Journal of Inequalities and Applications Volume 2009, Article ID 353801,22pages doi:10.1155/2009/353801
Research Article
Analytic Classes on Subframe and Expanded Disk
and the
R
s
Differential Operator in Polydisk
Romi F. Shamoyan
1and Olivera R. Mihi ´c
21Department of Mathematics, Bryansk University, 241036 Bryansk, Russia
2Faculty of Organizational Sciences, University of Belgrade, Jove Ili´ca 154, 11000 Belgrade, Serbia
Correspondence should be addressed to Olivera R. Mihi´c,[email protected]
Received 25 November 2008; Revised 17 September 2009; Accepted 28 September 2009
Recommended by Narendra Kumar Govil
We introduce and study new analytic classes on subframe and expanded disk and give complete description of their traces on the unit disk. Sharp embedding theorems and various new estimates concerning differentialRsoperator in polydisk also will be presented. Practically all our results
were known or obvious in the unit disk.
Copyrightq2009 R. F. Shamoyan and O. R. Mihi´c. 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.
1. Introduction and Main Definitions
Letn ∈ Nand Cn {z z
1, . . . , zn : zk ∈ C, 1 ≤ k ≤ n}be then-dimensional space of
complex coordinates. We denote the unit polydisk by
Un{z∈Cn:|zk|<1, 1≤k≤n}, 1.1
and the distinguished boundary ofUnby
Tn{z∈Cn:|zk|1, 1≤k≤n}. 1.2
We usem2nto denote the volume measure onUnandmnto denote the normalized Lebesgue
measure onTn.LetHUnbe the space of all holomorphic functions onUn.Whenn1,we
simply denoteU1byU, T1byT, m
2nbym2, mnbym.We refer to1,2for further details. We
denote the expanded disk by
Un∗ z r1ξ, . . . , rnξ∈Un:ξ∈T, rj∈0,1, j 1, . . . , n
and the subframe by
Un z∈Un :z
jr, r∈0,1, j 1, . . . , n
. 1.4
The Hardy spaces, denoted byHpUn0< p≤ ∞,are defined by
HpUn
f ∈HUn: sup
0≤r<1 Mp
f, r<∞ , 1.5
whereMppf, r
Tn|frξ1, . . . , rξn|
pdm
nξ, M∞f, r maxξ∈Tn|frξ1, . . . , rξn|, r ∈0,1,
f∈HUn.
Forαj >−1, j 1, . . . , n0 < p < ∞,recall that the weighted Bergman spaceA→p−αUn
consists of all holomorphic functions on the polydisk satisfying the condition
fp
A→p−α
Un
fz1, . . . , znp n
i1
1− |zi|2αidm2nz<∞. 1.6
Forαj>−1, j1, . . . , n, p∈0,∞,the Bergman class on expanded disk is defined by
A→p−αUn∗
⎧ ⎨
⎩f∈HUn:f
p A→p−αUn
∗
1
0
· · ·
1
0
T
f|z1|ξ, . . . ,|zn|ξpn j1
1−zj2
αj
dzjdmξ<∞
⎫ ⎬ ⎭,
1.7
and similarly the Bergman class on subframe denoted byApαUnis defined by
Apα
Un
f∈HUn:fp
ApαUn
Tn 1
0
f|z|ξ1, . . . ,|z|ξnp
1− |z|2αdmnξd|z|<∞ , 1.8
wherep∈0,∞, α >−1.
Throughout the paper, we write C sometimes with indexes to denote a positive
constant which might be different at each occurrence even in a chain of inequalities but
is independent of the functions or variables being discussed.
The notationA Bmeans that there is a positive constantCsuch thatB/C ≤A ≤
CB.We will write for two expressionsABif there is a positive constantCsuch thatA < CB.
This paper is organized as follows. In first section we collect preliminary assertions. In the second section we present several new results connected with so-called operator of
diagonal map in polydisk. Namely, we define two new maps Sbf and Edf from
subframe U and expanded disk Un
∗ to unit disk Uand, in particular, completely describe
expanded disk on usual unit diskU on the complex plane. Proofs are based among other things on new projection theorems for these classes.
A separate section will be devoted to the study ofRsdifferential operator in polydisk.
It is based in particular on results from the recent paper 3. We will use the dyadic
decomposition technique to explore connections between analytic classes on subframe, polydisk, and expanded disk. We also prove new sharp embedding theorems for classes on subframe and expanded disk. Last assertions of the final section generalize some
one-dimensional known results to polydisk and to the case ofRsoperators simultaneously.
2. Preliminaries
We need the following assertions.
Lemma Asee4. Letαbe a fixedn-tuple of nonnegative numbers and let{Bi}i∈Zbe an arbitrary family of α-boxes in Rn lying in the cubeQn −2π,2πn. There exists a setJ ⊂ I such that
Bi∩Bi∅, i, i∈Jand for alli∈Ithere existsj∈Jsuch thatBi⊂5Bj.
The following proposition is heavily based on ideas from4.
Proposition 2.1. a Let f be a nonnegative summable function on Tn T × · · · × T. Let
f∗x1, . . . , xn feix1, . . . , eixn. Let αj ∈ N, rj ∈ 0,1, j 1, . . . , n, x ∈ Rn, gr, x
supB∈Z1/mnB
Bf∗dmn,Z Z∩Zα,where Z {B Bx, y : yj ≥ 1−rj, 1 ≤ j ≤ n},
B→−ϕ,→−y {t∈Rn :|t
i−ϕi| ≤yi, 1 ≤i≤n},and whereZαis a family of allB→−ϕ,→−yboxes such
thaty1, . . . , ynis proportional to2−α1, . . . ,2−αn, αi ∈N, i1, . . . , nfor some fixed←α.− Then the
following statements hold:
μEαa μ
zreix∈U:g→−αr, x> a≤ C a
Tn
f∗ϕdmn
ϕ,
μ1E1αaμ1zr1eix, . . . , rneix
∈U∗n:g→−αr1, . . . , rn, x> a
≤ C1 a
Tnf∗
ϕdmn
ϕ,
2.1
whereμandμ1are any positive Borel measures onUandUn
∗ such that
μΔlξ∩U μΔlξ≤Cln, l∈0,1, ξ∈T,
μ1Δlξ∩Un∗≤Cl1· · ·ln, lj∈0,1, ξ∈Tn,
2.2
where Δlξ {z ∈ Un : 1−lj ≤ |zj| < 1,|argzj−argξj| ≤ lj/2, j 1, . . . , n}.b Let
μz 1− |z|n−2 dm2z, μ1z nk11− |zk|−1/n d|z1| · · ·d|zn|dmξ, n >1.Then
μEαa≤
C a
Tn
f∗ϕdmn
ϕ, μ1
E1
αa
≤ C
a
Tn
f∗ϕdmn
ϕ. 2.3
Proof. Proofs of all parts are similar. The first part of the lemma connected withμmeasure
and unit disk can be found in4. We will give the complete proof of the second part. To
prove the second part letE0
Fix a pointzin theE0
αaand associate anα- boxBzsuch that
1
|Bz|
Bz
f∗dmn> a, BzBz1× · · · ×Bzn, |Bz||Bz1| · · · |Bzn|, Bzj
≥1−rj. 2.4
We use standard covering Lemma A to construct a setJ ⊂E0αasuch thatα- boxesBz, z∈J
pairwise disjoint, we have
z∈J
|Bz|≤ 1 a
z∈J
Bz
f∗ξdmnξ≤
Tn
fdmn. 2.5
So the lemma will be proved ifμ1E1αa≤C
z∈J|Bz|.Let
Keiϕ, lξ∈U:ξ−eiϕ< l, l >0, E1αa z→−r eix ∈U∗n:g→−α→−r , x> a.
2.6
Let w ∈ E0
αa, then we will show w ∈
z∈JKeiϕ1, lz1× · · · × Ke
iϕn, l
zn, ϕj argzj. So E0
αa ⊂
z∈JKeiϕ1, lz1× · · · ×Keiϕn, lzn
z∈JKz1,where lzj C|Bzj|, j 1, . . . , n,
and constant Cwill be specified later. Hence we will have μ1E0
αa∩U∗n μ1Eα1a ≤
Cz∈Jμ1Kz1∩Un∗≤Cz∈J|Bz|.
The last estimate follows from inclusionKz
1 ∩Un∗ ⊂Δlzξ∩Un∗,
Δlzξ∩U
n ∗
z∈Un∗ :zj∈
1−lzj,1
, argξj
−argz≤ lzj
2 , j1, . . . , n . 2.7
It remains to show the inclusionE0
αa ⊂
z∈JKz1.To show this inclusion we note if ξ ξ1, . . . , ξn∈E0αathen we have|Bξi| ≥1− |ξi|, i1, . . . , n.Using covering Lemma A we haveBξi ⊂CB zi, i1, . . . , n, z z1, . . . , zn∈J.Hence
ξi−eiϕi≤ξi−eiargξi eiargξi−eiϕi≤1− |ξi| argξi−ϕi≤2C|Bzi|. 2.8
It remains to note that we put abovelzi |Bzi|2C.
bNote that for the case ofUn
∗ we can step by step repeat the same procedure with
E1
αainstead ofE0αaand the condition onμ1will be replaced by weaker condition
μ1Δlξ∩Un∗≤Cl1· · ·ln, lj∈0,1, ξ∈T. 2.9
Note forξ∈T
Δlξ∩Un∗
z∈Un
∗ : 1−lj≤zj<1,arg
zj
−argξ≤ minlj
2 , j1, . . . , n . 2.10
Remark 2.2. InProposition 2.1bnk11− |zk|−1/n can be replaced bynk11− |zk|−tk, t
j∈ 0,1, nj1tj1.
Lemma 2.3. Letzj ∈U, j1, . . . , n, 0< p≤1, s >1/p−2, f ∈HUand Fz1, . . . , zn CUfz1− |z|sdm2z/nj11−zzjs 2/n.Then
|Fz1, . . . , zn|p≤C
U
fwp1− |w|sp 2p−2dm2w
n
k1|1−zkw|
s 2/np . 2.11
Estimate2.11forn1 can be found in5and in1for general case. The following
lemma is well known.
Lemma 2.4. Let s > 0.Then forIs, Is ∞stβ−α−2dt/t 11 β one hasa Is∼sβ−α−1, β <1 α, s → 0;bIs∼ln 1/s,1 αβ, s → 0;cIs≤C, 1 α < β, s → 0.
Lemma 2.5see3. Letw|w|ξ, w, z∈Un,1−wzn
k11−wkzk, s∈N∪ {0}, β >0, p∈ 0,∞.Then one has
Tn Rs
1
1−ξ|w|zβ
p
dmnξ≤C
αj≥0,αjs n
k1
1
1− |wk||zk|pαk β−1
, p > 1
mink αk β. 2.12
Corollary 2.6. Let0< p <∞, s∈N∪ {0}, l∈0,∞, γ >1/p l, w∈Un.Then
Un
Rs 1 1−wzγ
p1− |z|pl−1dm2nz≤
αj≥0,αjs
n
k1
C
1− |wk|αk γp−pl−1. 2.13
We will need the following Theorems A and B.
Theorem Asee3. aLetf∈HUn, p∈0,∞, α
j>−1, j1, . . . , n, n∈N.Then
U
fz, . . . , zp1− |z|2
n j1αj n−1
dm2z
≤C
T
0,1n
f|z1|ξ, . . . ,|zn|ξp
n
k1
1− |zk|2αkd|z1| · · ·d|zn|dmξ.
2.14
bLetf ∈HUn, p∈0,∞, α >−1, n∈N.Then
U
fz, . . . , zp1− |z|2α n−1dm2z≤C
Tn 1
0
f|z|ξ1, . . . ,|z|ξnp
Theorem Bsee3. aLetf ∈HUn, p∈0,∞, α
j>−1, j1, . . . , n, n∈N.Then
T
0,1n
f|z1|ξ, . . . ,|zn|ξpn k1
1− |zk|αk n−1/n d|z1| · · ·d|zn|dmξ
≤C
Un
fz1, . . . , znp n
k1
1− |zk|2αkdm2nz.
2.16
bLetf ∈HUn, p∈0,∞, α >−1, n∈N.Then
1
0
Tn
f|z|ξ1, . . . ,|z|ξnp
1− |z|2αdmnξd|z|
≤C
Un
fz1, . . . , znp
1− |z1|2
α/n−1 1/n
· · ·1− |zn|2α/n−1 1/ndm2nz.
2.17
3. Analytic Classes on Subframe and Expanded Disk
Let us remind the main definition.
Definition 3.1. LetX ⊂HU,Y ⊂HUnbe subspaces ofHUandHUn.We say that the
diagonal ofYcoincides withXif for any functionf, f ∈ Y, fz, . . . , z∈ X, and the reverse is also true for every functiongfromXthere exists an expansionfz1, . . . , zn∈ Y,such that
fz, . . . , z gz.Then we write DiagY X.
Note when DiagY X,then
fXinf
Φ ΦfY, 3.1
whereΦfis an arbitrary analytic expansion offfrom diagonal of polydisk to polydisk.
The problem of study of diagonal map and its applications for the first time was also
suggested by Rudin in 2. Later several papers appeared where complete solutions were
given for classical holomorphic spaces such as Hardy, Bergman classes; see1,4,6,7and
references there. Recently the complete answer was given for so-called mixed norm spaces in
8. Partially the goal of this paper is to add some new results in this direction. Theorems on
diagonal map have numerous applications in the theory of holomorphic functionssee, e.g.,
9,10.
In this section we concentrate on the study of two maps closely connected with
diagonal mapping Sb from subframe into disk Sb : fz1, . . . , zn → fz, . . . , z where all
|zj| |z| ∈ 0,1and another map Ed : fz1, . . . , zn → fz, . . . , z from expanded disk
into disk wherez1 |z1|ξ, . . . , zn |zn|ξ, ξ ∈T,andf function is from a functional class on
subframeUnor expanded diskUn
∗.
Note that the study of maps which are close to diagonal mapping was suggested by
Theorem 3.2. Let0< p <∞.If a0< p≤1 and β >−1,or
b p > 1 and β > maxn/p− 1,n−1p−1−1, then for every function g ∈ ApβUn, g|z|ξ, . . . ,|z|ξ ∈Ap
β n−1Uand for every functionf ∈A
p
β n−1Uthere exist g ∈ApβUnsuch thatg|z|ξ, . . . ,|z|ξ fz.
Proof. Note that one part of the theorem was proved in3and follows from Theorem A. Let
us show the reverse. Let firstp≤1.Consider the followinggfunction:
grξ1, . . . , rξn Cα
U
fw1− |w|αdm2w
n
k11−rξkwα 2/n
, z rξ1, . . . , rξn, 3.2
whereα >0 can be large enough,Cαis a constant of Bergman from representation formula
see1, obviouslygrξ, . . . , rξ frξby Bergman representation formula in the unit disk.
It remains to note that the following estimate hold byLemma 2.3:
Tn 1
0
g|z|ξ1, . . . ,|z|ξnp1− |z|β|z|d|z|dmnξ
C
1
0
Tn
U
fwp
1− |w|pα 2p−21−rβdrdm2wdmnξ |1−rξ1w|α 2p/n· · · |1−rξnw|α 2p/n
,
3.3
whereαcan be large enough. Using Fubini’s theorem and calculating the inner integral we
get what we need.
We consider now 1< p <∞case. Letf∈Apβ n−1U, p >1.Then
fz Cα
U
fw1− |w|α
1−wzα 2 dm2w, z∈U, 3.4
by Bergman representation formula. ObviouslySbg f, as
grξ1, . . . , rξn Cα
U
fw1− |w|α
1−rξ1wα 2/n· · ·1−rξnwα 2/n
dm2w, 3.5
whereSbg grξ, . . . , rξ, rξ ∈ Uis a map from subframeU {z ∈ Un : |zj| |z|}to
diagonal. Using duality arguments and Fubini’s theorem we have
g
ApβUnCα
U
1− |w|αfw
1
0
Tn
hrξ1, . . . , rξn1−rβdr dmnξ
n
k11−rξkwα 2/n
dm2w,
3.6
We will need the following assertion. Leth∈LpβUn, β∈−1,∞, p∈1,∞.
Then let
gw1, . . . , wn
1
0
Tn
hrξ1, . . . , rξn1− |z|βdr dmnξ
n
j1
1−rξjwj
β 1 n/n , 3.7
wherewj∈U, wj |w|ϕj, j 1, . . . , n.We assert thatgbelongs toAp
βUn.
Indeed using H ¨older’s inequality we get
1
0
Tn
gw1, . . . , wnp
1− |w|βdmn
ϕd|w|
≤C
1
0
1
0
Tn
Tn
|hrξ1, . . . , rξn|p
1−rβp1− |w|β p/pnεp 1
n
k1|1−rξkwk|β n 1/n−2−εp
2
×dr dmnξdmn
ϕd|w| ≤ChLp βUn
, 1< p<∞,
3.8
whereε >0.We used above the estimate
Tn 1
0
1− |w|β p/pnεp 1dmn
ϕd|w|
n
k1|1−rξkwk|β 1/n−ε−1p
2 ≤C1−rβ−βp
, 3.9
r∈0,1, β >n−1p/p−1, β >n/p−1.
Returning to estimate forgAp
βwe have by H ¨older’s inequality
g
Apβ
UnC
U
fwp
1− |w|β n−1dm2w
!1/p
× U
|Φw, . . . , w|p1− |w|β n−1dm2w
!1/p
CfAp β n−1.
3.10
We used the fact that for allΦ∈Apβ,1< p<∞, SbΦ∈Apβn−1proved before.
The complete analogue ofTheorem 3.2is true for Bergman classes on expanded disk
we defined previously.
Theorem 3.3. Let0≤p <∞.If
ap∈0,1,→−β β1, . . . , βn, βk>−1, k1, . . . , n, |β|nk1βk,or
bp > 1,→−β β, . . . , β, β > 0, β > 1/n−1p/p−1,then the following assertion holds. For every functionf, f ∈ A→p−
βU n
∗, Edfz fz, . . . , z, z ∈ Ubelongs to
Ap|β| n−1and the reverse is also true, for any functionffromAp|β| n−1there exists a function
g ∈A→p− βU
n
Proof. We give a short sketch of proof ofTheorem 3.3and omit details. Note that the half of the theorem the inclusion EdA→p−
βU n
∗⊂ Ap|β| n−1 was proved in3and follows directly from
TheoremA.
Forp ≤ 1 we have to use againLemma 2.3and Fubini’s theorem. Forp > 1 we first
provegw1, . . . , wn ∈A→p− βU
n
∗ifh∈ A→p− βU
n
∗and if|β|p/n > supkβk > minkβk > 1/n−
1p/p−1
gw1, . . . , wn C
T
1
0
· · ·
1
0
hz1, . . . , znnk11− |zk||β|/nd|zk|dξ
n
k1|1−zkwk||β| 1 n/n
. 3.11
Indeed by H ¨older’s inequality we have
gw1, . . . , wnp
C
⎛ ⎝
T
1
0
· · ·
1
0
|hz1, . . . , zn|p
n
k11− |zk||β|p/nd|zk|dξ
n
k1|1−zkwk||β| 1 n/np−2 εp 2
⎞ ⎠
×
T
1
0
· · ·
1
0
d|zk|dξ
n
k1|1−zkwk|2 εp
p/p
Ah
T
dξ
n
k11−ξwkϕ1 εp
p/p
Ah
n
k1
1
1− |wk|1 εp−1/np/p
, where ε >0.
3.12
Hence calculating integrals we finally havegAp
− →
βU n
∗ChA→p− βU
n ∗.
We used the estimate
T
1
0
· · ·
1
0
n
k11− |wk|βk 1/n−1 εp
p/p
d|wk|dξ
n
k1|1−zkwk||β| 1 n/np−2 εp 2
≤C
n
k1
1− |zk|−|β|p/n βk, 3.13
which is true under the conditions on indexes we have in formulation of theorem and can
be obtained by using H ¨older’s inequality for n functions. Using this projection theorem
and repeating arguments of proof of the previous theorem we will complete the proof of
Remark 3.4. Note that Theorems3.2and3.3are obvious forn1.
Remark 3.5. Note that estimates between expanded disk, unit disk, and polydisk can be also obtained directly from Liuville’s formula
1
0
· · ·
1
0
ϕv1· · ·vn n
k1
1−vkpk−1vp21v
p1 p2
3 · · ·v
p1 ···pn−1
n dv1· · ·dvn
n
k1Γ
pk
Γp1· · ·pn
1
0
ϕu1−uni1pi−1du, p
i>0, ϕis continuous function on0,1. 3.14
Remark 3.6. The complete description of traces of classes with fA∞
αUn∗ and fA∞αUn quasinorms on the unit disk can be obtained similarly by small modification of the proof ofTheorem 3.2,
fp
A∞αUn 1
0
M∞· · ·M∞f, rp1−rαdr, α >−1, p∈0,∞,
fp A∞αUn∗
1
0
1
0
M∞f, r1, . . . , rn
p
1−r1α· · ·1−rnαdr1· · ·drn, α >−1, p∈0,∞ 3.15
Let
ΛαU∗n
f ∈HUn: sup rj∈0,1,ξ∈T
fr1ξ, . . . , rnξ n
k1
1−rkα<∞ , α >0,
Λα
Un
f ∈HUn sup r∈0,1,ξi∈T
frξ1, . . . , rξn1−rα<∞ , α >0.
3.16
We formulate complete analogues of Theorems3.2and3.3for classesΛαU∗n andΛαUn.
Theorem 3.7. Let α > 0 and f ∈ ΛαUn. ThenSbf fz, . . . , z is in ΛαU and any
g ∈ΛαUcan be expanded tof, f ∈ΛαUnsuch thatfz, . . . , z gz.The same statement is
true for pairsEd,ΛαUn∗,EdΛαUn∗ ΛnαU.
Note that one part of statement is obvious. If, for example,f ∈ ΛαUn∗,then Edf ∈
ΛnαU.On the other side, letg ∈ΛnαU.Then define as above that
fr1ξ, . . . , rnξ Cβ
U
gz1− |z|β
n
k11−zrkξβ 2/n
dm2z, 3.17
βis big enough,Cβis a Bergman constant of Bergman representation formula.
Obviously frξ, . . . , rξ grξ, z rξ, z ∈ U. Using H ¨older’s inequality for n
functions we getfΛ
It is natural to question about discrete analogues of operators we considered previously.
LetCk>0, rk1, k → ∞, β≥0.Let
Bn,Ck,rk,β ⎧ ⎨
⎩f∈HUn∗:fz1, . . . , zn ∞
k1 Ck
n
j1
1
rk−zj
1 β/n <∞,
zj|z| ∈0,1, zj∈U, j1, . . . , n⎫⎬ ⎭.
3.18
We have for such a function
1
0
1−r1 αM∞f, rdr
1
0
1−r1 α sup
|z1|r,...,|zn|r
fzdr
∞
k1 Ck
1
0
1−r1 α
rk−r1 β
dr
∞
k1
Ckrk−1α−β 1
∞
rk−1
tβ−α−2
t 11 βdt.
3.19
As a consequence of these arguments and using Lemma 2.4 we have the following
proposition, a discrete copy of assertions we proved above.
Proposition 3.8. Letf ∈ Bn,Ck,rk,β andα > −1.Then 1
01−r 1 αM
∞f, rdr < ∞if and only
if∞k1Ck < ∞ if 1 α > β;∞k1Ckln 1/rk −1 < ∞ if1 α β;∞k1Ck/rk−1β 1 < ∞ if1 α < β.
4. Sharp Embeddings for Analytic Spaces in Polydisk with
R
sOperators and Inequalities Connecting Classes on
Polydisk, Subframe, and Expanded Disk
The goal of this section is to present various generalizations of well-known one-dimensional results providing at the same time new connections between standard classes of analytic
functions with quazinorms on polydisk and Rs differential operator with corresponding
classes on subframe and expanded disk.
In this section we also study another two maps connected with the diagonal mapping from polydisk to subframe and expanded disk using, in particular, estimates for maximal
functions fromLemma 2.3which are of independent interest. Note that for the first time the
study of such mappings which are close to diagonal mapping was suggested by Rudin in2.
Later Clark studied such a map in11.
In this section we also introduce the Rs differential operator as follows see
3, 12, 13. Rsf
k1,...,kn≥0k1 · · · kn 1
sa k1,...,knz
k1
1 · · ·z
kn
n, s ∈ R, where fz
k1,...,kn≥0ak1,...,knz
k1
1 · · ·z
kn
n ∈HUn.
In the case of the unit ball an analogue ofRsoperator is a well-known radial derivative
which is well studied. We note that in polydisk the following fractional derivative is well
studiedsee1:
Dαfz k1,...,kn≥0
k1 1α· · ·kn 1αak1,...,knz
k1
1 · · ·z
kn
n , 4.1
whereα∈R, f ∈HUn, andD:HUn → HUn.Apparently theRsoperator was studied
in 12for the first time. Then in 13, the second author studied some properties of this
operator. In this section we also continue to study theRs operator. We need the following
simple but vital formula which can be checked by easy calculation:
fτξ1, . . . , τξn Cs
1
0
Rsfτξ
1ρ, . . . , τξnρ log
1
ρ
!s−1
dρ, 4.2
where s > 0, τ ∈ 0,1, Cs > 0, ξj ∈ T, j 1, . . . , n.This simple integral representation of
holomorphic fz, z ∈ Un functions in polydisk will allow us to consider them in close
connection with functional spaces on subframeUn.
The following dyadic decomposition of subframe and polydisk was introduced in1
and will be also used by us:
Uk,l1,...,ln Uk,l1× · · · ×Uk,ln
&
τξ1, . . . , τξn:τ∈ 1−
1 2k,1−
1 2k 1
'
,
k0,1,2, . . .; πlj 2k < ξj ≤
πlj 1
2k , lj−2
k, . . . ,2k−1, j 1, . . . , n ,
mIk,lj
m
ξ∈T : πlj 2k < ξj≤
πlj 1
2k
2−k, m2n
Uk,l1,...,ln
2−2kn,
Uk1,...,kn,l1,...,ln Uk1,l1× · · · ×Ukn,ln
&
τ1ξ1, . . . , τnξn:τj∈ 1− 1
2kj,1− 1 2kj 1
'
,
kj0,1,2, . . .;
πlj
2kj < ξj≤
πlj 1
2kj , lj−2
kj, . . . ,2kj−1, j 1, . . . , n ,
4.3
Mpf, r, 0 < p ≤ ∞ averages in analytic spaces in polydisk can obviously have a mixed
form, for example,MpM∞f, r1, r2, rj∈0,1, j1,2, p <∞.In8Ren and Shi described
the diagonal of mixed norm spaces, but the above mentioned mixed case was omitted there.
Our approach is also different. It is based on dyadic decomposition we introduced previously.
Theorem 4.1. Let p ∈ 0,∞ and Hα,np,∞ {f ∈ HUn :
1
0· · ·
1
0
TM∞· · ·M∞f,
Proof. Using diadic decomposition of polydisk we have
M
U
fz, . . . , zp1− |z|αdm2z
j,k
Uj,k
fz, . . . , zp1− |z|αdm2z
≤C
j,k
max
Uj,k
fz, . . . , zp
2−2k2−αk
≤C
k1≥0
· · · kn≥0
2minkj−1
j−2minkj
max
Uk1,...,kn,j
fz1, . . . , znp
2−α 2k1/n· · ·2−α 2kn/n
k1≥0
· · · kn≥0
2minkj−1
j−2minkj
1−2−k1 2
1−2−k1−1
· · ·
1−2−kn 2
1−2−kn−1
×
Ij,k1,...,kn
fz1, . . . , znp
dr1· · ·drndξ
2k1τ· · ·2knτ
2minkj n
, τ −α 2 n 1.
4.4
We used above the following estimate which can be found, for example, in1
sup
z∈Uk1,...,kn,j
fz1, . . . , znp
2minkj
n
2k1 ···kn
U∗k
1,...,kn,j
fz1, . . . , znpdm2nz, 0< p <∞,
4.5
whereUk∗
1,...,kn,jare enlarged dyadic cubessee1and 0< p <∞, f∈HU
n, and
Ij,k1,...,kn
ξ:zrξ∈Uk1,...,kn,j,
πj
2minkj ≤ξ <
πj 1 2minkj ,
Ij,k∗
1,...,kn
ξ:zrξ∈Uk∗
1,...,kn,j,
πj−1/2 2minkj ≤ξ <
πj 1/2
2minkj .
4.6
Note further sincemnIj,k1,...,kn 2−n
minkj,
Ij,k∗1,...,kn
fz1, . . . , znp
dξC
I1 j,k1,...,kn
M∞· · ·M∞f, rpdξ
2−n−1minkj
,
Ij,k∗
1,...,kn I
1
j,k1,...,kn × · · · ×I
n
j,k1,...,kn, 0< p <∞.
4.7
We have
MC
k1≥0
· · · kn≥0
2minkj−1
j−2minkj
1−2−k1 2
1−2−k1−1
· · ·
1−2−kn 2
1−2−kn−1
I1 j,k1,...,kn
M∞· · ·M∞f, rpdξ2−k1α 1/n−1· · ·2−knα 1/n−1/n.
4.8
Hence using the fact thatIj,k1
1,...,kn is a finite covering ofT, finally we have for all 0 <
p <∞ MCfHp,∞
α,n.One part of theorem is proved.
To get the reverse statement we use the estimate fromLemma 2.3. Then we have
|Fz1, . . . , zn|C
U
fz1− |z|αdm2z
n j1
1−zzj
α 2/np
, zj∈U, 4.9
for anyα >−1.HenceFz, . . . , z fzand forp <1 byLemma 2.3
FHp,∞ α,n
1
0
1
0
T
M∞· · ·M∞
U
fzp1− |z|αp 2p−2dm2z
n
j11−zzjα 2/np
×n k1
1−rkα 1/n−1dr1· · ·drndξ1· · ·dξn
U
fzp1− |z|αp 2p−2
1
0
1
0
n
k11−rkα 1/n−1dr1· · ·drn
n−1
k11− |z|rkα 2/np1−rn|z|α 2/np−1
dm2z
U
fzp1− |z|αp 2p−21− |z|−α 2/np α 1/nn−1
×1− |z|−α 2/np α 1/n 1dm2z
U
fzp1− |z|αdm2z, α >−1, 0< p≤1, f ∈Apα.
4.10
Let 1 < p < ∞.Then we may assume that againsis large enough. Let 1/p 1/q
1, γi>0, i1,2, γ1 γ2 s 1/n.Then
⎛ ⎜ ⎝
U
fw 1− |w|2s−1dm2w
n
j11−zjws 1/n
⎞ ⎟ ⎠
p
≤C
U
fwp
1− |w|2s−1dm2w
n
j11−zjwγ1p
n
j1
1−zjγ1p−s 1/n.
4.11
We used estimate
⎛ ⎜ ⎝
U
1− |w|2s−1dm2w
n
j11−zjwγ2q
⎞ ⎟ ⎠
p/q
≤C
n
j1
Choosing appropriateγ1, γ2we repeat now arguments that we presented forp≤1 above to get what we need. The proof is complete.
Remark 4.2. The case ofk, 1 < k ≤ n, Mp averages can be considered similarly. Note in Theorem 4.1k1.Thus ourTheorem 4.1extends known description of diagonal of classical
Bergman classessee1,7.
In14Carleson as Rudin and Clark showed that in the case of the polydisk one cannot
expect so simple description of Carleson measures as one has for measures defined in the disk. We would like to study embeddings of the type
Un
fzpdμz
!1/p
≤CRsfX, s≥0, 4.13
whereμis a positive Borel measure onUnandXis a Bergman class on polydisk or subframe.
Theorem 4.3. Letγ1 β1 s−1p n 1p−1, γ1 > 0, β1 > −1, 0 < p ≤1, s > β1
1/n n−pn/p, s, n∈N, andμis a positive Borel measure onUn.If
Un
fzp1− |z|tdμzRsf Apβ
1U
n, t >−1, 4.14
then
μ
Un sup
w∈Un
Tn 1
0
1− |w|tdμw
n
k1|1−wkwk|p
1− |w|γ1 <∞, 4.15
and conversely if
μ
Un sup
w∈Un
Tn 1
0
1− |w|tdμw
n
k1|1−wkwk|p ε
1− |w|γ1 ε<∞ 4.16
holds for someε >0wherew |w|ξ1, . . . ,|w|ξn, |w| ∈0,1, ξj∈T, j1, . . . , nthen
Un
fzp1− |z|tdμzRsfAp β1U
n, t >−1. 4.17
Remark 4.4. With another “ε-sharp” embedding theorem, the complete analogue of
Theorem 4.3is true also when we replace the left side byUn|fz|p n
k11− |zk|tkdμz, tk> −1, k1, . . . , n.The proof needs small modification of arguments we present in the proof of
Theorem 4.3.
Proof ofTheorem 4.3. It can be checked by direct calculations based on formula4.2that the following integral representation holds:
fz1, . . . , zn Cs
Tn 1
0
Rsfρξ1, . . . , ρξ n log
1
ρ
!s−1 dρdm
nξ
n
k1
1−ρϕkrξk
, 4.18
s > 0, zk ϕkr, k 1, . . . , n, r ∈ 0,1,z1, . . . , zn ∈ Un.Using the fact thatMpf, r in
increasing byr, r ∈ 0,1andM1f, τ2 ≤ C1−τn1−1/pM
pf, τ, τ ∈ 0,1 see1we
get from above by using the estimate
1
0
fρ1−ραdρ
p
1
0
fρp1−ραp p−1dρ, 4.19
wherefis growing measurable,p≤1, α >−1,
Un
fwp1− |w|tdμw≤CμUn 1
0
Tn
Rsfwp
1− |w|β1dm
nξd|w|. 4.20
To obtain the reverse implication we use standard test functiongzw 1/nk11−wkzkγ,
zk, wk∈U, k1, . . . , n, z, w∈Un, γ >1 andLemma 2.5
Rsg
zwAp β1U
n 1
0
1− |w|β1
αj≥0,αjs
Gz, wd|w|, 4.21
whereGz, w nk11/1− |wk||zk|pαk γ−1, γ >1, p >1/γ. The rest is clear. The proof is complete.
Below we continue to study connections between standard classes in polydisk and corresponding spaces on subframe and expanded disk.
Let
RAps,v
f ∈HUn:Rsf∈Apν
Un,
DA→p−γ ,α
f∈HUn:Dγf∈ApαUn
, DγDγ1
z1· · · D
γn
zn,
n
j1
γjγ, γj≥0.
4.22
Let us note that Theorems3.2and3.7of3show that under some restrictions onγ, α, s, ν
the following assertion is true.
For every function f, f ∈ DApγ,α, 0 < p < ∞, f|z|ξ1, . . . ,|z|ξn belongs to RAps,v, 0< p < ∞,and the reverse is also true, for any functionf, f ∈RAps,vthere exists
“an extension”Fsuch that
The followingTheorem 4.6gives an answer for the same map from polydisk to expanded disk
fz1, . . . , zn−→f|z1|ξ, . . . ,|zn|ξ, ξ∈T, zj∈0,1, j1, . . . , n, 4.24
forffunctions fromHpUn, 1< p <∞.
We will develop ideas from 4 to get the following sharp embedding theorem for
classes on expanded disk.
Theorem 4.6. Letn >1, 1< p <∞, μbe a positive Borel measure onUn ∗.Then
1
0
· · ·
1
0
T
fr1ξ, . . . , rnξpdμ
→−
r ξ≤fHpUn 4.25
if and only if11−l 1· · ·
1
1−ln
minjlj/2
−minjlj/2dμ
−
→r ξ≤Cl1· · ·l
n, lj>0, j1, . . . , n.
Proof. Obviously ifKUn ∗|fz|
pdμz≤Cfp
Hp,then by putting
f
n
j1
fj, fjz
⎛ ⎜
⎝ 1−lj 1−zjlj
2
⎞ ⎟ ⎠
1/p
, lj1−lj, lj∈0,1, j 1, . . . , n, 4.26
we havefHp const andK ≥ 1/l1· · ·ln 1
1−l1· · · 1
1−ln τ
−τdμr1ξ, . . . , rnξ, τ minjlj, |1−
zjlj| 1− |zj|, |zj| ∈1−lj,1, argzj∈−τ, τ.Hence we get what we need. Now we will
show the sufficiency of the condition.
LetN >0 : 2N1−r
j≤π <2N 11−rj, j1, . . . , n.
Letz →−r eiϕ ∈ Un
∗, z r1eiϕ, . . . , rneiϕ.Let also Bzk {t t1, . . . , tn : |tj −ϕ| <
2kj1−r
j,1≤j ≤n}andWk1,...,kn Ik1× · · · ×Ikn,where
Ikj
t: 2kj1−r
j
≤t−ϕ<2kj 11−r
j
, 0≤kj ≤N−1,
IN
t: 2N1−rj
≤t−ϕ<2N 11−rj
, |t|< π, I−1
t:|t|<1−rj
.
4.27
In what follows we will use notations ofProposition 2.1. Consider the Poisson integral of a
functionf, f ∈H1T, uz C
TnPz, ξfξdmnξ, z∈Unsee2.
LetEa {z∈Un
∗ :Eduz> a}.We will show now as in4that
μEa≤ C a
Tn
Indeed, this will be enough, since the operatorf → EdTnPz, ξfξdmnξisL∞, L∞
operator we can apply the Marcinkiewicz interpolation theoremsee16, Chapter 1to assert
that
J
1
0
· · ·
1
0
T
fzpdμz≤CfHpUn, 1< p <∞. 4.29
We have as in4forz∈Un
∗
Eduz
Tn
Pz, wfwdmnw
N
k1,...,kn−1
Wk1,...,kn
Pz, wfwdmnw
≤C0
N
k1,...,kn0
1 4k1 ··· knn
k11−rk
Bk z
ftdt≤C
N
k1,...,kn0 1 2|k|gk
r1, . . . , rn, ϕ
≤ N k1,...,kn0
2−|k|gk
→−
r , ϕ≤C1supk2|k|/2gk
→−
r , ϕ,
4.30
where we used the standard partition of Poisson integral. HenceEa⊂α∈ZnEαC22|α|/2a, and so usingProposition 2.1we finally getμEa≤C/aTnftdt.
Indeed byProposition 2.1we have
μEa≤
α∈Zn
μEα
C22|α|/2a
≤C3
α∈Zn
C−21
2|α|/2a
Tn
ftdt≤ C4 a
Tn
ftdt.
4.31
Theorem 4.6is proved.
Theorem 4.7. Letf ∈HUn.Then
sup
|zj|<1
Dαfz1, . . . , z n
n
k1
1− |zk|α 1
≤C
Tn 1
0
Rsfρξ1, . . . , ρξ np
1−ρsp np−1−1dmnξdρ,
4.32
wheres >0, α >0, p≤1, sp np−1>0;
sup
z∈Un Dβfz
1, . . . , zn
⎛
⎝
αk≥0,αks
n
k1
1
1− |zk|β αk−2s−α/n ⎞ ⎠
−1
≤Csup
z∈Un
Rsfz1− |z|α,
where2s−α >0, β >1, α >0, s >0,
sup
|zk|<1
Tn
Rsf|z
1|ϕ1, . . . ,|zn|ϕnpdmn
ϕ
αj≥0,αj2s n
k1
1
1− |zk|pαk s 1−1 −1
≤C
Tn 1
0
D−sfwp
1− |w|td|w|dmn
ϕ,
4.34
wheres >0, 1/s 1< p≤1, tps−1 n 1p−1>−1.
Proof. We use systematically the integral representation4.18,Lemma 2.5, and it is corollary.
The proof of the estimate4.32follows from equality
Dαfz
1, . . . , zn Cs
Tn 1
0
Rsfρξ
1, . . . , ρξn log
1
ρ
!s−1n
k1
1
1−ξϕkρk
α 1dρdmnξ,
4.35
zkϕkρk, k1, . . . , n, s >0, α >0, f∈HUn,and estimate
Tn 1
0
Rsfρξ
1, . . . , ρξn1−ρ
α
dρdmnξ
p
≤C
Tn 1
0
Rsfρξ
1, . . . , ρξnp
1−ραp n 1p−1dρdmnξ,
4.36
p≤1, α >0, s >0, f ∈HUn,obtained during the proof ofTheorem 4.3.
The proof of the estimate4.33follows fromLemma 2.5and its corollary and integral
representation4.35.
The proof of the estimate4.34follows from equalityzjρjϕj, j 1, . . . , n
Rsfz1, . . . , z n
Cs
Tn 1
0
D−sfρξ
1, . . . , ρξn log
1
ρ
!s−1
R2sn k1
1
1−ξϕkρρk
s 1dρdmnξ
.
4.37
Indeed using4.36integrating both sides of4.37byTnand usingLemma 2.5we arrive at
Remark 4.8. All estimates inTheorem 4.7forn1 are well knownsee1, Chapter1.
We present below a complete analogue of Theorems 3.2 and 3.3 for a map from
polydisk to expanded disk. Note the continuation offfunction is done again from diagonal
z, . . . , z.Letzj∈U, j1, . . . , nand
fz1, . . . , zn Cs
U
fz, . . . , z1− |z|s
n
k11−zkzs 2/n
dm2z, 4.38
whereCsis a constant of Bergman representation formulasee1.
Proposition 4.9. 1 aLet n∈N, p∈0,∞, α >−1, f ∈HUn.Then
T
1
0
· · ·
1
0
f|z1|ξ, . . . ,|zn|ξpn k1
1− |zk|α n−1/nd|z1| · · ·d|zn|dmξ
≤C
Un fzp
n
k1
1− |zk|2αdm2nz.
4.39
And reverse is also true:
bLet0< p <∞, α >−1and
T
1
0
· · ·
1
0
f|z1|ξ, . . . ,|zn|ξpn k1
1− |zk|α n−1/nd|z1| · · ·d|zn|dmξ<∞, 4.40
then for all functionsfsuch that condition4.38holds fors > maxα 2n/p−2,−1we have
f∈ApαUn.
2 aLetf ∈HpUn, 1< p <∞, n >1.Then
fAp
−1/nU∗n
T
1
0
1
0
f|z1|ξ, . . . ,|zn|ξp
n
k1
1− |zk|−1/ndmξd|z1| · · ·d|zm|<∞,
4.41
and the reverse is also true:
bFor any functionfwith a finite quasinormfAp
−1/nUn∗such that condition4.38holds fors2n−2one hasf ∈HpUn, p >1.
Proof. 1 aProof of estimate4.39follows directly from Theorem B.
bIndeed from4.38and results of1on diagonal map in Bergman classes we have
fAp
αUn≤CDiagfAptU, tαn 2n−2.It remains to apply Theorem A.
2For the proof ofawe useTheorem 4.6and get the result we need.
For the proof ofbwe use the same argument as in the proof of part1. Namely first
from4.38and from a Diagonal map theorem onHp classes from1we get f
HpUn ≤
CfAp
Remark 4.10. NoteProposition 4.9is obvious forn1.
We give only a sketch of the proof of the following result. It is based completely on a technique we developed above.
Proposition 4.11. aLetα > 0, n ∈N, n >1, s > nαthen the following assertions are true: If
Dαf∈HpUnthen
Jp,α,sf
Tn 1
0
Rsf|z|ξ1, . . . ,|z|ξ
np1− |z|sp−npα−1d|z|dmnξ<∞, p≥2. 4.42
IfJp,α,sf<∞thenDαf∈HpUn, 1/α 1< p≤1.
bLetDαf∈HpUn, p≥2.Ifs > nα, α >0,thenJp,α,sf<∞.Moreover the reverse is
also true if condition4.38holds fors2n−2thenDαfp
Hp ≤CJp,α,sf.
The proof of first part ofProposition 4.11follows from4.35,4.36andLemma 2.5
directly. The reverse assertion follows from Theorem Bband estimate
Un
Dα1,...,αnfzp
n
k1
1− |zk|αkp−1dm2
nz≤CfHp, αk>0, k1, . . . , n, p≥2, 4.43
which can obtained from one dimensional result by induction.
The proof of second part ofProposition 4.11can be obtained from Theorem A and
results on diagonal map on Hardy classes HpUn from 1 similarly as the proof of
Proposition 4.9. For n 1 part a is well known e.g., see 1 and b follows froma
since forn1 condition4.38vanishes by Bergman representation formula.
Remark 4.12. Theorem 4.6, Propositions4.9and4.11 give an answer to a problem of Rudin
see2to find traces ofHpUnHardy classes on subvarieties other than diagonalz, . . . , z.
Note that in11Clark solved this problem for subvarietes ofUn based on finite Blaschke
products.
Acknowledgment
The authors sincerely thank Trieu Le for valuable discussions.
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