Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 353801, 22 pages
doi:10.1155/2009/353801
Research Article
Analytic Classes on Subframe and Expanded Disk
and the Rs Differential Operator in Polydisk
Romi F. Shamoyan1 and Olivera R. Mihić2
1
2
Department of Mathematics, Bryansk University, 241036 Bryansk, Russia
Faculty of Organizational Sciences, University of Belgrade, Jove Ilića 154, 11000 Belgrade, Serbia
Correspondence should be addressed to Olivera R. Mihić, oliveradj@fon.rs
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 differential Rs operator in polydisk also will be presented. Practically all our results
were known or obvious in the unit disk.
Copyright q 2009 R. F. Shamoyan and O. R. Mihić. 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
Let n ∈ N and Cn {z z1 , . . . , zn : zk ∈ C, 1 ≤ k ≤ n} be the n-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 of Un by
T n {z ∈ Cn : |zk | 1, 1 ≤ k ≤ n}.
1.2
We use m2n to denote the volume measure on Un and mn to denote the normalized Lebesgue
measure on T n . Let HUn be the space of all holomorphic functions on Un . When n 1, we
simply denote U1 by U, T 1 by T, m2n by m2 , mn by m. We refer to 1, 2 for further details. We
denote the expanded disk by
U∗n z r1 ξ, . . . , rn ξ ∈ Un : ξ ∈ T, rj ∈ 0, 1, j 1, . . . , n ,
1.3
2
Journal of Inequalities and Applications
and the subframe by
n z ∈ Un : zj r, r ∈ 0, 1, j 1, . . . , n .
U
1.4
The Hardy spaces, denoted by H p Un 0 < p ≤ ∞, are defined by
H U p
n
f ∈ HU : sup Mp f, r < ∞ ,
n
1.5
0≤r<1
p
where Mp f, r T n |frξ1 , . . . , rξn |p dmn ξ, M∞ f, r maxξ∈T n |frξ1 , . . . , rξn |, r ∈ 0, 1,
f ∈ HUn .
p
Un For αj > −1, j 1, . . . , n 0 < p < ∞, recall that the weighted Bergman space A→
−
α
consists of all holomorphic functions on the polydisk satisfying the condition
p
f p A
→
−
α
Un
n αi
fz1 , . . . , zn p
1 − |zi |2 dm2n z < ∞.
1.6
i1
For αj > −1, j 1, . . . , n, p ∈ 0, ∞, the Bergman class on expanded disk is defined by
p
A→
U∗n −
α
⎧
⎨
p
f ∈ HUn : f Ap Un ⎩
→
−
α
1
···
1
0
0
∗
⎫
n ⎬
2 αj p f|z1 |ξ, . . . , |zn |ξ
1 − zj dzj dmξ < ∞ ,
⎭
T
j1
1.7
n is defined by
and similarly the Bergman class on subframe denoted by Aα U
p
p
Aα
n
U
p
f ∈ HU : f p
n
n
Aα U
1
Tn
α
p 2
f|z|ξ1 , . . . , |z|ξn 1 − |z| dmn ξd|z| < ∞ ,
0
1.8
where p ∈ 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 notation A B means that there is a positive constant C such that B/C ≤ A ≤
CB. We will write for two expressions A B if there is a positive constant C such that A < 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
and expanded disk U∗n to unit disk U and, in particular, completely describe
subframe U
p n
p
and Aα U∗n , p ∈ 0, ∞ defined on subframe and
traces of Bergman classes Aα U
Journal of Inequalities and Applications
3
expanded disk on usual unit disk U 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 of Rs differential 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 onedimensional known results to polydisk and to the case of Rs operators simultaneously.
2. Preliminaries
We need the following assertions.
Lemma A see 4. Let α be a fixed n-tuple of nonnegative numbers and let {Bi }i∈Z be an arbitrary
family of α-boxes in Rn lying in the cube Qn −2π, 2πn . There exists a set J ⊂ I such that
Bi ∩ Bi
∅, i, i
∈ J and for all i ∈ I there exists j ∈ J such that Bi ⊂ 5Bj .
The following proposition is heavily based on ideas from 4.
Proposition 2.1. a Let f be a nonnegative summable function on T n T × · · · × T. Let
f ∗ x1 , . . . , xn feix1 , . . . , eixn . Let αj ∈ N, rj ∈ 0, 1, j 1, . . . , n, x ∈ Rn , gr, x Z ∩ Zα , where Z {B Bx, y : yj ≥ 1 − rj , 1 ≤ j ≤ n},
supB∈Z 1/mn B B f ∗ dmn , Z
→
−
→
−
−
−
n
B ϕ, y {t ∈ R : |ti − ϕi | ≤ yi , 1 ≤ i ≤ n}, and where Zα is a family of all B→
ϕ,→
y boxes such
− Then the
−α1
−αn
α.
that y1 , . . . , yn is proportional to 2 , . . . , 2 , αi ∈ N, i 1, . . . , n for some fixed ←
following statements hold:
C
−
f ∗ ϕ dmn ϕ ,
x
>
a
≤
μEα a μ z reix ∈ U : g→
r,
α
a Tn
C1 ∗
−
ϕ dmn ϕ ,
μ1 Eα1 a μ1 z r1 eix , . . . , rn eix ∈ U∗n : g→
nf
α r1 , . . . , rn , x > a ≤
T
a
2.1
where μ and μ1 are any positive Borel measures on U and U∗n such that
μΔl ξ ∩ U μΔl ξ ≤ Cln ,
μ1 Δl ξ ∩ U∗n ≤ Cl1 · · · ln ,
l ∈ 0, 1, ξ ∈ T,
lj ∈ 0, 1, ξ ∈ T n ,
2.2
where Δl ξ {z ∈ Un : 1 − lj ≤ |zj | < 1, | argzj − argξj | ≤ lj /2, j 1, . . . , n}. b Let
μz 1 − |z|n−2 dm2 z, μ1 z nk1 1 − |zk |−1/n d|z1 | · · · d|zn |dmξ, n > 1. Then
μEα a ≤
C
a
f ∗ ϕ dmn ϕ ,
Tn
C
f ∗ ϕ dmn ϕ .
μ1 Eα1 a ≤
a Tn
2.3
Proof. Proofs of all parts are similar. The first part of the lemma connected with μ measure
and unit disk can be found in 4. We will give the complete proof of the second part. To
→
−
−
prove the second part let Eα0 a {z reix ∈ Un : g→
α r , x > a}.
4
Journal of Inequalities and Applications
Fix a point z in the Eα0 a and associate an α- box Bz such that
1
|Bz |
f ∗ dmn > a, Bz Bz1 × · · · × Bzn , |Bz | |Bz1 | · · · |Bzn |, Bzj ≥ 1 − rj .
Bz
2.4
We use standard covering Lemma A to construct a set J ⊂ Eα0 a such that α- boxes Bz , z ∈ J
pairwise disjoint, we have
1
∗
f ξdmn ξ ≤
fdmn .
|Bz | ≤
a z∈J Bz
Tn
z∈J
So the lemma will be proved if μ1 Eα1 a ≤ C
K eiϕ , l ξ ∈ U : ξ − eiϕ < l ,
l > 0,
z∈J
2.5
|Bz |. Let
→
−r eix ∈ Un : g→
− −
Eα1 a z →
α r,x > a .
∗
2.6
Let w ∈ Eα0 a, then we will show w ∈ z∈J Keiϕ1 , lz1 × · · · × Keiϕn , lzn , ϕj argzj .
So Eα0 a ⊂ z∈J Keiϕ1 , lz1 × · · · × Keiϕn , lzn z∈J K1z , where lzj C|Bzj |, j 1, . . . , n,
and constant C will be specified later. Hence we will have μ1 Eα0 a ∩ U∗n μ1 Eα1 a ≤
C z∈J μ1 K1z ∩ U∗n ≤ C z∈J |Bz |.
The last estimate follows from inclusion K1z ∩ U∗n ⊂ Δlz ξ ∩ U∗n ,
Δlz ξ ∩
U∗n
z∈
U∗n
lzj
: zj ∈ 1 − lzj , 1 , arg ξj − argz ≤ , j 1, . . . , n .
2
2.7
It remains to show the inclusion Eα0 a ⊂ z∈J K1z . To show this inclusion we note if
ξ ξ1 , . . . , ξn ∈ Eα0 a then we have |Bξi | ≥ 1 − |ξi |, i 1, . . . , n. Using covering Lemma A we
zi , i 1, . . . , n, z z1 , . . . , zn ∈ J. Hence
have Bξi ⊂ CB
zi |.
ξi − eiϕi ≤ ξi − ei argξi ei argξi − eiϕi ≤ 1 − |ξi | argξi − ϕi ≤ 2C|B
2.8
It remains to note that we put above lzi |Bzi |2C.
b Note that for the case of U∗n we can step by step repeat the same procedure with
Eα1 a instead of Eα0 a and the condition on μ1 will be replaced by weaker condition
μ1 Δl ξ ∩ U∗n ≤ Cl1 · · · ln ,
lj ∈ 0, 1, ξ ∈ T.
2.9
Note for ξ ∈ T
Δl ξ ∩
U∗n
z∈
U∗n
min lj
, j 1, . . . , n .
: 1 − lj ≤ zj < 1, arg zj − argξ ≤
2
Now part b can be obtained by direct calculation.
2.10
Journal of Inequalities and Applications
Remark 2.2. In Proposition 2.1b
0, 1, nj1 tj 1.
n
k1 1
5
− |zk |−1/n can be replaced by
n
k1 1
− |zk |−tk , tj ∈
Lemma 2.3. Let zj ∈ U, j 1, . . . , n, 0 < p ≤ 1, s > 1/p − 2, f ∈ HU and Fz1 , . . . , zn C U fz1 − |z|s dm2 z/ nj1 1 − zzj s2/n . Then
p
fw
1 − |w|
sp2p−2 dm2 w
.
|Fz1 , . . . , zn | ≤ C
n
s2/np
U
k1 |1 − zk w|
p
2.11
Estimate 2.11 for n 1 can be found in 5 and in 1 for general case. The following
lemma is well known.
∞
Lemma 2.4. Let s > 0. Then for Is, Is s tβ−α−2 dt/t 11β one has a Is ∼ sβ−α−1 ,
β < 1 α, s → 0; b Is ∼ ln 1/s, 1 α β, s → 0; c Is ≤ C, 1 α < β, s → 0.
Lemma 2.5 see 3. Let w |w|ξ, w, z ∈ Un , 1 − wz 0, ∞. Then one has
n
k1 1
− wk zk , s ∈ N ∪ {0}, β > 0, p ∈
p
n
1
1
1
s
, p>
.
R
dmn ξ ≤ C
β
p
α
β
−1
min
k
n
k αk β
1 − ξ|w|z
T
αj ≥0, αj s k1 1 − |wk ||zk |
2.12
Corollary 2.6. Let 0 < p < ∞, s ∈ N ∪ {0}, l ∈ 0, ∞, γ > 1/p l, w ∈ Un . Then
U
p
s
1
pl−1
R
1 − wzγ 1 − |z| dm2n z ≤
n
αj ≥0,
n
αj s k1
C
1 − |wk |αk γp−pl−1
.
2.13
We will need the following Theorems A and B.
Theorem A see 3. a Let f ∈ HUn , p ∈ 0, ∞, αj > −1, j 1, . . . , n, n ∈ N. Then
fz, . . . , zp 1 − |z|2
U
≤C
T
0,1n
n
j1
αj n−1
dm2 z
n αk
f|z1 |ξ, . . . , |zn |ξp
1 − |zk |2 d|z1 | · · · d|zn |dmξ.
2.14
k1
b Let f ∈ HUn , p ∈ 0, ∞, α > −1, n ∈ N. Then
αn−1
fz, . . . , zp 1 − |z|2
dm2 z ≤ C
U
1
Tn
0
α
f|z|ξ1 , . . . , |z|ξn p 1 − |z|2 dmn ξd|z|.
2.15
6
Journal of Inequalities and Applications
Theorem B see 3. a Let f ∈ HUn , p ∈ 0, ∞, αj > −1, j 1, . . . , n, n ∈ N. Then
T
0,1n
n
f|z1 |ξ, . . . , |zn |ξp
1 − |zk |αk n−1/n d|z1 | · · · d|zn |dmξ
≤C
Un
k1
n αk
fz1 , . . . , zn p
1 − |zk |2 dm2n z.
2.16
k1
b Let f ∈ HUn , p ∈ 0, ∞, α > −1, n ∈ N. Then
1
0
Tn
α
f|z|ξ1 , . . . , |z|ξn p 1 − |z|2 dmn ξd|z|
≤C
Un
α/n−11/n α/n−11/n
fz1 , . . . , zn p 1 − |z1 |2
· · · 1 − |zn |2
dm2n z.
2.17
3. Analytic Classes on Subframe and Expanded Disk
Let us remind the main definition.
Definition 3.1. Let X ⊂ HU, Y ⊂ HUn be subspaces of HU and HUn . We say that the
diagonal of Y coincides with X if for any function f, f ∈ Y, fz, . . . , z ∈ X, and the reverse
is also true for every function g from X there exists an expansion fz1 , . . . , zn ∈ Y, such that
fz, . . . , z gz. Then we write DiagY X.
Note when DiagY X, then
f infΦf ,
X
Y
Φ
3.1
where Φf is an arbitrary analytic expansion of f from 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; see 1, 4, 6, 7 and
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 functions see, 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, 1 and another map Ed : fz1 , . . . , zn → fz, . . . , z from expanded disk
into disk where z1 |z1 |ξ, . . . , zn |zn |ξ, ξ ∈ T, and f function is from a functional class on
n or expanded disk U∗n .
subframe U
Note that the study of maps which are close to diagonal mapping was suggested by
Rudin in 2 and previously in 11 Clark studied such a map.
Journal of Inequalities and Applications
7
Theorem 3.2. Let 0 < p < ∞. If
a 0 < p ≤ 1 and β > −1, or
b p > 1 and β > maxn/p − 1, n − 1p − 1 − 1, then for every function g ∈
p n
p
p
, g|z|ξ, . . . , |z|ξ ∈ Aβn−1 U and for every function f ∈ Aβn−1 U there exist
Aβ U
p n
g ∈ A U
such that g|z|ξ, . . . , |z|ξ fz.
β
Proof. Note that one part of the theorem was proved in 3 and follows from Theorem A. Let
us show the reverse. Let first p ≤ 1. Consider the following g function:
grξ1 , . . . , rξn Cα
fw1 − |w|α dm2 w
,
n
α2/n
U
k1 1 − rξk w
z rξ1 , . . . , rξn ,
3.2
where α > 0 can be large enough, Cα is a constant of Bergman from representation formula
see 1, obviously grξ, . . . , rξ frξ by Bergman representation formula in the unit disk.
It remains to note that the following estimate hold by Lemma 2.3:
1
Tn
g|z|ξ1 , . . . , |z|ξn p 1 − |z|β |z|d|z|dmn ξ
0
C
p
1 fw 1 − |w|pα2p−2 1 − rβ drdm2 wdmn ξ
0
Tn
|1 − rξ1 w|α2p/n · · · |1 − rξn w|α2p/n
U
3.3
,
where α can be large enough. Using Fubini’s theorem and calculating the inner integral we
get what we need.
p
We consider now 1 < p < ∞ case. Let f ∈ Aβn−1 U, p > 1. Then
fw1 − |w|α
fz Cα
U
1 − wzα2
dm2 w,
z ∈ U,
3.4
by Bergman representation formula. Obviously Sbg f, as
grξ1 , . . . , rξn Cα
fw1 − |w|α
U 1
− rξ1 wα2/n · · · 1 − rξn wα2/n
dm2 w,
3.5
{z ∈ Un : |zj | |z|} to
where Sbg grξ, . . . , rξ, rξ ∈ U is a map from subframe U
diagonal. Using duality arguments and Fubini’s theorem we have
g p n Cα
A U β
β
1
hrξ
,
.
.
.
,
rξ
dr
dm
−
r
1
ξ
1
n
n
1 − |w| fw
dm2 w,
α2/n
n
n
1 − rξk w
0 T
U
α
k1
p
n , 1/p 1/p
1.
where α β n − 1, α > −1, h ∈ Lβ U
3.6
8
Journal of Inequalities and Applications
p
n , β ∈ −1, ∞, p
∈ 1, ∞.
We will need the following assertion. Let h ∈ Lβ U
Then let
gw1 , . . . , wn 1
0
hrξ1 , . . . , rξn 1 − |z|β dr dmn ξ
,
β1n/n
n Tn
j1 1 − rξj w j
3.7
p
n .
where wj ∈ U, wj |w|ϕj , j 1, . . . , n. We assert that g belongs to Aβ U
Indeed using Hölder’s inequality we get
1
0
Tn
gw1 , . . . , wn p 1 − |w|β dmn ϕ d|w|
1 1 |hrξ1 , . . . , rξn |p 1 − rβp 1 − |w|βp /pnεp1
≤C
n
βn1/n−2−εp
2
0 0 Tn Tn
k1 |1 − rξk w k |
× dr dmn ξdmn ϕ d|w| ≤ ChLp
U n , 1 < p
< ∞,
β 3.8
where ε > 0. We used above the estimate
1 − |w|βp /pnεp1 dmn ϕ d|w|
≤ C1 − rβ−βp ,
n
β1/n −ε−1p
2
0
k1 |1 − rξk w k |
1
Tn
3.9
r ∈ 0, 1, β > n − 1p/p
− 1, β > n/p − 1.
Returning to estimate for gAp we have by Hölder’s inequality
β
n
g p U
C
A
β
fwp 1 − |w|βn−1 dm2 w
!1/p
U
|Φw, . . . , w|p 1 − |w|βn−1 dm2 w
×
!1/p
U
p
Cf Ap
βn−1
3.10
.
p
We used the fact that for all Φ ∈ Aβ , 1 < p
< ∞, SbΦ ∈ Aβn−1 proved before.
The complete analogue of Theorem 3.2 is true for Bergman classes on expanded disk
we defined previously.
Theorem 3.3. Let 0 ≤ p < ∞. If
→
−
a p ∈ 0, 1, β β1 , . . . , βn , βk > −1, k 1, . . . , n, |β| nk1 βk , or
→
−
b p > 1, β β, . . . , β, β > 0, β > 1/n − 1p/p
− 1, then the following assertion
p
n
holds. For every function f, f ∈ A→
− U∗ , Edfz fz, . . . , z, z ∈ U belongs to
β
p
p
A|β|n−1 and the reverse is also true, for any function f from A|β|n−1 there exists a function
g∈
p
n
A→
− U∗ β
such that gz, . . . , z fz, for all z ∈ U.
Journal of Inequalities and Applications
9
Proof. We give a short sketch of proof of Theorem 3.3 and omit details. Note that the half of
p
p
n
the theorem the inclusion EdA→
− U∗ ⊂ A|β|n−1 was proved in 3 and follows directly from
β
Theorem A.
For p ≤ 1 we have to use again Lemma 2.3 and Fubini’s theorem. For p > 1 we first
p
p
n
n
prove gw1 , . . . , wn ∈ A→
− U∗ if h ∈ A→
− U∗ and if |β|p/n > supk βk > mink βk > 1/n −
β
1p/p
− 1
gw1 , . . . , wn C
β
hz1 , . . . , zn nk1 1 − |zk ||β|/n d|zk |dξ
···
.
n
|β|1n/n
0
0
z
w
−
|1
|
k
k
k1
1
1
T
3.11
Indeed by Hölder’s inequality we have
⎛
⎞
1 1
p n
|β|p/n d|z |dξ
p
,
.
.
.
,
z
−
|
1
|z
|hz
|
1
n
k
k
k1
gw1 , . . . , wn C⎝
⎠
···
n
|β|1n/np−2εp2
0
T 0
z
w
−
|1
|
k
k
k1
×
1
···
0
T
Ah
1
0
p/p
d|zk |dξ
n
k1 |1
− zk wk |2εp
dξ
1εp
n T
1 − ξwk ϕ
3.12
p/p
k1
Ah
n
1
k1
1 − |wk |1εp −1/np/p ,
where ε > 0.
Hence calculating integrals we finally have gA→p− U∗n ChA→p− U∗n .
β
β
We used the estimate
1
T
0
···
1 n
0
n
− |wk |βk 1/n−1εp p/p d|wk |dξ
≤ C 1 − |zk |−|β|p/n βk ,
n
|1 − z w ||β|1n/np−2εp2
k1
k1 1
k1
k
3.13
k
which is true under the conditions on indexes we have in formulation of theorem and can
be obtained by using Hölder’s inequality for n functions. Using this projection theorem
and repeating arguments of proof of the previous theorem we will complete the proof of
Theorem 3.3.
10
Journal of Inequalities and Applications
Remark 3.4. Note that Theorems 3.2 and 3.3 are obvious for n 1.
Remark 3.5. Note that estimates between expanded disk, unit disk, and polydisk can be also
obtained directly from Liuville’s formula
1
···
1
0
ϕv1 · · · vn 0
n
p p p
p ···p
1 − vk pk −1 v2 1 v3 1 2 · · · vn1 n−1 dv1 · · · dvn
k1
n 1
n
Γ pk
k1
ϕu1 − u i1 pi −1 du,
Γ p1 · · · pn 0
pi > 0, ϕ is continuous function on 0, 1.
3.14
Remark 3.6. The complete description of traces of classes with fA∞α U∗n and fA∞α U n quasinorms on the unit disk can be obtained similarly by small modification of the proof
of Theorem 3.2,
p
f ∞
n
Aα U
1
p
M∞ · · · M∞ f, r 1 − rα dr,
α > −1, p ∈ 0, ∞,
0
p
f ∞ n Aα U∗ 1 1
0
p
M∞ f, r1 , . . . , rn 1 − r1 α · · · 1 − rn α dr1 · · · drn ,
α > −1, p ∈ 0, ∞
0
3.15
Let
Λα U∗n n
Λα U
f ∈ HU :
n
sup
rj ∈0,1,ξ∈T
n
fr1 ξ, . . . , rn ξ 1 − rk α < ∞ ,
f ∈ HU n
α > 0,
k1
sup
r∈0,1,ξi ∈T
α
frξ1 , . . . , rξn 1 − r < ∞ ,
3.16
α > 0.
n .
We formulate complete analogues of Theorems 3.2 and 3.3 for classes Λα U∗n and Λα U
n . Then Sbf fz, . . . , z is in Λα U and any
Theorem 3.7. Let α > 0 and f ∈ Λα U
n such that fz, . . . , z gz. The same statement is
g ∈ Λα U can be expanded to f, f ∈ Λα U
n
n
true for pairs Ed, Λα U∗ , EdΛα U∗ Λnα U.
Note that one part of statement is obvious. If, for example, f ∈ Λα U∗n , then Edf ∈
Λnα U. On the other side, let g ∈ Λnα U. Then define as above that
fr1 ξ, . . . , rn ξ Cβ
U
n
gz1 − |z|β
k1 1
− zrk ξβ2/n
dm2 z,
3.17
β is big enough, Cβ is a Bergman constant of Bergman representation formula.
Obviously frξ, . . . , rξ grξ, z rξ, z ∈ U. Using Hölder’s inequality for n
functions we get fΛα U∗n ≤ C1 gΛnα U . Similarly fΛα U n ≤ C2 gΛα U .
Journal of Inequalities and Applications
11
It is natural to question about discrete analogues of operators we considered
previously.
Let Ck > 0, rk 1, k → ∞, β ≥ 0. Let
Bn,Ck ,rk ,β ⎧
⎨
f ∈ HU∗n : fz1 , . . . , zn ⎩
n
∞
Ck
k1
j1
1
rk − zj
1β/n < ∞,
3.18
⎫
⎬
zj |z| ∈ 0, 1, zj ∈ U, j 1, . . . , n .
⎭
We have for such a function
1
0
1 − r
1α M∞ f, r
dr 1
0
1 − r1α
sup
fzdr
|z1 |r,...,|zn |r
1
∞
∞
∞
tβ−α−2
1 − r1α
α−β1
Ck
dr
C
−
1
dt.
r
k
k
1β
1β
0 rk − r
rk −1 t 1
k1
k1
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.
1
Proposition 3.8. Let f ∈ Bn,Ck ,rk ,β and α > −1. Then 0 1 − r1α M∞ f, rdr < ∞ if and only
∞
∞
β1
<
if ∞
k1 Ck < ∞ if 1 α > β;
k1 Ck ln 1/rk − 1 < ∞ if 1 α β;
k1 Ck /rk − 1
∞ if 1 α < β.
4. Sharp Embeddings for Analytic Spaces in Polydisk with
Rs Operators 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 from Lemma 2.3 which 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 in 2.
Later Clark studied such a map in 11.
In this section we also introduce the Rs differential operator as follows see
kn
k1
s
3, 12, 13. Rs f k1 ,...,kn ≥0 k1 · · · kn 1 ak1 ,...,kn z1 · · · zn , s ∈ R, where fz kn
k1
n
k1 ,...,kn ≥0 ak1 ,...,kn z1 · · · zn ∈ HU .
Note it is easy to check that Rs acts from HUn into HUn .
12
Journal of Inequalities and Applications
In the case of the unit ball an analogue of Rs operator is a well-known radial derivative
which is well studied. We note that in polydisk the following fractional derivative is well
studied see 1:
Dα f z k1 1α · · · kn 1α ak1 ,...,kn zk1 1 · · · zknn ,
k1 ,...,kn ≥0
4.1
where α ∈ R, f ∈ HUn , and D : HUn → HUn . Apparently the Rs operator was studied
in 12 for the first time. Then in 13, the second author studied some properties of this
operator. In this section we also continue to study the Rs operator. We need the following
simple but vital formula which can be checked by easy calculation:
fτξ1 , . . . , τξn Cs
1
R f τξ1 ρ, . . . , τξn ρ
s
0
1
log
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
n .
connection with functional spaces on subframe U
The following dyadic decomposition of subframe and polydisk was introduced in 1
and will be also used by us:
k,l1 × · · · × U
k,ln
k,l1 ,...,ln U
U
&
τξ1 , . . . , τξn : τ ∈
'
1
1
1 − k , 1 − k1 ,
2
2
πlj
π lj 1
k
k
k 0, 1, 2, . . . ;
< ξj ≤
, lj −2 , . . . , 2 − 1, j 1, . . . , n ,
2k
2k
πlj
π lj 1
−k
k,l1 ,...,ln 2−2kn ,
2
m Ik,lj m ξ ∈ T : k < ξj ≤
,
m
U
2n
2
2k
Uk1 ,...,kn ,l1 ,...,ln Uk1 ,l1 × · · · × Ukn ,ln
&
τ1 ξ1 , . . . , τn ξn : τj ∈
'
1
1
,
1
−
,
2kj
2kj 1
πlj
π lj 1
kj
kj
kj 0, 1, 2, . . . ;
< ξj ≤
, lj −2 , . . . , 2 − 1, j 1, . . . , n ,
2kj
2kj
4.3
1−
Mp f, r, 0 < p ≤ ∞ averages in analytic spaces in polydisk can obviously have a mixed
form, for example, Mp M∞ f, r1 , r2 , rj ∈ 0, 1, j 1, 2, p < ∞. In 8 Ren 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.
1 1
p,∞
Theorem 4.1. Let p ∈ 0, ∞ and Hα,n {f ∈ HUn : 0 · · · 0 T M∞ · · · M∞ f,
p,∞
p
rp dξ nk1 1 − rk α1/n−1 dr1 · · · drn < ∞, α > −1}. Then DiagHα,n Aα U.
Journal of Inequalities and Applications
13
Proof. Using diadic decomposition of polydisk we have
fz, . . . , zp 1 − |z|α dm2 z M
U
≤C
···
k1 ≥0
Uj,k
Uj,k
j,k
≤C
j,k
p
maxfz, . . . , z 2−2k 2−αk
fz, . . . , zp 1 − |z|α dm2 z
−1
min kj
2
kn ≥0 j−2min kj
···
k1 ≥0
−1
min kj
2
p
max fz1 , . . . , zn 2−α2k1 /n · · · 2−α2kn /n
1−2−k1 2
1−2−k1 −1
kn ≥0 j−2min kj
···
1−2−kn 2
1−2−kn −1
n
fz1 , . . . , zn p dr1 · · · drn dξ 2k1 τ · · · 2kn τ 2min kj ,
×
4.4
Uk1 ,...,kn ,j
τ −
Ij,k1 ,...,kn
α2
1.
n
We used above the following estimate which can be found, for example, in 1
n
p fz1 , . . . , zn p dm2n z, 0 < p < ∞,
sup fz1 , . . . , zn 2min kj 2k1 ···kn
Uk∗
z∈Uk1 ,...,kn ,j
1 ,...,kn ,j
4.5
where Uk∗ 1 ,...,kn ,j are enlarged dyadic cubes see 1 and 0 < p < ∞, f ∈ HUn , and
π j 1
,
ξ : z rξ ∈ Uk1 ,...,kn ,j , min k ≤ ξ < min k
j
j
2
2
π j − 1/2
π j 1/2
∗
.
ξ : z rξ ∈ Uk1 ,...,kn ,j ,
≤ξ<
2min kj
2min kj
Ij,k1 ,...,kn
∗
Ij,k
1 ,...,kn
πj
4.6
Note further since mn Ij,k1 ,...,kn 2−n min kj ,
∗
Ij,k
fz1 , . . . , zn p dξ C
1 ,...,kn
∗
Ij,k
1 ,...,kn
p M∞ · · · M∞ f, r dξ 2−n−1 min kj ,
1
Ij,k
1 ,...,kn
× ··· ×
1
Ij,k
1 ,...,kn
n
Ij,k
,
1 ,...,kn
4.7
0 < p < ∞.
We have
MC
k1 ≥0
···
kn ≥0
−1 1−2−k1 2
min kj
2
min kj
j−2
M∞ · · · M∞ f, r
p
1−2−k1 −1
···
1−2−kn 2 1−2−kn −1
1
Ij,k
1 ,...,kn
dξ2−k1 α1/n−1 · · · 2−kn α1/n−1/n.
We used the fact that 2min kj < 2k1 ···kn /n .
4.8
14
Journal of Inequalities and Applications
1
is a finite covering of T , finally we have for all 0 <
Hence using the fact that Ij,k
1 ,...,kn
p,∞ . One part of theorem is proved.
p < ∞ M CfHα,n
To get the reverse statement we use the estimate from Lemma 2.3. Then we have
α
fz1
−
dm
z
|z|
2
|Fz1 , . . . , zn | C α2/np ,
n
U j1 1 − zzj
zj ∈ U,
4.9
for any α > −1. Hence Fz, . . . , z fz and for p < 1 by Lemma 2.3
p,∞
FHα,n
×
1 1
0
0
p
fz 1 − |z|αp2p−2 dm2 z
M∞ · · · M∞ α2/np
n T
U
j1 1 − zzj
n
1 − rk α1/n−1 dr1 · · · drn dξ1 · · · dξn
k1
fzp 1 − |z|αp2p−2
0
U
n
1 1
0
n−1
k1 1
k1 1
− rk α1/n−1 dr1 · · · drn
− |z|rk α2/np 1 − rn |z|α2/np−1
dm2 z
fzp 1 − |z|αp2p−2 1 − |z|−α2/n pα1/nn−1
U
×1 − |z|−α2/npα1/n1 dm2 z
fzp 1 − |z|α dm2 z, α > −1, 0 < p ≤ 1, f ∈ Apα .
U
4.10
Let 1 < p < ∞. Then we may assume that again s is large enough. Let 1/p 1/q 1, γi > 0, i 1, 2, γ1 γ2 s 1/n. Then
⎛ ⎞p
s−1
fw 1 − |w|2
dm2 w
⎜
⎟
⎝
⎠
s1/n
n U
j1 1 − zj w
s−1
fwp 1 − |w|2
n
dm2 w γ p−s1/n
≤C
.
1 − zj 1
γ 1 p
n U
j1
j1 1 − zj w
4.11
We used estimate
⎛
⎜
⎝
⎞p/q
s−1
n
1 − |w|2
dm2 w
γ p−s1/n
⎟
≤C
.
1 − zj 1
γ 2 q ⎠
n j1
j1 1 − zj w
U
4.12
Journal of Inequalities and Applications
15
Choosing appropriate γ1 , γ2 we repeat now arguments that we presented for p ≤ 1 above to
get what we need. The proof is complete.
Remark 4.2. The case of k, 1 < k ≤ n, Mp averages can be considered similarly. Note in
Theorem 4.1k 1. Thus our Theorem 4.1 extends known description of diagonal of classical
Bergman classes see 1, 7.
In 14 Carleson 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
n
U
fzp dμz
!1/p
≤ CRs f X ,
s ≥ 0,
4.13
n and X is a Bergman class on polydisk or subframe.
where μ is a positive Borel measure on U
Theorem 4.3. Let γ1 β1 s − 1p n 1p − 1, γ1 > 0, β1 > −1, 0 < p ≤ 1, s > β1 n . If
1/n n − pn/p, s, n ∈ N, and μ is a positive Borel measure on U
n
U
fzp 1 − |z|t dμzRs f n
Aβ U
p
,
t > −1,
4.14
1
then
μ n sup
U
n
w∈
U
1
1 − |w|t dμw
γ1 < ∞,
n
p 1 − |w|
w
−
w
k k|
0
k1 |1
4.15
1 − |w|t dμw
γ1 ε < ∞
n
pε 1 − |w|
w
−
w
|1
|
0
k k
k1
4.16
Tn
and conversely if
μ n sup
U
n
w∈
U
1
Tn
n , |w|
∈ 0, 1, ξj ∈ T, j 1, . . . , n then
holds for some ε > 0 where w
|w|ξ
1 , . . . , |w|ξ
n
U
fzp 1 − |z|t dμzRs f n
Aβ U
p
,
t > −1.
4.17
1
Remark 4.4. With another “ε-sharp” embedding theorem,
the complete analogue of
Theorem 4.3 is true also when we replace the left side by Un |fz|p nk1 1 − |zk |tk dμz, tk >
−1, k 1, . . . , n. The proof needs small modification of arguments we present in the proof of
Theorem 4.3.
Remark 4.5. For n 1 and s 1 Theorem 4.3 is known see 15.
16
Journal of Inequalities and Applications
Proof of Theorem 4.3. It can be checked by direct calculations based on formula 4.2 that the
following integral representation holds:
fz1 , . . . , zn Cs
1
Tn
R f ρξ1 , . . . , ρξn
s
1
log
ρ
0
!s−1
dρdmn ξ
,
n k1 1 − ρϕk rξk
4.18
n . Using the fact that Mp f, r in
s > 0, zk ϕk r, k 1, . . . , n, r ∈ 0, 1, z1 , . . . , zn ∈ U
n1−1/p
2
increasing by r, r ∈ 0, 1 and M1 f, τ ≤ C1 − τ
Mp f, τ, τ ∈ 0, 1 see 1 we
get from above by using the estimate
1
α
f ρ 1 − ρ dρ
p
1
0
p αpp−1
dρ,
f ρ
1−ρ
4.19
0
where f is growing measurable, p ≤ 1, α > −1,
Un
fwp 1 − |w|t dμw ≤ Cμ n
U
1
0
Tn
s
R fwp 1 − |w|β1 dmn ξd|w|.
To obtain the reverse implication we use standard test function gz w 1/
n , γ > 1 and Lemma 2.5
zk , wk ∈ U, k 1, . . . , n, z, w ∈ U
s
R gz w p
A
β1
1
n
U
1 − |w|β1
0
αj ≥0,
n
k1 1
Gz, wd|w|,
αj s
4.20
− wk zk γ ,
4.21
where Gz, w nk1 1/1 − |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
p
p n
,
RAs,v f ∈ HUn : Rs f ∈ Aν U
p
p
n
n
γ
DA→
−γ ,α f ∈ HU : D f ∈ Aα U ,
γ Dzγ1 · · · Dzγn ,
D
1
n
n
γj γ, γj ≥ 0.
4.22
j1
Let us note that Theorems 3.2 and 3.7 of 3 show that under some restrictions on γ, α, s, ν
the following assertion is true.
p
For every function f, f ∈ DAγ,α , 0 < p < ∞, f|z|ξ1 , . . . , |z|ξn belongs to
p
p
RAs,v , 0 < p < ∞, and the reverse is also true, for any function f, f ∈ RAs,v there exists
“an extension” F such that
F|z|ξ1 , . . . , |z|ξn f|z|ξ1 , . . . , |z|ξn ,
p
F ∈ DAγ,α .
4.23
Journal of Inequalities and Applications
17
The following Theorem 4.6 gives an answer for the same map from polydisk to expanded
disk
ξ ∈ T, zj ∈ 0, 1, j 1, . . . , n,
fz1 , . . . , zn −→ f|z1 |ξ, . . . , |zn |ξ,
4.24
for f functions from H p Un , 1 < p < ∞.
We will develop ideas from 4 to get the following sharp embedding theorem for
classes on expanded disk.
Theorem 4.6. Let n > 1, 1 < p < ∞, μ be a positive Borel measure on U∗n . Then
1
···
0
if and only if
1
1−l1
···
1
0
1
− fr1 ξ, . . . , rn ξp dμ →
r ξ ≤ f minj lj /2
−r ξ
dμ→
1−ln −minj lj /2
Proof. Obviously if K H p Un T
4.25
≤ Cl1 · · · ln , lj > 0, j 1, . . . , n.
p
U∗n
|fz|p dμz ≤ CfH p , then by putting
⎞1/p
⎛
n
⎜ 1 − lj ⎟
f
fj , fj z ⎝ 2 ⎠ ,
j1
1 − zj lj
lj 1 − lj , lj ∈ 0, 1, j 1, . . . , n,
4.26
1
1 τ
we have fH p const and K ≥ 1/l1 · · · ln 1−l1 · · · 1−ln −τ dμr1 ξ, . . . , rn ξ, τ minj lj , |1 −
zj lj | 1 − |zj |, |zj | ∈ 1 − lj , 1, argzj ∈ −τ, τ. Hence we get what we need. Now we will
show the sufficiency of the condition.
Let N > 0 : 2N 1 − rj ≤ π < 2N1 1 − rj , j 1, . . . , n.
−r eiϕ ∈ Un , z r eiϕ , . . . , r eiϕ . Let also Bk {t t , . . . , t : |t − ϕ| <
Let z →
1
n
1
n
j
∗
z
2kj 1 − rj , 1 ≤ j ≤ n} and Wk1 ,...,kn Ik1 × · · · × Ikn , where
Ikj t : 2kj 1 − rj ≤ t − ϕ < 2kj 1 1 − rj ,
0 ≤ kj ≤ N − 1,
IN t : 2N 1 − rj ≤ t − ϕ < 2N1 1 − rj , |t| < π , I−1 t : |t| < 1 − rj .
4.27
In what follows we will use notations
of Proposition 2.1. Consider the Poisson integral of a
function f, f ∈ H 1 T , uz C T n P z, ξfξdmn ξ, z ∈ Un see 2.
Let Ea {z ∈ U∗n : Eduz > a}. We will show now as in 4 that
C
μEa ≤
a
Tn
ftdmn t.
4.28
18
Journal of Inequalities and Applications
Indeed, this will be enough, since the operator f → Ed T n P z, ξfξdmn ξ is L∞ , L∞ operator we can apply the Marcinkiewicz interpolation theorem see 16, Chapter 1 to assert
that
1
J
···
1
0
0
fzp dμz ≤ Cf H p Un T
4.29
1 < p < ∞.
,
We have as in 4 for z ∈ U∗n
Eduz Tn
≤ C0
≤
N
P z, wfwdmn w P z, wfwdmn w
k1 ,...,kn −1
N
k1 ···kn
k1 ,...,kn 0 4
N
k1 ,...,kn 0
Wk1 ,...,kn
1
n
k1 1 − rk Bzk
ftdt ≤ C
N
1
g r , . . . , rn , ϕ
|k| k 1
2
k1 ,...,kn 0
− − 2−|k| gk →
r , ϕ ≤ C1 supk 2|k|/2 gk →
r,ϕ ,
4.30
where we used the standard partition of Poisson integral.
Hence Ea ⊂
and so using Proposition 2.1 we finally get μEa ≤ C/a T n ftdt.
Indeed by Proposition 2.1 we have
μEa ≤
α∈Zn
Eα C2 2|α|/2 a,
μ Eα C2 2|α|/2 a
α∈Zn
C2−1 C4
ftdt ≤
≤ C3
|α|/2 a
a
n
n2
T
α∈Z
4.31
ftdt.
Tn
Theorem 4.6 is proved.
Theorem 4.7. Let f ∈ HUn . Then
n
sup Dα fz1 , . . . , zn 1 − |zk |α1
k1
|zj |<1
1
s R f ρξ1 , . . . , ρξn p 1 − ρ spnp−1−1 dmn ξdρ,
≤C
Tn
4.32
0
where s > 0, α > 0, p ≤ 1, sp np − 1 > 0;
⎛
sup Dβ fz1 , . . . , zn ⎝
z∈Un
αk ≥0,
⎞−1
n
αk s k1
1
1 − |zk |
βαk −2s−α/n
⎠
≤ C sup Rs fz1 − |z|α ,
n
z∈U
4.33
Journal of Inequalities and Applications
19
where 2s − α > 0, β > 1, α > 0, s > 0,
sup
|zk |<1 T n
≤C
s R f |z1 |ϕ1 , . . . , |zn |ϕn p dmn ϕ
αj ≥0,
1
Tn
n
αj 2s
k1
−1
1
1 − |zk |pαk s1−1
4.34
−s
D fwp 1 − |w|t d|w|dmn ϕ ,
0
where s > 0, 1/s 1 < p ≤ 1, t ps − 1 n 1p − 1 > −1.
Proof. We use systematically the integral representation 4.18, Lemma 2.5, and it is corollary.
The proof of the estimate 4.32 follows from equality
Dα fz1 , . . . , zn Cs
1
1
Rs f ρξ1 , . . . , ρξn log
ρ
0
Tn
!s−1 n
k1
1
1 − ξϕk ρk
α1 dρdmn ξ,
4.35
zk ϕk ρk , k 1, . . . , n, s > 0, α > 0, f ∈ HUn , and estimate
1
Tn
≤C
s R f ρξ1 , . . . , ρξn 1 − ρ α dρdmn ξ
p
0
1
Tn
s R f ρξ1 , . . . , ρξn p 1 − ρ αpn1p−1 dρdmn ξ,
4.36
0
p ≤ 1, α > 0, s > 0, f ∈ HUn , obtained during the proof of Theorem 4.3.
The proof of the estimate 4.33 follows from Lemma 2.5 and its corollary and integral
representation 4.35.
The proof of the estimate 4.34 follows from equality zj ρj ϕj , j 1, . . . , n
s
R fz1 , . . . , zn !s−1
1
n
−s 1
1
2s
Cs
R
dρdm
D f ρξ1 , . . . , ρξn log
ξ
.
n
s1
ρ
n
0
T
k1 1 − ξϕk ρ
ρk
4.37
Indeed using 4.36 integrating both sides of 4.37 by T n and using Lemma 2.5 we arrive at
4.34.
20
Journal of Inequalities and Applications
Remark 4.8. All estimates in Theorem 4.7 for n 1 are well known see 1, 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 of f function is done again from diagonal
z, . . . , z. Let zj ∈ U, j 1, . . . , n and
fz1 , . . . , zn Cs
fz, . . . , z1 − |z|s
dm2 z,
n
s2/n
U
k1 1 − zk z
4.38
where Cs is a constant of Bergman representation formula see 1.
Proposition 4.9. 1 a Let n ∈ N, p ∈ 0, ∞, α > −1, f ∈ HUn . Then
1
···
1
0
T
n
f|z1 |ξ, . . . , |zn |ξp
1 − |zk |αn−1/n d|z1 | · · · d|zn |dmξ
0
≤C
Un
k1
n α
fzp
1 − |zk |2 dm2n z.
4.39
k1
And reverse is also true:
b Let 0 < p < ∞, α > −1 and
1
···
1
0
T
n
f|z1 |ξ, . . . , |zn |ξp
1 − |zk |αn−1/n d|z1 | · · · d|zn |dmξ < ∞,
0
4.40
k1
then for all functions f such that condition 4.38 holds for s > maxα 2n/p − 2, −1 we have
p
f ∈ Aα Un .
2 a Let f ∈ H p Un , 1 < p < ∞, n > 1. Then
f p
A
U∗n −1/n
1 1
T
0
0
n
f|z1 |ξ, . . . , |zn |ξp
1 − |zk |−1/n dmξd|z1 | · · · d|zm | < ∞,
k1
4.41
and the reverse is also true:
b For any function f with a finite quasinorm fAp U∗n such that condition 4.38 holds
−1/n
for s 2n − 2 one has f ∈ H p Un , p > 1.
Proof. 1 a Proof of estimate 4.39 follows directly from Theorem B.
b Indeed from 4.38 and results of 1 on diagonal map in Bergman classes we have
fApα Un ≤ CDiagfAp U , t αn 2n − 2. It remains to apply Theorem A.
t
2 For the proof of a we use Theorem 4.6 and get the result we need.
For the proof of b we use the same argument as in the proof of part 1. Namely first
from 4.38 and from a Diagonal map theorem on H p classes from 1 we get fH p Un ≤
CfAp U . It remains to apply Theorem A.
n−2
Journal of Inequalities and Applications
21
Remark 4.10. Note Proposition 4.9 is obvious for n 1.
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. a Let α > 0, n ∈ N, n > 1, s > nα then the following assertions are true: If
Dα f ∈ H p Un then
Jp,α,s f 1
Tn
s
R f|z|ξ1 , . . . , |z|ξn p 1 − |z|sp−npα−1 d|z|dmn ξ < ∞, p ≥ 2.
4.42
0
If Jp,α,s f < ∞ then Dα f ∈ H p Un , 1/α 1 < p ≤ 1.
b Let Dα f ∈ H p Un , p ≥ 2. If s > nα, α > 0, then Jp,α,s f < ∞. Moreover the reverse is
p
also true if condition 4.38 holds for s 2n − 2 then Dα fH p ≤ CJp,α,s f.
The proof of first part of Proposition 4.11 follows from 4.35, 4.36 and Lemma 2.5
directly. The reverse assertion follows from Theorem B b and estimate
Un
n
α ,...,α
D 1 n fzp
1 − |zk |αk p−1 dm2n z ≤ Cf Hp
, αk > 0, k 1, . . . , n, p ≥ 2,
4.43
k1
which can obtained from one dimensional result by induction.
The proof of second part of Proposition 4.11 can be obtained from Theorem A and
results on diagonal map on Hardy classes H p Un 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 from a
since for n 1 condition 4.38 vanishes by Bergman representation formula.
Remark 4.12. Theorem 4.6, Propositions 4.9 and 4.11 give an answer to a problem of Rudin
see 2 to find traces of H p Un Hardy classes on subvarieties other than diagonal z, . . . , z.
Note that in 11 Clark solved this problem for subvarietes of Un based on finite Blaschke
products.
Acknowledgment
The authors sincerely thank Trieu Le for valuable discussions.
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p
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