WEIGHTED INTEGRALS OF HOLOMORPHIC
FUNCTIONS IN THE UNIT POLYDISC
STEVO STEVI ´C
Received 28 September 2003
Let f be a measurable function defined on the unit polydiscUn in Cnand let ωj(zj), j=1,. . .,n, be admissible weights on the unit diskU, with distortion functionsψj(zj), ᏸp,q
ω,N(Un)= {f | fᏸωp,,qN <∞}, where f q ᏸp,q
ω,N =
[0,1)nMqp(f,r)nj=1ωj(rj)drj, and Ꮽp,q
ω,N(Un)=ᏸ p,q
ω,N(Un)∩H(Un). We prove the following result: if p,q∈[1,∞) and for all j=1,. . .,n,ψj(zj)(∂ f /∂zj)(z)∈ᏸωp,,qN, then f ∈Ꮽ
p,q
ω,Nand there is a positive constant C=C(p,q,ωj,n) such thatfᏭp,q
ω,N ≤C
|f(0)|+nj=1ψj(∂ f /∂zj)ᏸp,q
ω,N
.
1. Introduction
LetU1=U be the unit disk in the complex plane,dm(z)=(1/π)dr dθthe normalized Lebesgue measure onU,Unthe unit polydisc in complex vector spaceCnandH(Un) the space of all analytic functions onUn. Forz,w∈Cnwe writez·w=(z
1w1,. . .,znwn);eiθ is an abbreviation for (eiθ1,. . .,eiθn);dt=dt
1···dtn;dθ=dθ1···dθnandr,θare vectors inCn. If we write 0≤r <1, wherer=(r1,. . .,rn) it means 0≤rj<1 forj=1,. . .,n.
For f ∈H(Un) andp∈(0,∞) we usually write
Mp(f,r)=
1 (2π)n
[0,2π]n
f
r·eiθpdθ 1/ p
, for 0≤r <1 (1.1)
for the integral means of f.
Letω(s), 0≤s <1, be a weight function which is positive and integrable on (0, 1). We extendωonUby settingω(z)=ω(|z|). We may assume that our weights are normalized so that01ω(s)ds=1.
Letᏸωp=ᏸωp(Un) denotes the class of all measurable functions defined onUnsuch that
fᏸpp
ω=
Un
f(z)pn
j=1 ωj
zj
dmzj
<∞, (1.2)
whereωj(zj), j=1,. . .,n, are admissible weights on the unit diskU.
Copyright©2005 Hindawi Publishing Corporation
The weighted Bergman spaceᏭωpis the intersection ofᏸωp andH(Un). Forω j(zj)= (1− |zj|2)αj,αj>−1,j=1,. . .,n, we obtain the classical Bergman spaceᏭp(dVα), see [1, page 33].
Letᏸωp,,qN=ᏸp,q
ω,N(Un), p,q >0, denotes the class of all measurable functions defined onUnsuch that
fqᏸp,q
ω,N =
[0,1)nM q p(f,r)
n
j=1
ωjrjdrj<∞, (1.3)
andᏭωp,,qNbe the intersection ofᏸωp,,qNandH(Un). Whenp=qwe denoteᏭp,q
ω,NbyᏭ p
ω,N. In the casep=q, these two norms are equivalent on the spaceH(Un), but the later one is more suitable for calculations than the first one. The result is contained in the following lemma.
Lemma1.1. The norms · Ꮽp
ωand · Ꮽ p
ω,N are equivalent on the spaceH(U n).
Proof. By the polar coordinates it is easy to see thatfᏭp
ω≤2 nf
Ꮽp
ω,N for every f ∈ H(Un), moreoverf
ᏸp
ω≤2 nf
ᏸp
ω,N for every f measurable onU n.
Now we prove that there is a positive constantC, which is independent of f, such that
fᏭp
ω,N ≤CfᏭ p
ω, (1.4)
for every f ∈H(Un). Without loss of generality we may assume that n=2. Let f ∈ H(Un), then
fpᏭp
ω,N =
1/2
0
1/2
0 +
1/2
0
1
1/2+
1
1/2
1/2
0 +
1
1/2
1
1/2g
r1,r2
dr1dr2, (1.5)
whereg(r1,r2)=Mpp(f,r1,r2)ω1(r1)ω2(r2). Now we estimate these four integrals, which we denote byIi,i=1, 2, 3, 4.
Since f ∈H(U2) then the function f is analytic in each variable separately onUand consequentlyMpp(f,r1,r2) is nondecreasing function inr1andr2, see, for example, [3]. Let
Cωi=
1/2
0 ωi
ri
1
1/2ωi
ri, i=1, 2. (1.6)
Using the above mentioned facts and definitions, we have
I1≤Mpp(f, 1/2, 1/2)
1/2
0
1/2
0 ω1
r1 ω2 r2
dr1dr2
=Cω1Cω2M
p
p(f, 1/2, 1/2)
1
1/2
1
1/2ω1
r1 ω2 r2
dr1dr2
≤Cω1Cω2
1
1/2
1
1/2M p pf,r1,r2
ω1 r1 ω2 r2
dr1dr2
≤4Cω1Cω2
1
1/2
1
1/2M p p
f,r1,r2
ω1 r1 ω2 r2
r1r2dr1dr2,
(1.7)
I2≤
1
1/2M p
pf, 1/2,r2
1/2
0 ω1
r1
dr1ω2
r2
dr2
=Cω1
1
1/2
1
1/2M p p
f, 1/2,r2
ω1 r1 ω2 r2
dr1dr2
≤Cω1
1
1/2
1
1/2M p p
f,r1,r2
ω1 r1 ω2 r2
dr1dr2
≤4Cω1
1
1/2
1
1/2M p pf,r1,r2
ω1 r1 ω2 r2
r1r2dr1dr2.
(1.8)
Similarly
I3≤4Cω2
1
1/2
1
1/2M p pf,r1,r2
ω1 r1 ω2 r2
r1r2dr1dr2. (1.9)
Finally, it is clear that
I4≤4
1
1/2
1
1/2M p pf,r1,r2
ω1 r1 ω2 r2
r1r2dr1dr2. (1.10)
From (1.7)–(1.10), we obtain
fᏭpp
ω,N≤
Cω1+ 1
Cω2+ 1
fpᏭp
ω, (1.11)
as desired.
Following [8], for a given weightωwe define the function
ψ(r)=ψω(r)def= ω1
(r)
1
r ω(u)du, 0≤r <1, (1.12) and we call it thedistortion functionofω. We putψ(z)=ψ(|z|) forz∈U. Definition 1.2[8]. We say that a weightωisadmissibleif it satisfies the following condi-tions:
(i) there is a positive constantA=A(ω) such that
ω(r)≥ A
1−r
1
(ii)ωis differentiable and there is a positive constantB=B(ω) such that
ω(r)≤ B
1−rω(r), for 0≤r <1; (1.14)
(iii) for each sufficiently small positiveδthere is a positive constantC=C(δ,ω) such that
sup 0≤r<1
ω(r)
ωr+δψ(r)≤C. (1.15)
Observe that (i) impliesAψ(r)≤1−rthus for sufficiently small positiveδwe haver+ 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].
Theorem1.3. Suppose1≤p <∞andωis an admissible weight with distortion function ψ. Then
U
f(z)p
ω(z)dm(z)f(0)p+
U
f(z)p
ψ(z)pω(z)dm(z), (1.16)
for all analytic functions f on the unit discU, wheredm(z)=rd rdθ/πdenotes the normal-ized Lebesgue area measure onU.
The above means that there are finite positive constantsCandCindependent of f such that the left- and right-hand sidesL(f) andR(f) satisfy
CR(f)≤L(f)≤CR(f) (1.17)
for all analytic f.
Some generalizations ofTheorem 1.3in 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 weightω(r)=(1−r)α,α >−1, are presented in [1,2,3,4,7,13].
Using a Bergman type projectionBα:ᏸp(dVα)→Ꮽp(dVα), in [1] the authors proved the following theorem.
Theorem1.4. Letp∈[1,∞),αj>−1,j=1,. . .,nandmbe a fixed positive integer and let
k=(k1,. . .,kn)∈(Z+)n. Let f be a holomorphic function defined on the polydiscUninCn. Then forα=(α1,. . .,αn),f ∈Ꮽp(dVα)if and only if
n
j=1
1−zj 2kj
∂|k|f ∂zk1
1 . . . ∂zknn
(z)∈ᏸpdV
α
Moreover,
fᏭp(dV α)
m−1
|k|=0
∂
|k|f ∂zk1
1 ···∂zknn (0)
+
|k|=m
n
j=1
1−zj 2kj
∂mf ∂zk1
1 ···∂znkn
ᏸp(dV
α)
.
(1.19)
In [11] among other things we proved the following theorem which generalizes Theo-rem 1.4.
Theorem 1.5. Let k=(k1,. . .,kn)∈(Z+)n, f be a holomorphic function defined on the polydiscUninCnandωj(z
j), j=1,. . .,nare admissible weights on the unit diskU, with distortion functionsψj(zj). If f ∈Ꮽωpandp >0, then
n
j=1 ψkj
j
zj
∂|k|f
∂zk1
1 ···∂znkn
(z)∈ᏸωp. (1.20)
Moreover, letmbe a fixed positive integer. Then there is a positive constantC=C(p,ωj,m,n) such that
fᏭp
ω≥C
m−1
|k|=0
∂
|k|f ∂zk1
1 ···∂znkn
(0)+ |k|=m
n
j=1 ψkj
j
zj
∂mf ∂zk1
1 ···∂zknn
ᏸp
ω
. (1.21)
In the same paper, we formulate and give a sketch of a proof of the following partially converse ofTheorem 1.5.
Theorem1.6. Let f ∈H(Un)andωj(z
j), j=1,. . .,nare admissible weights on the unit diskU, with distortion functionsψj(zj). Ifp∈[1,∞)and for allj=1,. . .,n,ψj(zj)(∂ f /∂zj) (z)∈ᏸp
ω, thenf ∈Ꮽ p
ωand there is a positive constantC=C(p,ωj,n)such that
fᏭp
ω≤C
f(0)+n
j=1
ψj∂ f
∂zj
ᏸp
ω
. (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 de-tailed proof ofTheorem 1.6. Also, we prove the following generalization ofTheorem 1.6.
Theorem1.7. Let f ∈H(Un)andωj(z
j), j=1,. . .,nare admissible weights on the unit diskU, with distortion functions ψj(zj). If p,q∈[1,∞) and for all j=1,. . .,n,ψj(zj) (∂ f /∂zj)(z)∈ᏸp,q
ω,N, then f ∈Ꮽ p,q
ω,Nand there is a positive constantC=C(p,q,ωj,n)such that
fᏭp,q
ω,N≤C
f(0)+n
j=1
ψj
∂ f ∂zj
ᏸp,q
ω,N
2. An auxiliary results
In this section, we prove an auxiliary result which we use in the proof of the main result.
Lemma2.1. Suppose1≤p <∞and f ∈H(Un). Then
d dtM
p
p(f,tr)≤pMp− 1 p (f,tr)
n
i=1 riMp
∂ f
∂zi, tr
, (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 dtf
tr·eiθp=pftr·eiθp−1d dtf
tr·eiθ
≤pftr·eiθp−1d dt
ftr·eiθ
=pftr·eiθp−1∇ftr·eiθ,r·eiθ
≤pftr·eiθp−1 n
i=1 ri∂ f
∂zi
tr·eiθ.
(2.2)
From (2.2) and by the dominated convergence theorem we obtain
d drM
p
p(f,tr)≤ p (2π)n
n
i=1 ri
[0,2π]n
f
tr·eiθp−1∂ f ∂zi
tr·eiθdθ. (2.3)
Ifp=1 the assertion is clear. If p >1, applying in the last integral H¨older’s inequality with exponentsp/(p−1) andpwe obtain the result. Corollary2.2. Supposep,q∈[1,∞)and f ∈H(Un). Then
d dtM
q
p(f,tr)≤qMpq−1(f,tr) n
i=1 riMp
∂ f
∂zi, tr
, (2.4)
almost everywhere.
Proof. Computing (d/dt)Mqp(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.
complicated notations we useMpp(f,r1t,r2t) instead ofMpp(r1t,r2t). We have
fᏭpp
ω,N=
1 0 1 0 1 0 d dtM p p
r1t,r2t
dt ω1 r1 ω2 r2
dr1dr2
≤p 1 0 1 0 1 0M p−1 p
r1t,r2t
2
j=1 Mp
∂ f
∂zi,r1t,r2t
ridt ω1 r1 ω2 r2
dr1dr2
≤p 1 0 1 0 1 0M p−1 p r1t,r2
Mp
∂ f
∂z1,r1t,r2
r1dt
ω1 r1 ω2 r2
dr1dr2
+p 1 0 1 0 1 0M p−1 p
r1,r2t
Mp
∂ f
∂z2, r1,r2t
r2dt
ω1 r1 ω2 r2
dr1dr2
≤p 1 0 1 0 r1 0 M p−1 p s,r2
Mp
∂ f
∂z1,s,r2
ds ω1 r1 ω2 r2
dr1dr2
+p 1 0 1 0 r2 0 M p−1 p
r1,τMp
∂ f
∂z2, r1,τ
dτ ω1 r1 ω2 r2
dr1dr2
≤p 1 0 1 0
Mpp−1s,r2Mp
∂ f
∂z1,s,r2
1
s ω1
r1dr1
ω2r2ds dr2
+p 1 0 1 0
Mpp−1
r1,τMp
∂ f
∂z2, r1,τ
1
τω2
r2 dr2 ω1 r1
dτ dr1
=p 1 0 1 0M p−1
p s,r2Mp
∂ f
∂z1,s,r2
ψ1(s)ω1(s)ω2r2ds dr2
+p
1
0
1
0M p−1 p
r1,τMp
∂ f
∂z2, r1,τ
ψ2(τ)ω2(τ)ω1
r1
dτ dr1.
(3.1)
Ifp >1, by H¨older inequality, we get
1 0 1 0M p−1 p
s,r2
Mp
∂ f
∂z1,s,r2
ψ1(s)ω1(s)ω2
r2
ds dr2
≤ 1 0 1 0M p ps,r2
ω1(s)ω2
r2
ds dr2
(p−1)/ p , 1 0 1 0M p p ∂ f ∂z1 ,s,r2
ψ1p(s)ω1(s)ω2
r2
ds dr2
1/ p
= fᏭp−1p
ω,N
1 0 1 0M p p ∂ f
∂z1,s,r2
ψ1p(s)ω1(s)ω2r2ds dr2
1/ p . (3.2) Similarly, 1 0 1 0M p−1 p
r1,τMp
∂ f
∂z2, r1,τ
ψ2(τ)ω2(τ)ω1
r1
dτ dr1
≤ fᏭp−p1
ω,N
1 0 1 0M p p ∂ f ∂z2 ,r1,τ
ψ2p(τ)ω2(τ)ω1
r1
dτ dr1
1/ p .
From (3.1)–(3.3) we obtain the result in this case. Forp=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, whenρ→1.
Open question 3.1. DoesTheorem 1.6hold in the case 0< p <1?
Remark 3.2. In the case whenωj(zj), j=1,. . .,n, are the classical weights (1− |zj|)αj, j=1,. . .,n, a positive answer toOpen question 3.1was given in [12].
Proof ofTheorem 1.7. If f(0, 0)=0, is not constant and all integrals are finite, then by Corollary 2.2, as in the proof ofTheorem 1.6, we obtain
fqᏭp,q
ω,N≤ fqᏭ−p1,q
ω,N
1
0
1
0M q p
∂ f
∂z1, s,r2
ψ1q(s)ω1(s)ω2
r2
ds dr2
1/q
+fqᏭ−p1,q
ω,N
1
0
1
0M q p
∂ f
∂z2 ,r1,τ
ψ2q(τ)ω2(τ)ω1
r1
dτ dr1
1/q
,
(3.4)
that is,
fᏭp,q
ω,N ≤ 2
j=1
ψj∂ f
∂zj
ᏸp,q
ω,N
. (3.5)
The rest of the proof is similar to the proof ofTheorem 1.6.
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Stevo Stevi´c: Mathematical Institute of Serbian Academy of Science, Knez Mihailova 35/I, 11000 Beograd, Serbia
