On some properties of a differential operator on the polydisk

Banach J. Math. Anal. 3 (2009), no. 1, 68–84
Banach Journal <strong>of</strong> Mathematical Analysis
ISSN: 1735-8787 (electronic)
http://www.math-analysis.org
ON SOME PROPERTIES OF A DIFFERENTIAL OPERATOR
ON THE POLYDISK
ROMI SHAMOYAN 1∗ AND SONGXIAO LI 2
Communicated by J. M. Isidro
Abstract. We study the action and <strong>properties</strong> <strong>of</strong> a <strong>differential</strong> <strong>operator</strong> in
the polydisk, extending <strong>some</strong> classical results from the unit disk. Using so
called dyadic decomposition <strong>of</strong> the polydisk we find precise connections between
quazinorms <strong>of</strong> holomorphic function in the polydisk with quazinorms on the
subframe and the unit disk. All our results were previously well-known in the
unit disk.
1. Introduction and preliminaries
Let n ∈ N and C n = {z = (z 1 , ..., z n ) |z k ∈ C, 1 ≤ k ≤ n} be the n-dimensional
space <strong>of</strong> complex coordinates. We denote the unit polydisk by
U n = {z ∈ C n : |z k | < 1, 1 ≤ k ≤ n}
and the distinguished boundary <strong>of</strong> U n by
T n = {z ∈ C n : |z k | = 1, 1 ≤ k ≤ n}.
We use m 2n to denote the volume measure on U n and m n to denote the normalized
Lebesgue measure on T n . Let H(U n ) be the space <strong>of</strong> all holomorphic functions
on U n . When n = 1, we simply denote U 1 by U, T 1 by T , m 2n by m 2 , m n by m.
We refer to [15] for further details.
Date: Received: 29 July 2008; Accepted: 31 August 2008.
∗ Corresponding author.
2000 Mathematics Subject Classification. Primary 32A18; secondary 32A36.
Key words and phrases. Differentiation <strong>operator</strong>, weighted Bergman space, Hardy space,
polydisk.
The second author <strong>of</strong> this paper is supported by the NSF <strong>of</strong> Guangdong Province (No.
7300614).
68

PROPERTIES OF DIFFERENTIAL OPERATOR ON THE POLYDISK 69
The Hardy spaces, denoted by H p (U n )(0 < p ≤ ∞), are defined by
H p (U n ) = {f ∈ H(U n ) :
sup M p (f, r) < ∞},
0≤r −1, j = 1, · · · , n, 0 < p < ∞, recall that the weighted Bergman
space A p ⃗α (U n ) consists <strong>of</strong> all holomorphic functions on the polydisk satisfying the
condition
∫
∏ n
‖f‖ p = |f(z)| p (1 − |z
A p i | 2 ) α i
dm 2n (z) < ∞.
⃗α
U n
i=1
Throughout this paper, constants are denoted by C, C α , or C(α), they are
positive and may differ from one occurrence to other. The notation A ≍ B
means that there is a positive constant C such that B/C ≤ A ≤ CB.
Let z = (z 1 , . . . , z n ) ∈ U n , f j (z) ∈ H(U n ), j = 1, · · · , n. It is easy to see that
if
f j (z) =
∑
a (j)
k 1 ,...,k n
z k 1
1 · · · zn kn
, j = 1, · · · n
k 1 ,··· ,k n≥0
is a usual Taylor expansion in U n <strong>of</strong> f j , then
S(f 1 , · · · , f n ) = f 1 + · · · f n =
∑ ( n∑ )
a (j)
k 1 ,...,k n
z k 1
1 · · · zn kn
.
We consider a very particular case when
where a k1 ,...,k n
k 1 ,...,k n≥0
j=1
a (j)
k 1 ,...,k n
= k j a k1 ,...,k n
, j = 1, · · ·, n,
is a certain sequence. We have
S(f 1 , · · · , f n ) =
∑
k 1 ,...,k n≥0
(k 1 + · · · k n )a k1 ,...,k n
z k 1
1 · · · z kn
Motivated by the above expression we define a <strong>operator</strong> in the polydisk as
follows
Rf =
∑
(k 1 + · · · + k n + 1)a k1 ,...,k n
z k 1
1 · · · z kn
or more general form
R s f =
∑
where
It is easy to see that
k 1 ,...,k n≥0
k 1 ,...,k n≥0
f(z) =
n .
(k 1 + · · · + k n + 1) s a k1 ,...,k n
z k 1
1 · · · zn kn
, s ∈ R,
∑
k 1 ,··· ,k n≥0
Rf = R 1 f = f +
a k1 ,...,k n
z k 1
1 · · · z kn
n ∈ H(U n ).
n∑
j=1
n
z j
∂f
∂z j
. (1.1)

70 R. SHAMOYAN, S. LI
In the case <strong>of</strong> the unit ball an analogue <strong>of</strong> R s <strong>operator</strong> is a well known radial
derivative which is well studied (see [17]). We note that in polydisk the following
fractional derivative is well studied (see [4, 11]),
(D α f)(z) = ∑
(k 1 + 1) α · · · (k n + 1) α a k1 ,...,k n
z k 1
1 z kn
k 1 ,...,k n≥0
where α ∈ R, f ∈ H(U n ) and D α : H(U n ) → H(U n ). We also note that in
polydisk the following derivative was studied in [16],
n∏ (
∂
)
D = 2 + z k .
∂z k
k=1
Apparently the R s <strong>operator</strong> was studied in [6] for the first time. Then in [12],
the first author studied <strong>some</strong> <strong>properties</strong> <strong>of</strong> this <strong>operator</strong>. The aim <strong>of</strong> this paper
continue to study the R s <strong>operator</strong>.
We need the following vital formula which can be checked by easy calculation
∫ 1
f(τξ 1 , · · ·, τξ n ) = C s R s f(τξ 1 ρ, · · ·, τξ n ρ)(log 1 ρ )s−1 dρ, (1.2)
0
where s > 0, τ ∈ (0, 1), C s > 0, ξ j ∈ T, j = 1, . . . , n. The integral representation
<strong>of</strong> functions via these <strong>operator</strong>s will allow us to consider them in U n in close
connection with functional spaces on subframe
Ũ n = {z ∈ U n , |z j | = r, r ∈ (0, 1], j = 1, . . . , n}.
The following dyadic decomposition <strong>of</strong> subframe and polydisk were introduced
in [4] and will be used by us.
Ũ k,l1 ,··· ,l n
= Ũk,l 1
× · · ·Ũk,l n
= { (τξ 1 , . . . , τξ n ) : τ ∈ (1 − 1 2 , 1 − 1 ],
k 2k+1 πl j
k = 0, 1, 2, · · ·;
2 < ξ k j ≤ π(l j + 1)
, l
2 k j = −2 k , · · · , 2 k − 1, j = 1, · · · , n } ,
m(Ĩk,l j
) = m(ξ ∈ T : πl j
2 k < ξ j ≤ π(l j + 1)
2 k ) ≍ 2 −k , m 2n (Ũk,l 1 ,··· ,l n
) ≍ 2 −2kn ,
U k1 ,··· ,k n,l 1 ,··· ,l n
= U k1 ,l 1
× · · ·U kn,l n
= { (τ 1 ξ 1 , · · · , τ n ξ n ),
τ j ∈ (1 − 1
2 , 1 − 1
k j
2 ], k k j+1 j = 0, 1, · · · , j = 1, 2, · · · , n,
ξ j ∈ ( πl j
2 , π(l j + 1)
], l k j
2 k j j = −2 k j
, · · · , 2 k j
− 1, j = 1, · · · , n, } .
The goal <strong>of</strong> this paper is to extend <strong>some</strong> known assertions connected with fractional
derivative in the unit disk to the polydisk and use the diadic decomposition
<strong>of</strong> subframe and polydisk to study the action and <strong>properties</strong> <strong>of</strong> R s <strong>operator</strong> and
quasinorm connected with them on subframe. In section 2 we give preliminaries,
several useful inequalities for the study <strong>of</strong> R s <strong>operator</strong>s, and show connections
between R s and D s <strong>operator</strong>s in the polydisk. In section 3, we using the R s
<strong>operator</strong> establish <strong>some</strong> embedding theorems extending <strong>some</strong> known embeddings
for Hardy classes and weighted Bergman classes in the unit disk. In section 3 we
n ,

PROPERTIES OF DIFFERENTIAL OPERATOR ON THE POLYDISK 71
also establish connections between holomorphic spaces with quasinorms in the
subframe and polydisk.
2. Some observations concerning R s and D s <strong>differential</strong> <strong>operator</strong>s
and pro<strong>of</strong>s <strong>of</strong> preliminaries
When we look at R s <strong>operator</strong>s then we have the following natural problem:
Is it possible to reduce the study <strong>of</strong> R s <strong>operator</strong>s to the study <strong>of</strong> D s <strong>operator</strong>s?
which was studied by many authors (see for example [4] and references there).
Then we will be able to use known <strong>properties</strong> <strong>of</strong> D s to get new results for R s
<strong>operator</strong>s. The <strong>differential</strong> <strong>operator</strong> D s is much more convenient at least because
<strong>of</strong> the following property <strong>of</strong> g function. Let
We have
and
R s g(z) =
g(z) = 1
1 − z = ∑
∑
k 1 ,··· ,k n≥0
k 1 ,··· ,k n≥0
z k 1
1 · · · z kn
n .
(k 1 + · · · + k n + 1) s z k 1
1 · · · z kn
D s g(z) = ∑ 1 + 1)
k 1 ≥0(k s z k 1
1 · · · ∑
(k n + 1) s z kn
For D s we reduce things to one dimensional <strong>differential</strong> <strong>operator</strong>s for one function
in the unit disk, that why we will find ways to reduce the study <strong>of</strong> R s to D s .
Let
˜R s f(z) =
∑
k 1 ,··· ,k n≥0
k n≥0
n
n .
(k 1 + · · · + k n ) s a k1 ,··· ,k n
z k 1
1 · · · z kn
The following lemma is playing an important role in the study <strong>of</strong> R s <strong>operator</strong>. We
use <strong>some</strong> known facts in the pro<strong>of</strong> <strong>of</strong> Lemma 2.1 about action <strong>of</strong> one dimensional
D α <strong>operator</strong>, for example the following estimate (see [2])
∫
∫
|D α g(τξ)| p dξ ≤ C |D β g(τξ)| p dξ (2.1)
T
which can be transferred by induction to polydisk.
T
n .
(τ ∈ (0, 1), 0 < p < ∞, α ≤ β, g ∈ H(U))
Lemma 2.1. Let w = |w|ξ, w, z, ∈ U n , 1 − wz = ∏ n
k=1 (1 − w kz k ), s ∈ {0} ∪ N,
β > 0, p ∈ (0, ∞). Then we have
∫ ∣ ∣∣R s 1
∣ ∣∣
p
dmn (ξ)
T (1 − ξ|w|z) n β
≤
C ∑ α j ≥0
P αj =s
( n
∏
k=1
1
(1 − |w k ||z k |) p(α k+β)−1
)
, p >
1
min k α k + β .

72 R. SHAMOYAN, S. LI
Pro<strong>of</strong>. Let first β = 1. Since
(k 1 + · · · k n ) s = ∑ α j ≥0
P αj =s
C α k α 1
1 · · · k αn
n , k i ∈ N, i = 1, · · · , n,
and
n
1
1 − w = ∏ 1
,
1 − w k
k=1
k α i
i = (k i + 1 − 1) α i
=
1
1 − w = ∑ k i ≥0
i=1,··· ,n
∑α i
l=0
where C α = C(α 1 , · · · α n ) = s!
α 1 !···α n!
, we have
˜R s 1
1 − w = ∑ k i ≥0
(k 1 + · · · + k n ) s w k
i=1,··· ,n
= ∑ k i ≥0
(k 1 + · · · + k n ) s w k 1
1 · · · wn
kn
i=1,··· ,n
= ∑ k i ≥0
i=1,··· ,n
= ∑ k i ≥0
i=1,··· ,n
∑
α j ≥0
P αj =s
∑
α j ≥0
P αj =s
C α k α 1
1 · · · kn αn
w k 1
1 · · · wn
kn
C α
∏ n
∑α t
t=1
l=0
C l α i
(k i + 1) l (−1) α i−l ,
w k 1
1 · · · w kn
n , |w j | < 1, k j ∈ N, j = 1, · · · , n,
C l α t
(k t + 1) l (−1) αt−l w k 1
1 · · · w kn
n , |w j | < 1, j = 1, · · · , n.
Therefore using polydisk version <strong>of</strong> (2.1) we have
∫ ∣ ∣∣ 1
J = ˜Rs ∣ p dm n (ξ)
T (1 − ξ|w|z)
n
≤ C ∑ ∫ ∣ ∣ ∣∣∣ ∑
C α (k 1 + 1) α1 · · · (k n + 1) αn k1 k ˜w 1 · · · ˜w
n
∣∣∣
n
pdm n (ξ)
α j ≥0 T
P n k i ≥0
αj =s
i=1,··· ,n
∫
∣ ∑
≤ C α
∣ (k 1 + 1) α1 k
˜w 1
∣∣∣
1
pdm(ξ 1 ) · · ·
C ∑ α j ≥0
P αj =s
T
∣
k 1 ≥0
∫
×
T
∑
∣ ∣ (k n + 1) αn k ˜w
n
∣∣∣
n
pdm(ξ n ),
where ˜w j = ξ j |w j ||z j |, j = 1, · · · , n. Hence by the estimate
∫
dm(η)
|1 − τη| ≤ C(γ) , γ > 1, τ ∈ (0, 1), (2.2)
γ (1 − τ)
γ−1
T
k n≥0

we finally have
J ≤ C ∑ α j ≥0
P αj =s
≤
≤
PROPERTIES OF DIFFERENTIAL OPERATOR ON THE POLYDISK 73
C ∑ α j ≥0
P αj =s
C ∑ α j ≥0
P αj =s
C α
∫
T
(
∏ n ∫
n∏
k=1
k=1
p ∫
∣ Dα 1
g 1 (|w 1 ||z 1 |ξ 1 )
∣ dξ 1 · · · ×
T
)
dm(ξ k )
|1 − ˜w k | p(α k+1)
1
(1 − |w k ||z k |) p(α k+1)−1 ,
T
p
∣ Dαn g n (|w n ||z n |ξ n )
∣ dξ n
where p > 1/(min k α k + 1), |w k | ∈ (0, 1), |z k | ∈ (0, 1), k = 1, · · · , n, and
D αs g s (z) = ∑ k≥0(k + 1) αs z k , z ∈ U, s = 1, · · · , n.
The case <strong>of</strong> β ∈ (0, ∞) needs small modification since
1
(1 − z) = ∑ C β β k zk , C β k ≍ (k + 1)β−1 , β > 0.
k≥0
Lemma 2.1 is proved since
∫
∫
|R s f(τξ)| p dm n (ξ) ≤ C | ˜R s f(τξ)| p dm n (ξ),
T n T n
for f(z) = ∏ n 1
k=1 (1−z k
, α > 0, s ≥ 0, p ∈ (0, ∞), which follows from equality
) α s∑
(k 1 + · · · + k n + 1) s = Cs(k j 1 + · · · + k n ) j
and <strong>some</strong> calculations similar to those that we used above.
j=0
Corollary 2.2. Let 0 < p < ∞, s ∈ N ∪ {0}, l ∈ (0, ∞), γ > 1/p + l, w ∈ U n .
Then
∫
|R s 1
∑ n∏
U (1 − wz) n γ |p (1 − |z|) pl−1 C
dm 2n (z) ≤
(1 − |w k |) .
(α k+γ)p−pl−1
α j ≥0, P α j =s k=1
Pro<strong>of</strong>. The result follows directly from Lemma 2.1 and the following estimate (see
[13]),
∫ 1
0
(1 − ρτ) −λ (1 − τ) α dτ ≤ C(1 − ρ) −λ+α+1 , λ > α + 1, α > −1, ρ ∈ (0, 1).(2.3)
Remark 2.3. Our estimates in Lemma 2.1 and Corollary 2.2 coincide with well
known estimates in the unit disk for n = 1, they also known in ball, see for
example [8, 17].
For the pro<strong>of</strong> <strong>of</strong> our main results <strong>some</strong> additional lemmas will be needed.
□
□

74 R. SHAMOYAN, S. LI
Lemma 2.4. Let 1 ≤ p ≤ q ≤ ∞, p ′ =
on (0, 1). Then
(
∫ 1
0
( ∫ t ) ) qdt 1/q ( ∫ 1
u(t) ϕ(r)dr ≤ K 1
for <strong>some</strong> constant K 1 if and only if
sup(
t
0
∫ 1
t
u(τ)dτ) p/q (
∫ t
p , u(t), v(t), ϕ(t) are positive functions
p−1
0
0
) 1/p
ϕ p (t)v(t)dt
v(τ) 1−p′ dτ) p−1 < ∞.
Remark 2.5. These are well known so-called Hardy-type inequalities, see [9].
Lemma 2.6. Let γ > 1. Then
∫
T n 1
|1 − wz| γ dm n(ξ) ≤
C γ
(1 − |w||z|) γ−1 ,
where w = ξ|w|, z, w ∈ U n , 1 − wz = ∏ n
k=1 (1 − w kz k ).
Pro<strong>of</strong>. The pro<strong>of</strong> easily follows by (2.2), we omit the details.
□
We want to get the analogue <strong>of</strong> Bergman representation formula for the R s
<strong>operator</strong> (see [4]). Using (1.2) we get
f(τz 1 , · · · , τz n ) = f τ (z 1 , · · · , z n )
∫
f τ (ξ 1 , · · · , ξ n )
= C
T n (1 − ξ 1 z 1 ) · · · (1 − ξ n z n ) dm n(ξ)
∫ ∫ 1
= C s R s f(τξ 1 ρ, · · · , τξ n ρ)(log 1 dm n (ξ)
T n 0
ρ )s−1 dρ
(1 − ξ 1 z 1 ) · · · (1 − ξ n z n )
∫
= C s f(τξ 1 ρ, · · · , τξ n ρ)(log 1 (
∏ n )
ρ )s−1 R s 1
dρdm n (ξ),
(1 − ξ k z k )
∫ 1
T n 0
where τ ∈ (0, 1). Let τ → 1. Then
∫
f(z 1 , · · · , z n ) = C s
∫ 1
T n 0
k=1
f(ρξ)(log 1 ρ )s−1 R s 1
(1 − ξϕ˜ρ) dρdm n(ξ), (2.4)
for f ∈ H(U n ), s > 0, z ∈ U n , z = ˜ρϕ.
Note for n = 1, we get the well known representation formula (see [4, 17]) after
small modifications. The last estimate is equivalent to
∫
(R −s f)(z) = C s
∫ 1
T n 0
where R −s R s f = f, s > 0, z ∈ U n , f ∈ H(U n ).
The following theorem connect D s and R s <strong>operator</strong>s.
f(ρξ)(log 1 ρ )s−1 1
(1 − ξz) dρdm n(ξ), (2.5)

PROPERTIES OF DIFFERENTIAL OPERATOR ON THE POLYDISK 75
Theorem 2.7. Let 0 < p < ∞, α > −1, s ∈ N, f ∈ H(U n ). If γ > α+2 − 1 for
p
p ≤ 1 and γ > α+1 + 1 (1 − 1 ) for p > 1, v = sp + αn − γpn + n − 1, then
p n p
∫
|D γ f(z)| p (1 − |z| 2 ) α dm 2n (z)
U
∫ n 1 ∫
≤ C |R s f(w)| p (1 − |w| 2 ) v dm n (ξ)d|w| ,
0 T n
where w = |w|ξ.
Pro<strong>of</strong>. Let f ∈ H(U n ), p ≤ 1. Then (see [14])
M 1 (f, τ 2 ) ≤ C(1 − τ) n(1−1/p) M p (f, τ), τ ∈ (0, 1). (2.6)
Using (2.6) and the fact that M p (f, r) is increasing as a function <strong>of</strong> r (see [5]) we
easily get
( ∫ 1
) p
|f(w)|(1 − |w|)
∫T t dm n (ξ)d|w|
n
≤ C
0
∫ 1 ∫T n
0
|f(w)| p (1 − |w|) tp+(n+1)(p−1) dm n (ξ)d|w|, (2.7)
where t > n(1−p) − 1, f ∈ H(U n ), p ≤ 1. By the Cauchy formula,
p
∫
∣ |D γ f ρ (z 1 , · · · , z n )| p ≤ C
f ρ (ξ 1 , · · · , ξ n )dm n (ξ) ∣∣∣
p
∣
, ρ ∈ (0, 1).
T n (1 − ξz) γ+1
Using (1.2) or (2.4), (2.5) for large s and (2.7) we obtain
∫
∣ f ρ (ξ 1 , · · · , ξ n )dm n (ξ) ∣∣∣
p
∣ ∏ n
T n k=1 (1 − ξ kz k ) γ+1
( ∫ 1 ∫
)
|R s f ρ (w)|(1 − |w|) s−1 p
dm n (ξ)d|w|
≤ C
0 T |1 − wz| n γ+1
∫ 1
|R
≤ C
∫T s f ρ (w)| p (1 − |w|) (s−1)p (1 − |w|) (n+1)(p−1)
dm
|1 − wz| n (γ+1)p n (ξ)d|w|. (2.8)
0
Therefore, from (2.6)-(2.8) and using Fubini’s theorem we have
∫
|D γ f ρ (z)| p (1 − |z| 2 ) α dm 2n (z)
U n ∫ 1 ∫
|R
≤ C
∫U s f ρ (w)| p (1 − |w|) sp+n(p−1)−1 dm n (ξ)d|w|
(1 − |z| 2 ) α dm n 0 T |1 − wz| n (γ+1)p 2n (z)
∫ 1 ∫
(1 − |z|
≤ C
0
∫T 2 ) α dm 2n (z)
|R s f n U |1 − wz| n (γ+1)p ρ (w)| p (1 − |w|) sp+n(p−1)−1 dm n (ξ)d|w|.
Using the following estimate
∫
(1 − |z|) t dm 2n (z)
∏ n
k=1 |1 − z kw k | ≤ C
∏ t 1 n
k=1 (1 − |w k|) , t > −1, t t 1 > t + 2,
1−t−2
U n

76 R. SHAMOYAN, S. LI
we get
∫
U n (1 − |z| 2 ) α dm 2n (z)
|1 − wz| (γ+1)p ≤ C(1 − |w|) (2+α−pγ−p)n
(
γ > α + 2
p
)
− 1, α > −1, |w| ∈ (0, 1) .
Using limit argument we can get the desired result.
Consider now the case p > 1. The arguments are partially the same as in the
case p ≤ 1. Let ɛ be positive and small enough. Using (1.2), Cauchy formula,
Hölder inequality we have
≤
( ∫ 1 ∫
|D γ f(z)| p ≤
C
∫ 1
0
∫
= CM 1 M p/p′
2 .
0
)
|R s f(w)|(1 − |w|) s−1 p
dm n (ξ)d|w|
T |1 − wz| n γ+1
(
|R s f(w)| p (1 − |w|) (s−1)p ∫
dm n (w)d|w|
T |1 − wz| n (γ−1)p+2+pɛ
Furthermore by Lemma 2.6,
M 2 ≤
∫ 1
0
dR
∏ n
k=1 (1 − R|z k|) 1−ɛp′
( ∫ R0
≤
where R 0 = max 1≤k≤n |z k |. In addition,
Using (2.3),
∫ 1
=
≤
R 0
0
∫ 1
+
R 0
)
∫ 1
T n 0
dR
∏ n
k=1 (1 − R|z k|) 1−ɛp′ ≤ (1 − R 0) 1/n+···1/n
∏ n
k=1 (1 − |z k|) 1−ɛp′
∫ R0
0
∫ R0
C
0
∫ R0
0
≤
dR
∏ n
k=1 (1 − R|z k|) 1−ɛp′
(1 − R) 1 n −1 × (1 − R) n−1
n dR
∏ n
k=1 (1 − R|z k|) 1−ɛp′
) p/p ′
dm n (ξ)d|w|
|1 − wz| 2−p′ ɛ
dR
∏ n
k=1 (1 − R|z , (2.9)
k|) 1−ɛp′
C
∏ n
k=1 (1 − |z k|) 1−ɛp′ − 1 n
(1 − R|z 1 |) 1/n · · · (1 − R|z n−1 |) 1/n
(1 − R) 1/n−1 dR
(1 − R|z 1 |) 1−ɛp′ · · · (1 − R|z n |) 1−ɛp′
∫ R0
= ˜M. (2.10)
C
(1 − R) 1/n−1 dR
≤ ∏ n−1
k=1 (1 − |z ≤ C
k|) 1−ɛp′ − 1 n 0 (1 − R|z n |) ˜M. (2.11)
1−ɛp′
Combing (2.9) with (2.10), (2.11) and repeating the arguments that we used for
p ≤ 1, we can get the desired result at the case <strong>of</strong> p > 1.
□
Remark 2.8. As we see from Theorem 2.7, the R s <strong>operator</strong> are closely connected
with quasinorms defined on subframe. Theorem 2.7 is an extension <strong>of</strong> result on
the action <strong>of</strong> fractional derivatives on weighted Bergman spaces in the unit disk,
see [2].

PROPERTIES OF DIFFERENTIAL OPERATOR ON THE POLYDISK 77
3. Embedding theorems for <strong>some</strong> analytic spaces connected with
R s <strong>operator</strong>
In this section we state <strong>some</strong> new embedding theorems for various quasinorms
where the R s <strong>operator</strong> is participating, note that practically all results are well
known or obvious in the unit disk. Pro<strong>of</strong>s <strong>of</strong> our results are heavily based on
definitions and preliminaries from previous sections.
Theorem 3.1. i) Let 0 < q < ∞, α ∈ [0, ∞), s ∈ N, 1 < p < ∞, f ∈ H(U n ).
Then
( ∫ 1
q/p
(1 − |z|)
∫T α |f(z)| p |z| d|z|) sp dm n (ξ)
n
≤
0
( ∫ 1
q/p
C |R
∫T s f(uξ)| p (1 − u) du) sp+α dm n (ξ).
n
0
ii) Let 0 < q < ∞, α ∈ [0, ∞), s ∈ N, 1 < p < ∞, γ ∈ (−1/p, 1/p ′ ), 1/p ′ + 1/p =
1, f ∈ H(U n ). Then
≤
( ∫ 1
q/p
(1 − r)
∫T γ |f(rξ)| p r dr) p dm n (ξ)
n
0
( ∫ 1
q/p
C |R
∫T s f(rξ)| p (1 − r) dr) sp+γ dm n (ξ) .
n
Pro<strong>of</strong>. Using (1.2) we have
( ∫ 1
M = (1 − |z|) α |f(z)| p |z| sp d|z|
≤
= C
0
( ∫ 1 ( ∫ 1
C
0
∫ 1 ∫ 1
0
where ψ(|z|) ∈ L p′ (d|z|), 1/p ′ + 1/p = 1.
Changing the variables we have
∫ 1
0
0
0
0
) 1/p
) p(1 1/p
|R s f(ρz)|(1 − ρ) dρ) s−1 − |z|) α |z| sp d|z|
|R s f(ρz)|(1 − ρ) s−1 (1 − |z|) α/p ψ(|z|)|z| s d|z|dρ ,
|R s f(ρz)|(1 − ρ) s−1 dρ ≤
∫ |z|
Using the last inequality and Hölder inequality, we get
M ≤ C
≤
C
∫ 1 ∫ |z|
0
∫ 1
0
( ∫ 1
0
0
|R s s−1 du
f(uξ)|(1 − u)
|z| . s
|R s f(uξ)|(1 − u) s−1 (1 − |z|) α/p ψ(|z|)d|z|du
|R s f(vξ)|(1 − v) s
1 − v
∫ 1
v
) p ′
(1 − |z|) α/p ψ(|z|)d|z|dv
( ∫ 1 1
) 1/p ′( ∫ 1
1/p
≤ C
ψ(|z|)d|z| dv |R s f(uξ)| p (1 − u) du) α+ps .
0 1 − v v
0
From the last inequality and Lemma 2.4 we easily get the first part <strong>of</strong> the theorem.

78 R. SHAMOYAN, S. LI
Now we prove the second part. From (1.2) and changing the variables we have
f(τξ) = f(τξ 1 , · · · , τξ n ) =
=
≤
∫ τ
C
0
∫ τ
∫ 1
R s f(uξ 1 , · · · , uξ n )(log τ du
)s−1
u τ
0
R s f(uξ 1 , · · · , uξ n ) ×
0
R s f(τξ 1 ρ, · · · , τξ n ρ)(log 1 ρ )s−1 dρ
(1 − uτ)−1
(1 − u 2 ) s dτ, s ≥ 1 .
τ
Using duality argument, (2.3) and Hölder inequality we have
( ∫ 1
) 1/p
f(τξ) p τ p (1 − τ) γ dτ
≤
= C
≤
≤
0
( ∫ 1 ( ∫ 1
C
0
∫ 1 ∫ 1
0
( ∫ 1
C
0
0
0
∫ 1
) p(1 1/p
|R s f(uξ)|(1 − u) s (1 − τu) du) −1 − τ) γ dτ
|R s f(uξ)|(1 − u) s (1 − τu) −1 (1 − τ) γ ψ(τ)( 1 − u
1 − τ ) −1
pp ′ ( 1 − u
1 − τ ) 1
pp ′ dudτ
0
∫ 1
ψ p′ (τ)(1 − τu) −1 ( 1 − u
1 − τ
) −
1
p
(1 − τ) γ dudτ
) 1/p ′
( ∫ 1
( 1 − u
) )
1
1/p
× (1 − u) sp |R s f(uξ)| p p ′ (1 − τ) γ (1 − τu) −1 dudτ
0 0
1 − τ
( ∫ 1
) 1/p ′( ∫ 1
1/p
C ψ p′ (τ)(1 − τ) γ dτ (1 − u) sp+γ |R s f(uξ)| du) p ,
0
0
where ψ ∈ L p′ ((1 − τ) γ dτ), 1/p + 1/p ′ = 1. We can get the desired inequality by
integrating both sides on T n . The pro<strong>of</strong> is completed.
□
Remark 3.2. Our Theorem 3.1 extends the well-known Hardy-Littlwood theorems(case
n = 1 and p = q) with fractional derivative (see [2, 5]) to the case <strong>of</strong>
mixed norm spaces with R s <strong>operator</strong>s.
Next we give an application <strong>of</strong> Lemma 2.1. For measurable function f in the
disk we define
( ∫ |f(z)| p ) 1/p,
A p (f)(ξ) =
Γ δ (ξ) (1 − |z|) dm 2(z) p < ∞;
2
A ∞ (f)(ξ) = sup{|f(z)| : z ∈ Γ δ (ξ)}, ξ ∈ T, δ > 1;
( ∫ 1 |f(z)| p ) 1/p,
C p (f)(ξ) = sup
ξ∈I |I| S(I) 1 − |z| dm 2(z) p < ∞, ξ ∈ T,
where S(I) = {z ∈ U, z
|z|
1
∈ I, 1 − |z| ≤ |I|}, I ⊂ T,
2π
Γ δ (ξ) = {z ∈ U : |1 − ξz| < δ(1 − |z|), δ > 1}.
The following result can be found in [1] or [10]:

PROPERTIES OF DIFFERENTIAL OPERATOR ON THE POLYDISK 79
For any functions f(z) and g(z) measurable in the unit disk
∫
∫
|f(z)||g(z)|
dm 2 (z) ≤ C A p (f)(ξ)C p ′(g)(ξ)dm(ξ), 1 < p ≤ ∞, 1 1 − |z|
p + 1 = 1.(3.1)
p ′
U
T
The above result can be easily by iteration extended to polydisk.
Using the polydisk version <strong>of</strong> (3.1), by Cauchy formula in polydisk, Littlewood-
Paley inequality (see [1]) and 2.1 we have for any holomorphic function f in the
polydisk,
∫
|R s f ρ (ϕτ 2 )| = C∣
f(τξ)R s 1
T n 1 − τρξϕ dm n(ξ) ∣
∫
≤ C |D α R s 1
U 1 − wϕρ |f(w)|(1 − |w|)α−1 dm 2n (w)
n
≤
≤
∫
C
∫
C
T n
T n
sup |D α R s 1
w∈Γ t(ξ) 1 − wϕρ (1 − |w|)α dm n (ξ) × ‖C 1 (f)‖ L ∞
sup |R s 1
w∈Γ t(ξ) 1 − wϕρ |dm n(ξ) × ‖C 1 (f)‖ L ∞.
Here α > 0, ϕ, ξ ∈ T n , ρ ∈ [0, 1] n , ϕτ 2 = (ϕ 1 τ1 2 , · · ·, ϕ n τn), 2 τ j ∈ (1/2, 1), j =
1, · · · , n. We used in the last inequality a maximal theorem for D s <strong>operator</strong>s, see
[7]. Therefore
( ∑ n∏
−1
|R s f(ρϕ)| × (1 − ρ k ) k) −α ≤ C‖C1 (f)‖ L ∞
α j ≥0
P αj =s
k=1
and
( ∑
sup M ∞ (D −1 R s f, ρ)| ×
ρ∈[0,1] n
α j ≥0
P αj =s
n∏
(1 − ρ k ) −α k
k=1
) −1
where
≤ C‖C 1 (f × (1 − |z|))‖ L ∞, (3.2)
C 1 (f × (1 − |z|))(ξ 1 , · · · , ξ n ) = sup
ξ 1 ∈I
∫
1
· · · sup
|I| S(I) ξ n∈I
∫
1
|f(z)|dm 2n (z).
|I| S(I)
Remark 3.3. Estimates (3.2) extends known one dimensional inequality which
can be found in [10]. We just give one application <strong>of</strong> Lemma 2.1. Various other
generalizations can be obtained with the help <strong>of</strong> Lemma 2.1. Estimate (3.2) is
just an example.
We see that the R s <strong>operator</strong> are connected with the quasinorms on subframe
via integral representations (2.4) and (2.5). So it is natural to find new estimates
for quasinorms on subframe. In the polydisk we consider the following three
expressions
∫ 1 ∫
|f(rξ 1 , · · · , rξ n )| p (1 − r) α rdrdm n (ξ), 0 < p < ∞, α > −1;
0 T n

80 R. SHAMOYAN, S. LI
∫T n ∫
∏ n
|f(τ 1 ξ 1 , · · · , τ n ξ n )| p (1 − τ k ) α k
τ 1 · · · τ n dτ 1 · · · dτ n dm n (ξ);
[0,1] n
k=1
and ∫ ∫
|f(τ 1 ξ, · · · , τ n ξ)| p (1 − τ 1 ) α1 · · · (1 − τ n ) αn τ 1 · · · τ n dτ 1 · · · dτ n dm(ξ),
T [0,1] n
where 0 < p < ∞, α j > −1, j = 1, · · · , n, f ∈ H(U n ). Our intension is to show
<strong>some</strong> connection between these quasinorms using so-called dyadic decomposition
<strong>of</strong> the polydisk and subframe, see [4]. First, we have the following result.
Theorem 3.4. Let f ∈ H(U n ), 0 < p < ∞, α > −1, n ∈ N. Then
∫ 1 ∫
|f(z)| p (1 − |z| 2 ) α dm n (ξ)d|z|
0 T
∫
n
≤ C |f(z)| p (1 − |z 1 | 2 ) α n −1+ 1 n · · · (1 − |zn | 2 ) α n −1+ 1 n dm2n (z) .
U n
Pro<strong>of</strong>. Using diadic decomposition that were introduced in introduction and the
subharmonicity <strong>of</strong> the |f(z)| p (0 < p < ∞) function, we have
∫ 1 ∫
|f(z)| p (1 − |z| 2 ) α dm n (ξ)d|z|
0 T n
= ∑ ∫ 1−2 −k−1
∑
∫ ∫
· · · |f(z)| p (1 − |z| 2 ) α dm n (ξ)d|z|
k≥0
1−2 −k l 1 ,··· ,l
I n k,l1 I k,ln
≤ C ∑ ∑
max |f(z)| p · 2 −k · 2 −kα · (2 −k · · · 2 −k )
eU
k≥0 l 1 ,··· ,l k,l1 ,··· n
,ln
≤ C ∑ · · · ∑ ∑
n∏ n∏ n∏
max |f(z)| p · 2 − k i
n · 2 − k i α
n 2 −k i
U k1 ,··· ,kn,l
k 1 ≥0 k n≥0 l 1 ,··· ,l 1 ,··· ,ln
n
i=1 i=1 i=1
∫
≤ C |f(z)| p (1 − |z 1 | 2 ) α n −1+ 1 n · · · (1 − |zn | 2 ) α n −1+ 1 n dm2n (z).
U n
In the last inequality we used two facts. The first is that we used the following
inequality
∫
max |f(z)| p ≤ C2 2(k 1+···+k n)
|f(z)| p dm 2n (z) ,
z∈U k1 ,··· ,kn,l 1 ,··· ,ln
U ∗ k 1 ,··· ,kn,l 1 ,··· ,ln
which follows from the subharmonicity <strong>of</strong> |f(z)| p (0 < p < ∞, f ∈ H(U n ) (see
[4]). Here Uk ∗ 1 ,··· ,k n,l 1 ,··· ,l n
= Uk ∗ 1 ,l 1
× · · · × Uk ∗ n,l n
is a family <strong>of</strong> enlarged dyadic
cubes, see [4]. The second is that the family <strong>of</strong> enlarged dyadic cubes is a finite
covering <strong>of</strong> polydisk (see [4])
( ∑ )( ∑
|f(z)| p (1 − |z| 2 ) γ dm 2n (z)
≤
) ∫
k 1 ,··· ,k n≥0 l 1 ,··· ,l
U ∗ n≥0 k 1 ,··· ,kn,l 1 ,··· ,ln
∫
C |f(z)| p (1 − |z| 2 ) γ dm 2n (z), 0 < p < ∞, γ > −1.
U n

PROPERTIES OF DIFFERENTIAL OPERATOR ON THE POLYDISK 81
The pro<strong>of</strong> <strong>of</strong> the theorem is completed.
Remark 3.5. It is easy to see that the assertion <strong>of</strong> Theorem 3.4 is true for n = 1.
The following theorem can be obtained using the same ideas by small modification
<strong>of</strong> methods that we used above.
Theorem 3.6. Let f ∈ H(U n ). Then the following assertions are true.
i) Let p ∈ (0, ∞), α j > −1, j − 1, · · · , n, n ∈ N. Then
∫
|f(z, · · · , z)| p (1 − |z| 2 ) P n
j=1 αj+n−1 dm 2 (z)
≤
U
∫
C
T
∫
∏ n
|f(|z 1 |ξ, · · · , |z n |ξ)| p (1 − |z k | 2 ) α k
d|z 1 | · · · d|z n |dm(ξ).
[0,1] n
k=1
ii) Let 0 < p < ∞, α > −1. Then we have
∫
|f(z, · · · , z)| p (1 − |z| 2 ) α+n−1 dm 2 (z)
≤
C
U
∫ 1 ∫T n
0
|f(|z|ξ 1 , · · · , |z|ξ n )| p (1 − |z| 2 ) α dm n (ξ)d|z|.
iii) Let p ∈ (0, ∞), α j > −1, j − 1, · · · , n, n ∈ N. Then
∫ ∫
∏ n
|f(|z 1 |ξ, · · · , |z n |ξ)| p (1 − |z k |) α k+ n−1
n d|z1 | · · · d|z n |dm(ξ)
T [0,1] n k=1
∫
∏ n
≤ C |f(z)| p (1 − |z k | 2 ) α k
dm 2n (z).
U n
k=1
Pro<strong>of</strong>. We prove only the second inequality. We use diadic decomposition <strong>of</strong> U
and subframe and the same ideas that we used in the pro<strong>of</strong> <strong>of</strong> Theorem 3.4. We
have
∫
|f(z, · · · , z)| p (1 − |z| 2 ) α+n−1 dm 2 (z)
U
= ∑ ∑
∫
|f(z, · · · , z)| p (1 − |z| 2 ) α+n−1 dm 2 (z)
k≥0 j U j,k
≤ C ∑ ∑
max |f(z, · · · , z)| p 2 −2k 2 −k(α+n−1)
U j,k
k≥0 j
≤ C ∑ ∑
max |f(z, · · · , z)| p 2 −k(α+n+1)
z∈U k,j1 ,··· ,jn
k≥0 j 1 ,···j n
≤ C ∑ ∑
2 2kn M2 −k(α+n+1) ,
k≥0 j 1 ,···j n
where
M =
∫
eU ∗ k,j 1 ,··· ,jn
|f(z 1 , · · · , z n )| p dm 2n (z),
□

82 R. SHAMOYAN, S. LI
Ũk,j ∗ 1 ,··· ,j n
is a enlarged version <strong>of</strong> dyadic cube on dyadic decomposition <strong>of</strong> subframe
(see [4])
{
Ũk,j ∗ 1 ,··· ,j n
= Ũ k,j ∗ 1
× · · · × Ũ k,j ∗ n
= (τξ 1 , · · · , τξ n ), τ ∈ Ĩk = (1 − 1
2 , 1 − 1 ],
k−1 2k+2 ξ i ∈ Ĩk,j i
= [ π(j i − 1/2)
, π(j i + 1/2)
}
) .
2 k 2 k
We have
( ∫ 1−2 −k−2 ∫ ) ( ∫ 1−2 −k−2 ∫ )
M =
· · ·
|f(z 1 , · · · , z n )| p dm 2n (z)
1−2 −k+1 eI k,j1 1−2 −k+1 eI k,jn
∫
≤ C2 −k · · · 2
∫e −k · · · |f( ˜τ 1 ξ 1 , · · · , ˜τ n ξ n )| p dm n (ξ),
Ik,j1 eI k,jn
where |f( ˜τ 1 ξ 1 , · · · , ˜τ n ξ n )| = max τj ∈e Ik ,j=1,··· ,n. |f(τ 1ξ 1 , · · · , τ n ξ n )|. Therefore,
∫
|f(z, · · · , z)| p (1 − |z| 2 ) α+n−1 dm 2 (z)
Since
≤
U
C ∑ k≥0
2 ∑ ∫ ∫
∫ 1−2 −k−3
−kα · · · |f( ˜τ 1 ξ 1 , · · · , ˜τ n ξ n )| p dm n (ξ) dτ.
j 1 ,···,j
eI n k,j1
eI k,jn 1−2 −k−2
∫ ∫
|f( ˜τ 1 ξ 1 , · · · , ˜τ n ξ n )| p dm n (ξ) ≤ |f(τξ 1 , · · · , τξ n )| p dm n (ξ),
T n T n
for 0 < p < ∞, τ j ∈ [1 − 2 −k−2 , 1 − 2 −k−3 ), j = 1, · · · , n, we obtain
∫
|f(z, · · · , z)| p (1 − |z| 2 ) α+n−1 dm 2 (z)
≤
≤
U
C ∑ k≥0
C
∫ 1−2 −k−3 ∫
2 −kα
∫T n ∫ 1
0
1−2 −k−2
The pro<strong>of</strong> <strong>of</strong> this theorem is finished.
T n |f(τξ 1 , · · · , τξ n )| p dm n (ξ)dτ
|f(|z|ξ 1 , · · · , |z|ξ n )| p (1 − |z| 2 ) α dm n (ξ)d|z|.
Remark 3.7. The assertions <strong>of</strong> Theorem 3.6 are obvious for n = 1.
From Theorem 3.6, we easily get the following result, which was proved in [4].
Corollary 3.8. Let p ∈ (0, ∞), α > −1, n ∈ N, f ∈ H(U n ). Then
∫
|f(z, · · · , z)| p (1 − |z|) αn+2n−2 dm 2 (z)
U∫
≤ C |f(z 1 , · · · , z n )| p (1 − |z 1 |) α · · · (1 − |z n |) α dm 2n (z).
U n
□

PROPERTIES OF DIFFERENTIAL OPERATOR ON THE POLYDISK 83
Corollary 3.9. Let u ∈ H(U n ), q ∈ (0, ∞), α j > 0, j = 1, · · · , n, n ∈ N. Then
∫
|u(z, · · · , z)| q (1 − |z| 2 ) P n
j=1 αj−1 dm 2 (z)
≤
U
∫
C
T
∫
Γ t(ξ)
Pro<strong>of</strong>. Fix ξ ∈ T , then (see [3])
∫
· · · |u(z 1 , · · · , z n )| q
Γ t(ξ)
∫ 1
0
n
∏
k=1
|ũ(ρξ)| q
1 − ρ
∫Γ dρ ≤ C |ũ(z)| q dm 2 (z)
.
t(ξ) (1 − |z| 2 ) 2
(1 − |z k | 2 ) α k−2 dm 2n (z)dm(ξ).
Choosing ũ = u(ρξ)(1 − ρ) α/q , u ∈ H(U), α, q ∈ (0, ∞), we have
∫ 1
∫
|u(ρξ)| q (1 − ρ) α−1 dρ ≤ C |u(z)| q (1 − |z| 2 ) α−2 dm 2 (z).
0
Γ t(ξ)
Using the last inequality, by each variable we get the following
≤
∫ 1
0
∫
C
· · ·
Γ t(ξ)
∫ 1
0
∫
· · ·
|u(ρ 1 ξ, · · · , ρ n ξ)| q
Γ t(ξ)
|u(z)| q
n
∏
k=1
n
∏
k=1
(1 − ρ k ) α k−1 dρ 1 · · · dρ n
(1 − |z k | 2 ) α k−2 dm 2n (z).
Integrating both sides by T and using i) <strong>of</strong> Theorem 3.6, we have
∫
|u(z, · · · , z)| q (1 − |z| 2 ) P n
j=1 αj−1 dm 2 (z)
≤
≤
U
∫
C
∫
C
T
T
∫ 1
∫
0
· · ·
Γ t(ξ)
∫ 1
0
∫
· · ·
|u(ρ 1 ξ, · · · , ρ n ξ)| q
Γ t(ξ)
|u(z 1 , · · · , z n )| q
This completes the pro<strong>of</strong> <strong>of</strong> Corollary 3.9.
n
∏
k=1
∏ n
(1 − ρ k ) α k−1 dρ 1 · · · dρ n dm(ξ)
k=1
(1 − |z k | 2 ) α k−2 dm 2n dm(ξ).
Remark 3.10. For n = 1, the assertion <strong>of</strong> Corollary 3.9 is contained in [3, 18].
References
1. K. L. Avetisyan and R. F. Shamoyan, Some generalization <strong>of</strong> Littlewood-Paley inequality
in the polydisk, Matematicki Vesnik, 58 (3-4) (2006), 97–110.
2. S. M. Buckley, P. Koskela and D. Vukotic, Fractional integration, differentiation, and
weighted Bergman spaces, Math. Proc. Cambridge Phil. Soc., 126 (1999), 369–385.
3. W. Cohn, Weighted Bergman projections and tangential area integrals, Studia Math., 106
(1993), 59–76.
4. A. E. Djrbashian and F. A. Shamoian, Topics in the Theory <strong>of</strong> A p α Spaces, Leipzig, Teubner,
1988.
5. P. L. Duren, Theory <strong>of</strong> H p Spaces, Academic Press, New York, 1970.
6. V. S. Guliev and P. I. Lizorkin, Spaces <strong>of</strong> holomorphic functins and harmonic functions in
the polydisk and their connections with boundary values, Trudy Mat. Inst. Steklov RAN,
204 (1993), 137–159.
□

84 R. SHAMOYAN, S. LI
7. M. I. Gvaradze, Multipliers <strong>of</strong> a class <strong>of</strong> analytic functions defined on a polydisc, Akad.
Nauk Gruzin. SSR Trudy Tbiliss. Mat. Inst. Razmadze, 66 (1980), 15–21.
8. H. Hedenmalm, B. Korenblum and K. Zhu, Theory <strong>of</strong> Bergman Spaces, Graduate Texts in
Mathematics, 199, Springer, New York, 2000.
9. V. G. Mazya, Sobolev Spaces, Springer-Verlag, Berlin, 1985.
10. J. M. Ortega and J. Fàbrega, Hardy’s inequality and embeddings in holomorphic Triebel-
Lizorkin spaces, Illinois J. Math., 43 (1999), 733–751.
11. R. F. Shamoyan, <strong>On</strong> multipliers from spaces <strong>of</strong> Bergman type to Hardy spaces in polydisk,
Ukrainian Math. J., 52 (10)(2000), 1606–1617.
12. R. F. Shamoyan, <strong>On</strong> quasinorms <strong>of</strong> functions from holomorphic Lizorkin Triebel type spaces
on subframe and polydisk, Sympozium Fourier series and application, Conference materials,
Rostov Na Donu, in Russian, 2002, 54-55.
13. A. L. Shields and D. L. Williams, Bounded projections, duality, and multipliers in spaces
<strong>of</strong> analytic functions, Trans. Amer. Math. Soc., 162 (1971), 287–302.
14. R. M. Trigub, Multipliers in the Hardy spaces H p (D m ) with p ∈ (0, 1] and approxiamation
<strong>properties</strong> <strong>of</strong> summability methods for power series, Mat. Sb., 188 (4) (1997), 145–160.
15. W. Rudin, Function Theory in the Polydisk, Benjamin, New York, 1969.
16. K. Zhu, Weighted Bergman projections on the polydisk, Houston J. Math., 20 (2) (1994),
275–292.
17. K. Zhu, Spaces <strong>of</strong> Holomorphic Functions in the Unit Ball, New York, 2005.
18. A. Zygmund, Trigonometric Series, (2nd.ed.), Vol.2, Cambridge Univ. Press, Cambridge,
1959.
1 Department <strong>of</strong> Mathematics, Bryansk State Pedagogical University, Russian
E-mail address: rsham@mail.ru
2 Department <strong>of</strong> Mathematics, JiaYing University, 514015, Meizhou, Guang-
Dong, China
E-mail address: jyulsx@163.com; lsx@mail.zjxu.edu.cn