynomial p(z, w) such
that
p(z, w)Rα,N f (w)
dμβ (w), z ∈ D.
f (z) =
(1 − zw)2+β−N
D
Differentiating N times, we have
|Rα,N f (w)|dμβ (w)
(N )
, z ∈ D.
|f (z)| C
|1 − zw|2+β
D
Note that pN − 2 + λ > −1, hence for 1 p < ∞ due to Theorem 2.10
from [15] we have
f (N ) Ap
pN−2+λ (D)
CRα,N f Ap
pN−2+λ (D)
.
That is, (1 − |z|2 )N f (N ) ∈ Lp (D, dνλ ), which implies f ∈ Bpλ (D).
If 0 < p < 1, we set β = α + K = 2+α
p − 2, where K is such that
α > pN − 2 + λ. According to Lemma 2.15 from [15] we have
|Rα,N f (w)|p
|Rα,N f (w)|p
(N )
dμ
(w)
C
dμα (w).
|f (z)| C
1
β
|1 − zw|p(2+β)
|1 − zw|p(2+β)
D
D
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A. Karapetyants, F. Kodzoeva
Multiplying both sides by (1 − |z|2 )pN −2+λ and integrating with respect to
dμ, we have
|f (N ) (z)|(1 − |z|2 )pN −2+λ dμ(z)
D
C1
⎛
⎝
D
D
⎞
|Rα,N f (w)|p
|1 − zw|p(2+β)
dμα (z)⎠ (1 − |z|2 )pN −2+λ dμ(z)
|Rα,N f (w)|p dμα (z)
C1
D
D
(1 − |z|2 )pN −2+λ
dμ(z).
|1 − zw|p(2+β)
Here we used Fubini’s theorem. Since
p(2 + β) − (pN − 2 + λ) − 2 = 2 + α − pN − λ > 0,
then due to Theorem 1.9 from [6] there exists C > 0 such that
(1 − |z|2 )pN −2+λ
dμ(z) (1 − |w|2 )−2−α +pN +λ .
p(2+β)
|1 − zw|
D
Hence,
f pAp
pN−2+λ (D)
= C2
C1
|Rα,N f (w)|p
D
1
dμα (w)
(1 − |w|2 )2+α −pN −λ
|Rα,N f (w)|p (1 − |z|2 )pN −2+λ dμ(w) = C2 Rα,N f pAp
D
pN−2+λ (D)
.
2
Let us prove now the following general result.
Theorem 4.3. Let 0 < p < ∞, −1 < λ < ∞, and 1 + α be not
negative integer, and f be analytic in D. Then the following conditions are
equivalent:
(1) f ∈ Bpλ (D);
(2) (1 − |z|2 )t Rα,t f (z) belongs to Lp (D, dνλ ) for some t > 1−λ
p , where
1 + α + t is not negative integer;
(3) (1 − |z|2 )t Rα,t f (z) belongs to Lp (D, dνλ ) for any t > 1−λ
p , where
1 + α + t is not negative integer.
P r o o f. In accordance with Theorem 4.2, the third condition implies
the first one, which, in its own turn, implies the second condition. To
prove that the second condition implies the third one we show that the
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CHARACTERIZATION OF WEIGHTED ANALYTIC . . .
905
norms of the functions (1 − |z|2 )t Rα,t f and (1 − |z|2 )s Rα,s f in Lp (D, dνλ )
are equivalent for all f ∈ H(D), with s, t being fixed positive numbers
1−λ
s > 1−λ
p , t >
p , and 1 + α + t, 1 + α + s are not negative integers.
Write s = t + σ, where σ > 0. The norm of the function (1 − |z|2 )t Rα,t f (z)
in Lp (D, dνλ ) coincides with the norm of Rα,t f (z) in Appt−2+λ (D). Since
pt − 2 + λ > −1, then due to Theorem 2.19 from [15] for some c, C > 0 we
have
c |Rα,t f (z)|p dμpt−2+λ (z)
D
D
|(1 − |z|2 )σ Rα+t,σ Rα,t f (z)|p dμpt−2+λ (z)
|Rα,t f (z)|p dμpt−2+λ (z).
C
D
Having in mind that Rα+t,σ Rα,t = Rα,s we get
c |Rα,t f (z)|p dμpt−2+λ (z)
D
D
|(1 − |z|2 )s Rα,s f (z)|p dμpt−2+λ (z)
|Rα,t f (z)|p dμpt−2+λ (z).
C
D
Hence, the norms of the functions (1 − |z|2 )t Rα,t f (z), (1 − |z|2 )s Rα,s f (z)
2
in Lp (D, dνλ ) are equivalent.
The author expresses deep gratitude to Professor S.G. Samko and Professor V.S. Kiryakova for the valuable comments and suggestions.
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1
Mathematics, Mechanics and Computer Sciences Department
Southern Federal University Miltchakova, 8a
Rostov-on-Don– 344090, RUSSIAN Federation
e-mail: [email protected]
Received: April 11, 2014
2
Mathematics and Physics Department, Ingush State Unuiversity
Magistralnaya, 39, Nazran – 386132, RUSSIAN Federation
e-mail: [email protected]
Please cite to this paper as published in:
Fract. Calc. Appl. Anal., Vol. 17, No 3 (2014), pp. 897–906;
DOI: 10.2478/s13540-014-0204-2
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