WEIGHTED INTEGRALS OF HOLOMORPHIC
FUNCTIONS IN THE UNIT POLYDISC
STEVO STEVIĆ
Received 28 September 2003
Let f be a measurable function defined on the unit polydisc U n in Cn and let ω j (z j ),
j = 1,...,n, be admissible weights on the unit disk U, with distortion functions ψ j (z j ),
p,q
q
q
ᏸω,N (U n ) = { f | f ᏸ p,q < ∞}, where f ᏸ p,q = [0,1)n M p ( f ,r) nj=1 ω j (r j )dr j , and
p,q
Ꮽω,N (U n )
,N
ω
=
,N
ω
p,q
ᏸω,N (U n ) ∩ H(U n ).
We prove the following result: if p, q ∈ [1, ∞) and for
p,q
p,q
all j = 1,...,n, ψ j (z j )(∂ f /∂z j )(z) ∈ ᏸω,N , then f ∈ Ꮽω,N and there is a positive constant
n
C = C(p, q,ω j ,n) such that f Ꮽ p,q ≤ C | f (0)| + j =1 ψ j (∂ f /∂z j )ᏸ p,q .
,N
ω
,N
ω
1. Introduction
Let U 1 = U be the unit disk in the complex plane, dm(z) = (1/π)dr dθ the normalized
Lebesgue measure on U, U n the unit polydisc in complex vector space Cn and H(U n ) the
space of all analytic functions on U n . For z,w ∈ Cn we write z · w = (z1 w1 ,...,zn wn ); eiθ
is an abbreviation for (eiθ1 ,...,eiθn ); dt = dt1 · · · dtn ; dθ = dθ1 · · · dθn and r, θ are vectors
in Cn . If we write 0 ≤ r < 1, where r = (r1 ,...,rn ) it means 0 ≤ r j < 1 for j = 1,...,n.
For f ∈ H(U n ) and p ∈ (0, ∞) we usually write
1
M p ( f ,r) =
(2π)n
[0,2π]n
f r · eiθ p dθ
1/ p
,
for 0 ≤ r < 1
(1.1)
for the integral means of f .
Let ω(s), 0 ≤ s < 1, be a weight function which is positive and integrable on (0,1). We
extend ω on U by setting ω(z) = ω(|z|). We may assume that our weights are normalized
1
so that 0 ω(s)ds = 1.
p
p
Let ᏸω = ᏸω (U n ) denotes the class of all measurable functions defined on U n such
that
p
f ᏸ p
ω
=
Un
n
f (z) p
ω j z j dm z j < ∞,
j =1
where ω j (z j ), j = 1,...,n, are admissible weights on the unit disk U.
Copyright © 2005 Hindawi Publishing Corporation
Journal of Inequalities and Applications 2005:5 (2005) 583–591
DOI: 10.1155/JIA.2005.583
(1.2)
584
Weighted integrals of holomorphic functions
p
p
The weighted Bergman space Ꮽω is the intersection of ᏸω and H(U n ). For ω j (z j ) =
(1 − |z j |2 )α j , α j > −1, j = 1,...,n, we obtain the classical Bergman space Ꮽ p (dVα ), see [1,
page 33].
p,q
p,q
Let ᏸω,N = ᏸω,N (U n ), p, q > 0, denotes the class of all measurable functions defined
on U n such that
q
f ᏸ p,q
,N
ω
=
p,q
q
[0,1)n
M p ( f ,r)
n
ω j r j dr j < ∞,
(1.3)
j =1
p,q
p,q
p
and Ꮽω,N be the intersection of ᏸω,N and H(U n ). When p = q we denote Ꮽω,N by Ꮽω,N .
In the case p = q, these two norms are equivalent on the space H(U n ), but the later one
is more suitable for calculations than the first one. The result is contained in the following
lemma.
Lemma 1.1. The norms · Ꮽ p and · Ꮽ p are equivalent on the space H(U n ).
ω
,N
ω
Proof. By the polar coordinates it is easy to see that f Ꮽ p ≤ 2n f Ꮽ p for every f ∈
,N
ω
ω
H(U n ), moreover f ᏸ p ≤ 2n f ᏸ p for every f measurable on U n .
,N
ω
ω
Now we prove that there is a positive constant C, which is independent of f , such that
f Ꮽ p ≤ C f Ꮽ p ,
,N
ω
(1.4)
ω
for every f ∈ H(U n ). Without loss of generality we may assume that n = 2. Let f ∈
H(U n ), then
p
f Ꮽ p =
,N
ω
1/2 1/2 1/2 1
+
0
0
1 1/2 1 1
+
0
1/2
+
1/2 0
1/2 1/2
g r1 ,r2 dr1 dr2 ,
(1.5)
p
where g(r1 ,r2 ) = M p ( f ,r1 ,r2 )ω1 (r1 )ω2 (r2 ). Now we estimate these four integrals, which
we denote by Ii , i = 1,2,3,4.
Since f ∈ H(U 2 ) then the function f is analytic in each variable separately on U and
p
consequently M p ( f ,r1 ,r2 ) is nondecreasing function in r1 and r2 , see, for example, [3].
Let
Cωi =
1/2
0
1
ωi ri
1/2
ωi ri ,
i = 1,2.
(1.6)
Note that Cωi , i = 1,2, are well defined and finite numbers since ωi are positive integrable
functions on (0,1).
Stevo Stević 585
Using the above mentioned facts and definitions, we have
p
I1 ≤ M p ( f ,1/2,1/2)
1/2 1/2
0
0
p
= Cω1 Cω2 M p ( f ,1/2,1/2)
≤ Cω1 Cω2
1 1
1/2 1/2
≤ 4Cω1 Cω2
I2 ≤
1
1/2
= Cω1
≤ Cω1
M p f ,1/2,r2
ω1 r1 ω2 r2 dr1 dr2
(1.7)
1/2
0
ω1 r1 dr1 ω2 r2 dr2
p
M p f ,1/2,r2 ω1 r1 ω2 r2 dr1 dr2
1 1
p
(1.8)
M p f ,r1 ,r2 ω1 r1 ω2 r2 dr1 dr2
1 1
≤ 4Cω1
1/2 1/2
p
1/2 1/2
1/2 1/2
1 1
M p f ,r1 ,r2 ω1 r1 ω2 r2 r1 r2 dr1 dr2 ,
p
1/2 1/2
p
M p f ,r1 ,r2 ω1 r1 ω2 r2 dr1 dr2
1 1
1 1
ω1 r1 ω2 r2 dr1 dr2
p
1/2 1/2
M p f ,r1 ,r2 ω1 r1 ω2 r2 r1 r2 dr1 dr2 .
Similarly
I3 ≤ 4Cω2
1 1
p
1/2 1/2
M p f ,r1 ,r2 ω1 r1 ω2 r2 r1 r2 dr1 dr2 .
(1.9)
Finally, it is clear that
I4 ≤ 4
1 1
p
1/2 1/2
M p f ,r1 ,r2 ω1 r1 ω2 r2 r1 r2 dr1 dr2 .
(1.10)
From (1.7)–(1.10), we obtain
p
p
f Ꮽ p ≤ Cω1 + 1 Cω2 + 1 f Ꮽ p ,
,N
ω
ω
(1.11)
as desired.
Following [8], for a given weight ω we define the function
def
ψ(r) = ψω (r) =
1
ω(r)
1
r
ω(u)du,
0 ≤ r < 1,
and we call it the distortion function of ω. We put ψ(z) = ψ(|z|) for z ∈ U.
(1.12)
Definition 1.2 [8]. We say that a weight ω is admissible if it satisfies the following conditions:
(i) there is a positive constant A = A(ω) such that
ω(r) ≥
A
1−r
1
r
ω(u)du,
for 0 ≤ r < 1;
(1.13)
586
Weighted integrals of holomorphic functions
(ii) ω is differentiable and there is a positive constant B = B(ω) such that
ω (r) ≤
B
ω(r),
1−r
for 0 ≤ r < 1;
(1.14)
(iii) for each sufficiently small positive δ there is a positive constant C = C(δ,ω) such
that
ω(r)
≤ C.
ω
r
+
δψ(r)
0≤r<1
(1.15)
sup
Observe that (i) implies Aψ(r) ≤ 1 − r thus for sufficiently small positive δ we have r +
δψ(r) < 1 and the quantity in the denominator of the fraction in (iii) is well defined.
For a list of examples of admissible weights, see [8, pages 660–663]. The following
theorem was proved in [8].
Theorem 1.3. Suppose 1 ≤ p < ∞ and ω is an admissible weight with distortion function
ψ. Then
U
f (z) p ω(z)dm(z) f (0) p +
U
p
f (z) ψ(z) p ω(z)dm(z),
(1.16)
for all analytic functions f on the unit disc U, where dm(z) = rd rdθ/π denotes the normalized Lebesgue area measure on U.
The above means that there are finite positive constants C and C independent of f
such that the left- and right-hand sides L( f ) and R( f ) satisfy
CR( f ) ≤ L( f ) ≤ C R( f )
(1.17)
for all analytic f .
Some generalizations of Theorem 1.3 in many directions can be found in [10, 11]. In
[5, 6, 9] was also investigated Bergman space of analytic functions with weights other
than classical. Closely related results for the classical weight ω(r) = (1 − r)α , α > −1, are
presented in [1, 2, 3, 4, 7, 13].
Using a Bergman type projection Bα : ᏸ p (dVα ) → Ꮽ p (dVα ), in [1] the authors proved
the following theorem.
Theorem 1.4. Let p ∈ [1, ∞), α j > −1, j = 1,...,n and m be a fixed positive integer and let
k = (k1 ,...,kn ) ∈ (Z+ )n . Let f be a holomorphic function defined on the polydisc U n in Cn .
Then for α = (α1 ,...,αn ), f ∈ Ꮽ p (dVα ) if and only if
n j =1
2 k j
1 − z j ∂|k| f
p
α
kn (z) ∈ ᏸ dV
k1
∂z1 ...∂zn
∀|k| = m.
(1.18)
Stevo Stević 587
Moreover,

m
−1
f Ꮽ p (dVα ) 
|k|=0
n
2 k j
∂|k| f
∂m f
k
(0)
+
1
−
z
j
kn k1
∂z 1 · · · ∂zkn
∂z · · · ∂z
n
1
|k|=m
j =1
n
1

.
ᏸ p (dVα )
(1.19)
In [11] among other things we proved the following theorem which generalizes Theorem 1.4.
Theorem 1.5. Let k = (k1 ,...,kn ) ∈ (Z+ )n , f be a holomorphic function defined on the
polydisc U n in Cn and ω j (z j ), j = 1,...,n are admissible weights on the unit disk U, with
p
distortion functions ψ j (z j ). If f ∈ Ꮽω and p > 0, then
n
kj ψj zj
j =1
∂|k| f
p
.
kn (z) ∈ ᏸω
k1
∂z1 · · · ∂zn
(1.20)
Moreover, let m be a fixed positive integer. Then there is a positive constant C = C(p,ω j ,m,n)
such that

f Ꮽ p ≥ C 
ω
m
−1
|k|=0
n
∂|k| f
∂m f
kj k
+
(0)
ψ
z
j
j
k
k
k
n
n
1
1
∂z · · · ∂z
∂z · · · ∂z n
1
|k|=m
j =1
n
1

.
(1.21)
p
ᏸω
In the same paper, we formulate and give a sketch of a proof of the following partially
converse of Theorem 1.5.
Theorem 1.6. Let f ∈ H(U n ) and ω j (z j ), j = 1,...,n are admissible weights on the unit
disk U, with distortion functions ψ j (z j ). If p ∈ [1, ∞) and for all j = 1,...,n, ψ j (z j )(∂ f /∂z j )
p
p
(z) ∈ ᏸω , then f ∈ Ꮽω and there is a positive constant C = C(p,ω j ,n) such that
n ∂f ψ j
f Ꮽ p ≤ C f (0) +
∂z ω
j =1
j
p
ᏸω
.
(1.22)
In communication with other specialists in this field it has turned out that the proof is
more complicated than we have expected. Hence in this note we will present a clear detailed proof of Theorem 1.6. Also, we prove the following generalization of Theorem 1.6.
Theorem 1.7. Let f ∈ H(U n ) and ω j (z j ), j = 1,...,n are admissible weights on the unit
disk U, with distortion functions ψ j (z j ). If p, q ∈ [1, ∞) and for all j = 1,...,n, ψ j (z j )
p,q
p,q
(∂ f /∂z j )(z) ∈ ᏸω,N , then f ∈ Ꮽω,N and there is a positive constant C = C(p, q,ω j ,n) such
that
f Ꮽ p,q
,N
ω
n ∂f
ψ j
≤ C f (0) +
∂z
j =1
j
p,q
ᏸω,N
.
(1.23)
588
Weighted integrals of holomorphic functions
2. An auxiliary results
In this section, we prove an auxiliary result which we use in the proof of the main result.
Lemma 2.1. Suppose 1 ≤ p < ∞ and f ∈ H(U n ). Then
n
∂f
d p
p −1
,tr ,
M p ( f ,tr) ≤ pM p ( f ,tr) ri M p
dt
∂z
i
i =1
(2.1)
almost everywhere.
Proof. For f ≡ 0 the result is obvious. If f ≡ 0, at points where f is not zero, we have
d f tr · eiθ p = p f tr · eiθ p−1 d f tr · eiθ dt
dt
iθ p−1 d
iθ f
tr
·
e
≤ p f tr · e
dt
p−1 = p f tr · eiθ ∇ f tr · eiθ ,r · eiθ n ∂f p−1 ri tr · eiθ ≤ p f tr · eiθ .
i =1
(2.2)
∂zi
From (2.2) and by the dominated convergence theorem we obtain
n
p d p
ri
M p ( f ,tr) ≤
n
dr
(2π) i=1
[0,2π]
f tr · eiθ p−1 ∂ f tr · eiθ dθ.
∂z
n
(2.3)
i
If p = 1 the assertion is clear. If p > 1, applying in the last integral Hölder’s inequality
with exponents p/(p − 1) and p we obtain the result.
Corollary 2.2. Suppose p, q ∈ [1, ∞) and f ∈ H(U n ). Then
n
∂f
d q
q −1
,tr ,
M p ( f ,tr) ≤ qM p ( f ,tr) ri M p
dt
∂zi
i =1
(2.4)
almost everywhere.
q
Proof. Computing (d/dt)M p ( f ,tr) and then using Lemma 2.1 we prove the corollary.
3. Proofs of the theorem
In this section, we prove the main results in this paper.
Proof of Theorem 1.6. Without loss of generality, we may assume that n = 2, and f (0,0) =
0. Also we assume that f is not constant and all integrals are finite. In order to avoid some
Stevo Stević 589
p
p
complicated notations we use M p ( f ,r1 t,r2 t) instead of M p (r1 t,r2 t). We have
p
f Ꮽ p =
1 1 1
,N
ω
0
≤p
0
0
d p
M p r1 t,r2 t dt ω1 r1 ω2 r2 dr1 dr2
dt
1 1 1
0
0
0
p −1 Mp
r1 t,r2 t
2
Mp
j =1
1 1 1
Mp
∂f
,r1 t,r2 r1 dt ω1 r1 ω2 r2 dr1 dr2
∂z
0 0
0
1
1 1 1
∂f
p −1 +p
M p r1 ,r2 t M p
,r1 ,r2 t r2 dt ω1 r1 ω2 r2 dr1 dr2
∂z2
0 0
0
1 1 r1
∂f
p −1 ≤p
M p s,r2 M p
,s,r2 ds ω1 r1 ω2 r2 dr1 dr2
∂z1
0 0
0
1 1 r2
∂f
p −1
+p
M p r1 ,τ M p
,r1 ,τ dτ ω1 r1 ω2 r2 dr1 dr2
∂z2
0
0 0
11
1
∂f
p −1 ≤p
M p s,r2 M p
,s,r2
ω1 r1 dr1 ω2 r2 dsdr2
∂z
0 0
s
1
11
1
∂f
p −1 +p
M p r1 ,τ M p
,r1 ,τ
ω2 r2 dr2 ω1 r1 dτ dr1
∂z2
0 0
τ
11
∂f
p −1 =p
M p s,r2 M p
,s,r2 ψ1 (s)ω1 (s)ω2 r2 dsdr2
∂z1
0 0
11
∂f
p −1
+p
M p r1 ,τ M p
,r1 ,τ ψ2 (τ)ω2 (τ)ω1 r1 dτ dr1 .
∂z2
0 0
≤p
p −1 ∂f
,r1 t,r2 t ri dt ω1 r1 ω2 r2 dr1 dr2
∂zi
r1 t,r2 M p
(3.1)
If p > 1, by Hölder inequality, we get
11
0
p −1 Mp
0
≤
1 1
0
0
s,r2 M p
1 1
0
0
∂f
,s,r2 ψ1 (s)ω1 (s)ω2 r2 dsdr2
∂z1
p
(p−1)/ p
M p s,r2 ω1 (s)ω2 r2 dsdr2
,
1/ p
∂f
p
,s,r2 ψ1 (s)ω1 (s)ω2 r2 dsdr2
∂z1
1 1
1/ p
p −1
p ∂f
p
= f Ꮽ p
Mp
,s,r2 ψ1 (s)ω1 (s)ω2 r2 dsdr2
.
∂z1
,N
ω
0 0
p
Mp
(3.2)
Similarly,
11
0
0
p −1 Mp
∂f
,r1 ,τ ψ2 (τ)ω2 (τ)ω1 r1 dτ dr1
∂z2
1 1
1/ p
p −1
p ∂f
p
≤ f Ꮽ p
Mp
,r1 ,τ ψ2 (τ)ω2 (τ)ω1 r1 dτ dr1
.
∂z2
,N
ω
0 0
r1 ,τ M p
(3.3)
590
Weighted integrals of holomorphic functions
From (3.1)–(3.3) we obtain the result in this case. For p = 1 the result it follows from
(3.3). If f is constant the result is clear. To remove the restriction of the finiteness of
the integrals we consider holomorphic functions fρ (z) = f (ρz), ρ ∈ (0,1) and use the
Monotone Convergence theorem, when ρ → 1.
Open question 3.1. Does Theorem 1.6 hold in the case 0 < p < 1?
Remark 3.2. In the case when ω j (z j ), j = 1,...,n, are the classical weights (1 − |z j |)α j ,
j = 1,...,n, a positive answer to Open question 3.1 was given in [12].
Proof of Theorem 1.7. If f (0,0) = 0, is not constant and all integrals are finite, then by
Corollary 2.2, as in the proof of Theorem 1.6, we obtain
q
f Ꮽ p,q
,N
ω
1 1
1/q
∂f
q
,s,r2 ψ1 (s)ω1 (s)ω2 r2 dsdr2
∂z1
0 0
1 1
1/q
q −1
q ∂f
q
+ f Ꮽ p,q
Mp
,r1 ,τ ψ2 (τ)ω2 (τ)ω1 r1 dτ dr1
,
∂z2
,N
ω
0 0
q −1
≤ f Ꮽ p,q
,N
ω
q
Mp
(3.4)
that is,
f
p,q
Ꮽω,N
2 ∂f
ψ j
≤
∂z
j =1
j
p,q
ᏸω,N
.
The rest of the proof is similar to the proof of Theorem 1.6.
(3.5)
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Stevo Stević: Mathematical Institute of Serbian Academy of Science, Knez Mihailova 35/I, 11000
Beograd, Serbia
E-mail addresses: sstevic@ptt.yu; sstevo@matf.bg.ac.yu