An. Şt. Univ. Ovidius Constanţa
Vol. 19(1), 2011, 75–92
HÖLDER WEIGHT ESTIMATES OF
RIESZ-BESSEL SINGULAR INTEGRALS
GENERATED BY A GENERALIZED SHIFT
OPERATOR
I. Ekincioglu, C. Keskin and S. Er
Abstract
Riesz-Bessel singular integral operators generated by generalized shift
γ
γ
operator in weighted Hölder space Hα,β
is studied. The Hα,β
boundedness of this operator is established in certain cases.
1
Introduction
Singular integral operators are playing an important role in Harmonic Analysis, theory of functions and partial differential equations. Singular integrals
associated with the ∆Bn Laplace-Bessel differential operator
∆Bn =
n−1
X
i=1
∂2
+ Bn ,
∂x2i
Bn =
∂2
γn ∂
+
, γn > 0,
2
∂xn
xn ∂xn
which is known as an important operator in analysis and its applications, have
been the research areas of many mathematicians such as B. Muckenhoupt and
E.M. Stein [12, 13], I. Kipriyanov and M. Klyuchantsev [1, 2, 11], K. Trimeche
[15], K. Stempak [14], I.A. Aliev and A.D. Gadjiev [3], V.S. Guliyev [9], A.D.
Gadjiev and Emin V. Guliyev [8] and others.
Key Words: Generalized Shift Operator, Riesz-Bessel Singular Integral, Hölder spaces.
Mathematics Subject Classification: 47G10, 45E10, 47B37
Received: December, 2009
Accepted: January, 2010
75
76
I. Ekincioglu, C. Keskin and S. Er
The multidimentional singular integral operators generated by generalized shift operator are studied by Klyuchantsev in [1] and Kipriyanov and
Klyuchantsev in [2]. Aliev and Gadjiev [3], and Ekincioglu and Serbetci
[7] have studied the boundedness of certain singular integrals generated by
generalized shift operator in weighted Lp -spaces with radial weights.
Klyuchantsev and Kipriyanov [2] have firstly introduced and investigated
the Lp,γ -boundedness of singular integrals generated by the ∆Bn –LaplaceBessel diffe-rential operator (Bn –singular integrals). Aliev and Gadjiev [3]
have studied the boundedness of the Bn –singular integrals in weighted Lp,ω,γ spaces with radial weights. Gadjiev and Guliyev [8] and Ekincioglu [18] have
investigated the boundedness of the Bk -singular integrals in weighted Lp,ω,γ spaces with general weights.
Matrix singular integral operators and singular integral operators are investigated in weighted Hölder spaces by Kapanadze in [16] and Abdullayev
[17].
The paper is organized as follows. In the next section, we collect the
background materials. In the Section 3, we recall the definition of weighted
Hölder spaces and give some inequalities in weighted Hölder spaces. Finally,
we give the boundedness of Riesz-Bessel singular integral operators in weighted
Hölder spaces will be studied in certain cases.
Throughout this paper we use the convention that c denotes a generic
constant, depending on k, γ or other fixed parameters, its value may change
from line to line.
In this paper, Riesz-Bessel singular integral operators generated by generalized shift operator associated with the ∆Bk Laplace-Bessel differential operator
∆Bk =
n−k
X
i=1
k
X γi
∂2
∂
∂2
+
+
, k = 1, . . . , n.
2
2
∂xi
x
∂xn−k+i
∂xn−k+i
i=1 n−k+i
is studied in weighted Hölder spaces.
2
Preliminaries
Let Rn be n–dimensional Euclidean space and x = (x1 , ..., xn ), ξ = (ξ1 , ..., ξn )
be vectors in Rnk,+ , then x · ξ = x1 ξ1 + . . . + xn ξn , |x| = (x · x)1/2 , x =
n−k
(x′ , x′′ ), x′ = (x1 , . . . , xn−k ) ∈ Rk,+
, x′′ = (xn−k+1 , . . . , xn ) ∈ Rkk,+ , x′′′ =
k
′
n
(xn+1 , . . . , xn+k ) ∈ Rk,+ and x̃ = (x , x′′ , x′′′ ) ∈ Rn+k . Denote , Sk,+
= {x ∈
n
Rk,+ : |x| = 1}, γ = (γ1 , . . . , γk ), γ1 > 0, . . . , γk > 0, |γ| = γ1 + . . . + γk ,
γn−k
.
(x′ )γ = xγ11 · . . . · xn−k
HÖLDER WEIGHT ESTIMATES OF RIESZ-BESSEL BESSEL SINGULAR
INTEGRALS GENERATED BY A GENERALIZED SHIFT OPERATOR
77
Let Rnk,+ = {x : x = (x1 , x2 , . . . , xn ), xn−k+i > 0, . . . , xn > 0, 1 ≤ i ≤ k}.
We now introduce the generalized shift operator is defined by
Z π
Z π
T y f (x) = cγ
f (x′ − y ′ , (xn−k+1 , yn−k+1 )α1 , . . . , (xn , yn )αk ) dγ(α),
...
0
0
k
Q
Γ(γi + 12 )
sinγi−1 αi dαi and x′ , y ′ ∈
, dγ (α) =
Γ(γi )
i=1
i=1
1
2
]2 ,
Rn−k and (xn−k+i , yn−k+i )αi = [x2n−k+i − 2xn−k+i yn−k+i cos αi + yn−k+i
1 ≤ i ≤ k, [18].
Note that the generalized shift operator is closely connected with LaplaceBessel differential operator (see [4],[1]). The shift T y generates the corresponding convolution
Z
(f ∗ g)(x) =
f (y) T y g(x) (y ′′ )γ dy
k
where cγ = π − 2
k
Q
Rk,+
n
which satisfies the property (f ∗ g) = (g ∗ f ).
The Riesz-Bessel singular integral operator generated by a generalized shift
operator (RBk )f (1 ≤ k ≤ n) is defined by
Z
Pk (θ) y
T f (x) (y ′′ )γ dy
(RBk )f (x) ≡ p.v. ck
Q+k
|y|
Rn
k,+
Z
Pk (y) y
(2.1)
≡ ck lim
T f (x) (y ′′ )γ dy
Q+k
ε→0
|y|
{y∈Rn
k,+ : |y|>ε}
≡ lim (RBk )ε f (x),
ε→0
k −1
, k = 1, 2, . . . , n, θ = y/|y|,
2
2
and the characteristic Pk (θ) belongs to some function space on the hemisphere
n
Sk,+
such that Pk (x) = Pk (x1 , . . . , xn−k , x2n−k+1 , . . . , x2n ) is a homogeneous
polynomial with order k which holds ∆Bk Pk = 0 and satisfies the cancellation
condition
Z
Pk (θ)(θ′′ )γ dθ = 0.
(2.2)
where Q = n + |γ|, ck = 2Q/2 Γ
Q + k
Γ
n
Sk,+
Unless stated otherwise we will assume throughout that Pk (θ) satisfies a unin
form Hölder condition on Sk+
, i.e., if θ1 and θ2 are unit vectors, then for some
positive constants A and 0 < δ ≤ 1,
|Pk (θ1 ) − Pk (θ2 )| ≤ cA|θ1 − θ2 |δ
(2.3)
78
I. Ekincioglu, C. Keskin and S. Er
n
where |θ1 − θ2 | denotes the geodesic distance on Sk+
from θ1 to θ2 , c is a
constant and sup |Pk (θ)| = A ≤ ∞.
Lemma 2.1. Let α and β be are arbitrary numbers such that 0 < α < β ≤ +∞
then for any point x ∈ Rnk,+ the following equality holds
y
Z
Z
Pk |y|
k
y
′′ γ
Pk (θ̃) ˜ Q
|sn−k+2i |γi −1 ds̃,
f (s)
T
f
(x)
(y
)
dy
=
c
γ
Q+k
Q+k
|y|
|x̃ − s̃|
i=1
α<|y|<β
α<|x̃−s̃|<β
1
1
where f˜(s) = f s′ , [s2n−k+1 + s2n−k+2 ] 2 . . . , [s2n+k−1 + s2n+k ] 2 , ds̃ = ds′ ds′′ ds′′′ ,
1
1
2
2
x′ − s′ [(x
[(xn+k−1 − sn+k−1 )2 + s2n+k ] 2 n−k+1 − sn−k+1 ) + sn−k+2 ] 2
θ̃ =
,
,
,...,
|x̃ − s̃|
|x̃ − s̃|
|x̃ − s̃|
n+k
and x̃ = (x′ , x′′ , 0, . . . , 0) ∈ Rk+
. (see [1])
Proof. We denote the first part by I
I
= cγ
Z
y
Pk ( |y|
)
|y|Q+k
= cγ
Z
|y|>ǫ
...
Zπ
Zπ . . . f x′ − y ′ , x2n−k+1 − 2xn−k+1 yn−k+1 cos α1 + (yn−k+1 cos α1 )2
y
Pk ( |y|
)
Q+k
|y|
0
f (x′ − y ′ , (xn−k+1 , yn−k+1 )α1 , . . . , (xn , yn )αk ) (y ′′ )γ dγ(α)dy
0
0
|y|>ǫ
Zπ
Zπ
0
1
1
+(yn−k+1 sin α1 )2 ] 2 , . . . , [x2n − 2xn yn cos αk + (yn cos αk )2 + (yn sin αk )2 ] 2
×(y ′′ )γ sinγi −1 αi dαi dy
Zπ
Zπ Z
y
Pk ( |y|
)
1
. . . f x′ − y ′ , [(xn−k+1 − yn−k+1 cos α1 )2 + (yn−k+1 sin α1 )2 ] 2 ,
= cγ
|y|Q+k
0
0
|y|>ǫ
1
. . . , [(xn − yn cos αk )2 + (yn sin αk )2 ] 2 (y ′′ )γ sinγi −1 αi dαi dy.
Performing a change of variables in the last integral to x′ −y ′ = s′ , sn−k+(2i−1) =
xn−k+i − yn−k+i cos αi , sn−k+2i = yn−k+i sin αi , 0 ≤ αi < π and yn−k+i >
0, i = 1, 2, . . . , k. Since the jacobian of the transformation is equal to (y ′′ )−1 ,
we have
y
Z
Z
Pk |y|
k
y
′′ γ
Pk (θ̃) ˜ Q
T
f
(x)
(y
)
dy
=
c
|sn−k+2i |γi −1 ds̃.
f (s)
γ
Q+k
Q+k
|y|
|x̃ − s̃|
i=1
α<|y|<β
This completes the proof.
|x̃−s̃|>ǫ
HÖLDER WEIGHT ESTIMATES OF RIESZ-BESSEL BESSEL SINGULAR
INTEGRALS GENERATED BY A GENERALIZED SHIFT OPERATOR
79
Lemma 2.2.
Z
f (y)(y ′′ )γ dy
= cγ
Rn
k+
Z
Rn+k
k+
k
Q
|sn−k+2i |γi −1 ds̃.
f˜(s)
i=1
Proof. Changing variables y ′ = s′ , sn−k+(2i−1) = yn−k+i cos αi , sn−k+2i =
yn−k+i sin αi ,
0 ≤ αi < π and yn−k+i > 0, i = 1, 2, . . . , k show that
Z
Z
k
Q
f (y)(y ′′ )γ dy = cγ
|sn−k+2i |γi −1 ds̃,
f˜(s)
Rn
k,+
Rn+k
k,+
i=1
which completes the proof.
Setting f (x) = 1 in Lemma 2.1 (T y f (x) = 1) and using the polar coordinates show that
Z
Z
k
k
Q
Q
γi
θn−k+i dθ = cγ
|θn−k+2i |γi−1 dθ̃.
Pk (θ)
Pk (θ̃)
(2.4)
i=1
i=1
n
Sk,+
3
n+k
Sk,+
ν
Holder Space with weight Hα,β
(Rnk,+ )
Let ν > 0, α > 0, β be a real number,
ρ(x) =
k
X
β−α
xα
,
n−k+i (1 + |x|)
x ∈ Rnk,+ .
i=1
If lim f (x)ρ(x) = 0,
x→∞
ν
Hα,β
(Rnk,+ )
lim
xn−k+i →0
f (x)ρ(x) = 0,
1 ≤ i ≤ k, then f ∈
ν
is defined by the equality
and the norm in the space Hα,β
ν
||f ||Hα,β
= sup
|f (x)ρ(x) − f (y)ρ(y)|d−ν (x, y) ,
x,y∈Rn
k,+
|x − y|
(see [6]).
(1 + |x|)(1 + |y|)
If the contrary is not stipulated, then later on we will assume that 0 < ν <
1, 0 < α − ν < 1, 0 < β + ν < n. Let x ∈ Rnk,+ . We introduce the following
notation,
where d(x, y) =
Ψν (x) =
k
X
i=1
ν−α
(1 + |x|)−ℓ ≡ ρ−1 (x)
xn−k+i
k
X
i=1
xνn−k+i (1 + |x|)−2ν
80
I. Ekincioglu, C. Keskin and S. Er
where ℓ = 2ν + β − α. Let E ′ and E be defined as above
E′ =
n+k
: |x̃−s̃| <
s̃ ∈ Rk,+
k
X
xn−k+i
i=1
2
,
E=
y ∈ Rnk,+ : |y−x| <
k
X
xn−k+i
.
2
i=1
Remark 3.1. Note that if s̃ ∈ E ′ , then |x̃| < 2|s̃| < 3|x̃|; E ′ ⊂ E × (sn−k+2i ) :
Pk xn−k+i
n+k
|sn−k+2i | <
, 1 ≤ k ≤ n and at s̃ ∈ Rk,+
\ E ′ , |x̃ − s̃| ≈
i=1
2
P
P
P
k
k
k
|x′ − s′ | + i=1 xn−k+i + i=1 |sn−k+i | + i=1 |sn−k+2i | .
ν
The following lemma gives an explicit expression for Hα,β
.
ν
Lemma 3.2. Let 0 < γ < α, β + γ > 0. f ∈ Hα,β
(Rnk,+ ) if and only if there
exits C1 and C2 depend on f such that
(i) |f (x)| ≤ C1 (f )Ψγ (x), ∀x ∈ Rnk,+
(ii) |f (x̃) − f (s̃)| ≤ C2 (f )ρ−1 (x)dγ (x̃, s̃) ∀x ∈ Rnk,+ and ∀s̃ ∈ E ′ .
ν
≈ (max C1 (f ) + max C2 (f )) (see [5]).
Moreover ||f ||Hα,β
We have the following result.
ν
Corollary 3.3. If f ∈ Hα,β
, then
ν−α
Pk
ν
(i) |f˜(s)| ≤ ||f ||Hα,β
(|1 + |s|)−ℓ
i=1 |sn−k+i | + |sn−k+2i |
ν−α
Pk
Pk
ν
≤ ||f ||Hα,β
1 + |s′ | + i=1 |sn−k+i | +
i=1 |sn−k+i | + |sn−k+2i |
−ℓ
|sn−k+2i |
ν
ν
ρ−1 (x)|x̃ − s̃|ν (1 +
ρ−1 (x)dν (x̃, s̃) ≤ ||f ||Hα,β
(ii) f˜(s) − f (x) ≤ ||f ||Hα,β
|x|)−2ν for ∀x ∈ Rnk,+ and ∀s̃ ∈ E ′ .
4
Riesz-Bessel Singular Integral Operators
It is well known that the Riesz-Bessel singular integral operators RBk (1 ≤
k ≤ n) are bounded in Lp (Rn ). The basic goal of this paper is to establish
an estimate in the weighted Hölder spaces for singular integral operators generated by a generalized shift operator such as Riesz-Bessel singular integral
operators RBk . We obtain sufficient conditions for α, β, ν so that RBk opν
erators are bounded from the weighted Hα,β
(Rnk,+ ) spaces into the weighted
ν
n
Hα,β (Rk,+ ) spaces.
HÖLDER WEIGHT ESTIMATES OF RIESZ-BESSEL BESSEL SINGULAR
INTEGRALS GENERATED BY A GENERALIZED SHIFT OPERATOR
81
Theorem 4.1. Let α > 0, ν > 0 and β be a real number such that 0 < ν < 1,
0 < α − ν < 1, 0 < β + ν < n. Suppose that the characteristic Pk (θ) of the
singular integral (2.1) is bounded and satisfies the condition (2.2). Then there
exists Riesz-Bessel singular integrals generated by generalized shift operator
such that
RBk f (x)
Z
= p.v. ck
y
|y|
Pk
Rn
= ck,γ
Zk,+
|y|−Q−k T y f (x) (y ′′ )γ dy
|sn−k+2i |γi −1 ds̃
i=1
|x̃−s̃|≥η
Z
+ck,γ
k
Q
Pk (θ̃)|x̃ − s̃|−Q−k f˜(s)
ǫ<|x̃−s̃|<η
k
Q
|sn−k+2i |γi −1 ds̃
Pk (θ̃)|x̃ − s̃|−Q−k f˜(s) − f (x)
i=1
(4.1)
ν
for any f ∈ Hαβ
and for a.e. x ∈ Rnk,+ .
Proof. From Lemma 2.1
RBk f (x)
Z
= p.v. ck
Z
= ck
+ck
Rk,+
n
Pk
|y|≥η
Z
y
|y|
Z
ǫ<|x̃−s̃|<η
= ck,γ
Rn+k \E ′
|y|−Q−k T y f (x) (y ′′ )γ dy
|y|−Q−k T y f (x) (y ′′ )γ dy
|y|−Q−k T y f (x) − f (x) (y ′′ )γ dy
k
Q
|sn−k+2i |γi −1 ds̃
i=1
Z
Z
Pk (θ̃)|x̃ − s̃|−Q−k f˜(s)
|x̃−s̃|≥η
+ck,γ
y
|y|
Pk
ǫ<|y|<η
= ck,γ
y
|y|
Pk
k
Q
|sn−k+2i |γi −1 ds̃
Pk (θ̃)|x̃ − s̃|−Q−k f˜(s) − f (x)
Pk (θ̃)|x̃ − s̃|−Q−k f˜(s)
i=1
k
Q
|sn−k+2i |γi −1 ds̃
i=1
Zk,+
k
Q
|sn−k+2i |γi −1 ds̃
+ck,γ Pk (θ̃)|x̃ − s̃|−Q−k f˜(s) − f (x)
E′
i=1
82
I. Ekincioglu, C. Keskin and S. Er
where E ′ (ǫ) =
′
n+k
: ǫ < |x̃ − s̃| < η, η =
s̃ ∈ Rk,+
definition E (ǫ)
RBk f (x)
Z
= p.v. ck
Pk
Rk,+
n
Z
= ck,γ
y
|y|
+ck,γ
E ′ (ǫ)
i=1
xn−k+i
. By
2
|y|−Q−k T y f (x) (y ′′ )γ dy
Pk (θ̃)|x̃ − s̃|−Q−k f˜(s)
k
Q
|sn−k+2i |γi −1 ds̃
i=1
′
Rn+k
k,+ \E (ǫ)
Z
Pk
k
Q
|sn−k+2i |γi −1 ds̃.
Pk (θ̃)|x̃ − s̃|−Q−k f˜(s) − f (x)
i=1
(4.2)
If we calculate the second integral in (4.2), we get
Z
E ′ (ǫ)
k
Q
|sn−k+2i |γi −1 ds̃
Pk (θ̃)|x̃ − s̃|−Q−k f˜(s) − f (x)
i=1
Z
=
Pk (θ̃)|x̃ − s̃|−Q−k f˜(s)
|sn−k+2i |γi −1 ds̃
k
Q
|sn−k+2i |γi −1 ds̃.
i=1
ǫ<|x̃−s̃|<η
−f (x)
k
Q
Z
Pk (θ̃)|x̃ − s̃|−Q−k
ǫ<|x̃−s̃|<η
(4.3)
i=1
Passing to the limit of (4.3) as ǫ → 0 and polar coordinates, it can be easily
see that the second integral in the right-hand side is vanish.
Z
lim
ǫ→0
ǫ<|x̃−s̃|<η
= lim
k
Q
|sn−k+2i |γi −1 ds̃
Pk (θ̃)|x̃ − s̃|−Q−k f˜(s) − f (x)
i=1
Z
ǫ→0
ǫ<|x̃−s̃|<η
Pk (θ̃)|x̃ − s̃|−Q−k f˜(s)
k
Q
|sn−k+2i |γi −1 ds̃.
i=1
Hence, we get
lim RBk ǫ f (x) = ck,γ i1 (x̃, E ′ ) + i2 (x̃, G)
ǫ→0
n+k
where E ⊆ E ′ and G ⊆ Rk,+
\ E ′ . This completes the proof of the theorem.
HÖLDER WEIGHT ESTIMATES OF RIESZ-BESSEL BESSEL SINGULAR
INTEGRALS GENERATED BY A GENERALIZED SHIFT OPERATOR
83
n+k
We just need to the estimates for i1 (x̃; Ẽ) and i2 (x̃; G) when a.e. x̃ ∈ Rk,+
n+k
ν
and E ⊆ E ′ and G ⊆ Rk,+
\ E ′ . Firstly, let f ∈ Hα,β
and Pk (θ) be bounded
n
for θ ∈ Sk,+ . For 0 < ν < 1, 0 < α − ν < 1 and 0 < β + ν < n, then we prove
the absolute convergence of the following integrals
i1 (x̃; E ′ )
=
k Q
Pk (θ̃) ˜
sn−k+2i γi −1 ds̃
f (s) − f (x)
Q+k
|x̃ − s̃|
i=1
Z
E′
i2 (x̃, G)
=
Z
G
γi −1
Pk (θ̃) ˜ Qk f (s) i=1 sn−k+2i ds̃
|x̃ − s̃|Q+k
To estimate i1 (x̃; E ′ ) we use Corollary 3.3 which states that
i1 (x̃; E ′ )
Z
γi −1 Qk Pk (θ̃) ˜
f (s) − f (x)
ds̃
=
i=1 sn−k+2i
|x̃ − s̃|Q+k
E′
≤
Z
E′
Pk (θ̃) Q
f˜(s) − f (x) k sn−k+2i γi −1 ds̃
i=1
|x̃ − s̃|Q+k
Z
Qk
γi −1
ν
|x̃ − s̃|−Q−k ds̃
ρ−1 (x)(1 + |x|)−2ν A
≤ c||f ||Hα,β
i=1 |sn−k+2i |
E′
where ||Pk (θ̃)|| = sup |Pk (θ)| = A < ∞. Since
n
θ∈Sk,+
Z
E′
k
Q
|sn−k+2i |
i=1
γi −1
|x̃ − s̃|
−Q−k
ds̃ ≤
Pk
i=1
xn−k+i
2
ν
= η,
we conclude that
i1 (x̃; E ′ ) ≤ cA||f ||Ψν (x).
(4.4)
We use the Corollary 3.3 to get the i2 (x̃, G) estimate. Then we introduce the
84
I. Ekincioglu, C. Keskin and S. Er
following notation
Qk
ν−α
|sn−k+2i |γi −1 Pk
L(x̃, s̃) = i=1
(1 + |s̃|)−ℓ
i=1 |sn−k+i | + |sn−k+2i |
|x̃ − s̃|Q+k
ν−α
Pk
Qk
γi −1
i=1 |sn−k+i | + |sn−k+2i |
i=1 |sn−k+2i |
=
Q+k
Pk
Pk
Pk
|x′ − s′ | + i=1 xn−k+i + i=1 |sn−k+i | + i=1 |sn−k+2i |
−ℓ
Pk
1 + |s′ | + i=1 |sn−k+i | + |sn−k+2i |
×
Q+k .
Pk
Pk
Pk
|x′ − s′ | + i=1 xn−k+i + i=1 |sn−k+i | + i=1 |sn−k+2i |
(4.5)
Now we shall obtain the estimate i2 (x̃; G). In the case, let us introduce the
spaces
|s̃|
|s̃|
n+k
n+k
, D2 = s̃ ∈ R
:
< |x̃ − s̃| ≤ 3|s̃| ,
D1 = s̃ ∈ R
: |x̃ − s̃| <
2
2
D3
n+k
= s̃ ∈ R
: 3|s̃| < |x̃ − s̃| .
Thus i2 (x̃, G) can be written as
n+k
\ E′
≤ ||f || ||Pk (θ)||i2 x̃, Rk,+
= ||f || ||Pk (θ)|| i2 x̃, D1 + i2 x̃, D2 + i2 x̃, D3 .
|i2 (x̃, G)|
(4.6)
Now we get the estimates i2 x̃, D1 , i2 x̃, D2 and i2 x̃, D3 in (4.6). Let us
take s̃ ∈ D1 . We know that (1 + |x̃|) ≈ (1 + |s̃|). Hence we obtain
i2 (x̃, D1 )
≤ c.(1 + |x|)
×
×
Z∞
...
0Z
Z∞
0
−ℓ
...
0
Pk
|y ′ | +
≤ (1 + |x|)−ℓ
Z∞
0
Pk
i=1
|sn−k+2i |γi −1 ds′′′
|sn−k+i | + |sn+i |
i=1
Rn−k
k,+
i2 (x̃, D1 )
Z∞
Pk
i=1
Pk
i=1
ν−α
ds′′
|sn−k+i | + |sn−k+2i | + xn−k+i
xn−k+i
To get estimate i2 (x̃; D2 ), we define
i2 x̃, D2 =
Z
ν−α
= Ψν (x).
L(x̃, s̃)ds̃.
G∩D2
−Q−k
dy ′
HÖLDER WEIGHT ESTIMATES OF RIESZ-BESSEL BESSEL SINGULAR
INTEGRALS GENERATED BY A GENERALIZED SHIFT OPERATOR
85
|x|
Let us take first |x| ≥ 1 and s̃ ∈ D1 . We see that |x̃ − s̃| ≈ |s̃| and |s̃| ≥
.
4
Hence, we get
|x̃ − s̃| ≈ |s̃| + |x| ≈ |s̃| + 1.
We obtain
i2 x̃, D2 =
Z
Pk
i=1
D2
ν−α Qk
γi −1
|sn−k+i | + |sn−k+2i |
i=1 |sn−k+2i |
ds̃.
Q+k+ℓ
|s̃| + |x|
Suppose µ = (ν + β) + (1 + ν − α). By virtue of 0 < ν < 1, 0 < α − ν < 1, and
0 < β + ν < n), µ > 0. Taking into account that n + 2ν + ℓ = (n − 1) + µ + 2ν,
we obtain
i2 (x̃, D2 ) <
×
Z
Z∞
...
0
0
|y ′ | +
Rn−k
k,+
Z∞
Pk
i=1
Qk
i=1
|sn−k+2i |
γi −1
ds
′′′
Z∞
...
0
Z∞
0
|sn−k+i | + |sn−k+2i | + xn−k+i
Pk
i=1
|sn−k+i | + |sn−k+2i |
(n−1)+µ+2ν
ν−α
dy ′ < Ψν (x), (|x| ≥ 1).
Let use take second |x| < 1 and s̃ ∈ D2 . Then it follows from |s̃| ≥ 1,
|s̃| ≈ |s̃| + 1 ≈ |s̃| + 1 + |x| and |s̃| < 1, 1 + |s̃| ≈ 1 that |x̃ − s̃| ≈ |s̃| +
Pk
Pk
Pk
′
i=1 |xn−k+i | ≈ |s | +
i=1 |sn−k+i | + |sn−k+2i | +
i=1 xn−k+i .
Therefore, we deduce that
ν−α Qk
Pk
Z
γi −1
i=1 |sn−k+i | + |sn−k+2i |
i=1 |sn−k+2i |
i2 (x̃, D2 ) ≤
ds̃
P
Q+k+ℓ
k
′| +
|s
|s
|
+
|s
|
+
x
n−k+i
n−k+2i
n−k+i
i=1
{s̃∈Rn+k
k,+ : |s̃|<1}
ν−α Qk
Pk
Z
γi −1
i=1 |sn−k+i | + |sn−k+2i |
i=1 |sn−k+2i |
ds̃
+
Q+k+ℓ
|s̃|
+
1
+
|x|
n+k
{s̃∈Rk,+ : |s̃|≥1}
≤
Pk
i=1
ν−α
xn−k+i
+
1
< Ψν (x) (|x| < 1).
(1 + |x|)β+ν
Thus, we prove that
i2 (x̃, D2 ) ≤ Ψν (x).
(4.7)
Finally, i2 (x̃, D3 ) estimate can be proved by similar way. Thus we prove that
i2 x̃, G < Ψν (x)
and
ds′′
|i2 (x̃, G)| < ||f || ||Pk (θ)||Ψν (x).
(4.8)
Consequently, the four estimates show that the absolute convergence of integrals i1 (x̃, E ′ ) and i2 (x̃, G).
86
I. Ekincioglu, C. Keskin and S. Er
Theorem 4.2. Under the assumption (2.3), above if f ∈ Lp,ν , and f satisfies
the Hölder condition
|f (x) − f (s)| ≤ M |x − s|δ ,
(4.9)
ν < δ, then for the integral operator
RBk ǫ f (x) = ck
Z
y
)
Pk ( |y|
|y|Q+k
T y f (x)(y ′′ )γ dy
(4.10)
{y∈Rn
k,+ : |y|>ǫ}
the limit RBk f (x) = lim RBk ǫ f (x) exists, and this limit satisfies the inequalǫ→0
ity
| RBk f (x) − RBk f (s)| ≤ ck M A|x − s|δ .
(4.11)
Proof. We show first of all that the limit of RBk ǫ f (x) exists as ǫ → 0 and
equal to RBk f (x). The explicit formula of (4.10) shows that
Z
y −Q−k y
|y|
T f (x)(y ′′ )γ dy
RBk ǫ f (x) = I1 + I2 = ck
Pk
|y|
|y|>η
Z
y −Q−k y
+ck
Pk
|y|
T f (x) − T 0 f (x) (y ′′ )γ dy
|y|
ǫ<|y|<η
so that the Riesz-Bessel singular integral operator can be written as a sum of
two terms. We consider the term I2 . By Lemma 2.1, we have
Z
k
Q
Pk (θ̃) ˜
γi −1
I 2 = ck
sn−k+2i
ds̃.
f
(s)
−
f
(x)
|x̃ − s̃|Q+k
i=1
ǫ<|x̃−s̃|<η
From the condition (4.9) and the triangle inequality we obtain for I2 the
estimate
|I2 | ≤ ck M A
R
ǫ<|x̃−s̃|<η
k
Q
ck M A δ
1
|sn−k+2i |γi −1 ds̃ ≤
(η − ǫδ ),
Q+k−δ
|x̃ − s̃|
δ
i=1
whence there exits the limit of the function RBk ǫ f (x) as ǫ → 0.
We now state the main theorem.
Theorem 4.3. Let Pk (θ) satisfy condition (2.2) and (2.3). If 0 < ν < δ ≤ 1
and 0 < α − ν < 1, β + ν < n, then Riesz-Bessel Singular integral operator
ν
generated by generalized shift operator is bounded for each f ∈ Hα,β
(Rnk,+ ) in
ν
the weighted space Hαβ .
HÖLDER WEIGHT ESTIMATES OF RIESZ-BESSEL BESSEL SINGULAR
INTEGRALS GENERATED BY A GENERALIZED SHIFT OPERATOR
87
Proof. By virtue of Theorem 4.1, from (4.4) and (4.8) we obtain
| RBk f (x)| ≤ ck,γ |i1 (x̃, E ′ )| + |i2 (x̃, G)| .
By virtue of Lemma 3.2, to prove the inequality it suffices to show that ∀x ∈
Pk xn−k+i
n
Rk+
and |h| ≤ i=1
8
|(RBk )f (x) − (RBk )f (x + h)| ≤ ck .M.A|h|ν
where ck is independent of x and h.
(RBk )f (x) − (RBk )f (x + h)
Z
= ck
y
|y|
Pk
Rn
k,+
−ck
Z
Pk
y
|y|
Rn
k+
|y|−Q−k T y f (x) (y ′′ )γ dy
|y|−Q−k T y f (x + h) (y ′′ )γ dy.
By Lemma 2.1,
(RBk )f (x) − (RBk )f (x + h)
k
Pk θ̃
˜(s) Q |sn−k+2i |γi −1 ds̃
f
= ck,γ
|x̃ − s̃|Q+k
i=1
Rn
k+
Z
k
Q
Pk θ˜1
−ck,γ
|sn−k+2i |γi −1 ds̃
f˜(s)
Q+k
^
i=1
|
x
+
h
−
s̃|
Rn
k+
Z
k
Q
^
= ck,γ
|sn−k+2i |γi −1 ds̃
K(s̃, x̃) − K(s̃, x
+ h) f˜(s) − f (x)
Z
+ck,γ
E′ Z
′
Rn+k
k+ \E
i=1
^
K(s̃, x̃) − K(s̃, x
+ h) f˜(s)
k
Q
|sn−k+2i |γi −1 ds̃
i=1
where
Pk θ̃
.
K(s̃, x̃) =
|x̃ − s̃|Q+k
Pk xn−k+i
n+k
n+k
it can be easily seen that
For x̃ ∈ Rk,+
, s̃ ∈ Rk,+
\ E ′ and |h| < i=1
8
^
|x̃ − s̃| ≈ |x
+ h − s̃|.
(4.12)
It is well known that if |x − (x + h)| ≤ max{|x|, |x + h|} then
K(s̃, x̃) − K(s̃, x
^
+ h) ≤
ck .A.|h|δ
.
|x̃ − s̃|Q+k+δ
(4.13)
88
I. Ekincioglu, C. Keskin and S. Er
^
^
Suppose E1 (x) = E(x̃, 2|h|), E2 (x) = E(x
+ h, 3|h|) and E3 (x) = E(x
+ h,
Pk
i=1
|h|). Obviously E1 (x) ⊂ E2 (x) ⊂ E3 (x) ⊂ E ′ . Subject to (2.4) and (2.2), it
can easily prove that
(RBk )f (x) − (RBk )f (x + h) = I1 (x; h) + I2 (x; h) + I3 (x; h) + I4 (x; h),
where
I1 (x; h)
Z
=
+
I2 (x; h)
=−
E2
I3 (x; h)
Z
=
i=1
i=1
E ′ \E3
I4 (x; h)
k
Q
|sn−k+2i |γi −1 ds̃,
K(s̃, x̃) f˜(s) − f (x)
k
Q
^
|sn−k+2i |γi −1 ds̃,
K(s̃, x
+ h) f˜(s) − f (x)
Z
=−
E ′ \E2
E2
Z
Z
′
Rn+k
k,+ \E
k
Q
^
|sn−k+2i |γi −1 ds̃,
K(s̃, x
+ h) f˜(s) − f (x)
i=1
k
Q
^
|sn−k+2i |γi −1 ds̃.
K(s̃, x̃) − K(s̃, x
+ h) f˜(s)
i=1
Let us |Ii (x; h)| for i = 1, 2, 3, 4. Taking into account ii) of Corollary 3.3 and
also (4.13), we get
I1 (x; h)
=
Z
≤
Z2
E
|I1 (x; h)|
E2
+
Z
k
Q
|sn−k+2i |γi −1 ds̃,
K(s̃, x̃) f˜(s) − f (x)
Z
E ′ \E2
+
E ′ \E2
i=1
k
|Pk (θ̃)| −1
|s − x̃| ν Q
ρ (x)
|sn−k+2i |γi −1 ds̃
Q+k
2
|s̃ − x̃|
(1 + |x|)
i=1
≤ Aρ−1 (x)(1 + |x|)−2ν
Z
E2
+
Z
E ′ \E
k
Q
|sn−k+2i |γi −1
i=1
^
|x
+ h − s̃|Q+k−ν
ds̃
2
< Aρ−1 (x)(1 + |x|)−2ν |h|ν
(4.14)
xn−k+i
−
2
HÖLDER WEIGHT ESTIMATES OF RIESZ-BESSEL BESSEL SINGULAR
INTEGRALS GENERATED BY A GENERALIZED SHIFT OPERATOR
89
The following expressions are proved similarly:
I2 (x; h)
|I2 (x; h)|
=−
≤
Z
Z E2
E2
k
Q
^
|sn−k+2i |γi −1 ds̃,
K(s̃, x
+ h) f˜(s) − f (x)
i=1
k
Q
|Pk (θ̃)|
|sn−k+2i |γi −1 ds̃
ρ−1 (x)(1 + |x|)−2ν
Q+k−ν
^
i=1
|x + h − s̃|
< Aρ−1 (x)(1 + |x|)−2ν
Z
E2
k
Q
|sn−k+2i |γi −1
i=1
^
|x
+ h − s̃|Q+k−ν
ds̃
< Aρ−1 (x)(1 + |x|)−2ν |h|ν .
(4.15)
I3 (x; h)
Z
=−
E ′ \E3
|I3 (x; h)|
≤
Z
E ′ \E3
−1
< Aρ
k
Q
^
|sn−k+2i |γi −1 ds̃,
K(s̃, x
+ h) f˜(s) − f (x)
i=1
k
Q
|Pk (θ̃)|
|sn−k+2i |γi −1 ds̃
ρ−1 (x)(1 + |x|)−2ν
Q+k−ν
^
i=1
|x + h − s̃|
(x)(1 + |x|)
−2ν
k
Q
Z
E ′ \E
|sn−k+2i |γi −1
i=1
^
|x
+ h − s̃|Q+k−ν
ds̃
3
< Aρ−1 (x)(1 + |x|)−2ν |h|ν .
(4.16)
90
I. Ekincioglu, C. Keskin and S. Er
According to (4.12) and (4.13), we obtain
I4 (x; h)
Z
=
′
Rn+k
k,+ \E
|I4 (x; h)|
Z
≤
′
Rn+k
k,+ \E
< A||f˜||
k
Q
^
|sn−k+2i |γi −1 ds̃
K(s̃, x̃) − K(s̃, x
+ h) f˜(s)
k
Q
^
|sn−k+2i |γi −1 ds̃
|K(s̃, x̃) − K(s̃, x
+ h)||f˜(s)|
i=1
Z
Rn+k \E ′
×
i=1
Pk,+
k
k
Q
|h|δ
|sn−k+2i |γi −1
|x̃ − s̃|Q+k+δ i=1
|sn−k+i | + |sn−k+2i |)ν−α
ℓ ds̃
Pk
1 + |s′ | + ( i=1 |sn−k+i | + |sn−k+2i |)
(
i=1
Pk
n+k
≤ A|h|δ ||f˜|| i=1 (xn−k+i )−δ i2 (x, Rk,+
\ E′)
δ
ν
|h|
|h|
Ψν (x) ≤ A||f˜|| Pk
Ψν (x)
≤ A||f˜|| Pk
i=1 xn−k+i
i=1 xn−k+i
≤ A||f˜||ρ−1 (x)(1 + |x|)−2ν |h|ν
(4.17)
Thus, the estimates (4.14)-(4.17) show that the theorem is proved.
Acknowledgement. We are very grateful to the referee for his careful
review and helpful comments which contributed to improve the paper.
References
[1] Klyuchantsev, M.I., On singular integrals generated by the generalized
shift operator I, Sibirsk. Math. Zh., 11(1970), 810–821; translation in
Siberian Math. J., 11(1970), 612–620.
[2] Kipriyanov, I.,A., and Klyuchantsev, M.I., On singular integrals generated
by the generalized shift operator II, Sibirsk. Mat. Zh., 11(1970), 1060–
1083; translation in Siberian Math. J., 11(1970), 787–804.
[3] Aliev, I.A. and Gadzhiev, A.D., Weighted estimates for multidimensional singular integrals generated by the generalized shift operator, Mat.
Sb., 183(1992), no. 9, 45–66; translation in Russian Acad. Sci. Math.
77(1994), no. 1, 37–55.
HÖLDER WEIGHT ESTIMATES OF RIESZ-BESSEL BESSEL SINGULAR
INTEGRALS GENERATED BY A GENERALIZED SHIFT OPERATOR
91
[4] B.M. Levitan, Bessel function expansions in series and Fourier integrals,
Uspekhi Mat. Nauk., 42, No 2, 102-143, (1951). (Russian)
[5] S.K. Abdullayev, Multidimensional Singular Integral equations in wiegted
Hölder spaces. Inst. of Fhysics of AS of Azerb. SSR, Preprint, No.8, p.50.
(Russian), (1988).
[6] S.K. Abdullayev, Multidimensional Singular Integral equations in wiegted
Hölder spaces with the weigted degenerated on noncompat sets. Soviet
Math. Dokl., 308, No.6, p.1289-1292. (Russian), (1989).
[7] Ekincioglu, I. and Serbetci, A., On weighted estimates of high order RieszBessel transformations generated by the generalized shift operator, Acta
Mathematica Sinica, English Series, 21(2005), no.1, 53–64.
[8] A.D. Gadjiev and E.V. Guliyev, Two-weighted inequality for singular
integrals in Lebesgue spaces, associated with the Laplace-Bessel differential
operator. Proceedings of A. Razmadze Mathematical Institute, 138, 1-15,
(2005).
[9] V.S. Guliev, Sobolev theorems for B–Riesz potentials. Dokl. RAN, 358,
No. 4, p.450-451, (1998). (Russian)
[10] I. Ekincioglu, A. Serbetci, On weighted estimates of high order RieszBessel transformations generated by the generalized shift operator, Acta
Mathematica Sinica, 21, 1, 53-64, (2005).
[11] I.A. Kiprijanov, Fourier-Bessel transformations and imbedding theorems.
Trudy Math. Inst. Steklov, 89, p. 130-213, (1967).
[12] B. Muckenhoupt and E.M. Stein, Classical expansions and their relation
to conjugate harmonic functions, Trans. Amer. Math. Soc., 118, p. 17-92,
(1965).
[13] E.M. Stein, Singular Integrals And Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N.J., (1970).
[14] K. Stempak, The Littlewood-Paley theory for the Fourier-Bessel transform. Mathematical Institute of Wroslaw (Poland), Preprint No. 45,
(1985).
[15] K. Trimeche, Inversion of the Lions transmutation operators using generalized wavelets, Applied and Computational Harmonic Analysis, 4, 97112, (1997).
92
I. Ekincioglu, C. Keskin and S. Er
[16] R. Kapanadze, Singular Integral Operators on Manifolds with a Boundary. Georgian Math. J., 1(94), No. 4, p.377-393, (1993).
[17] S.K. Abdullayev, Hölder Weighted Estimates of Singular Integrals Generated by Generalized Shift Operator. Transactions of NAS Of Azerb.,
p.9-18, (2003).
[18] I. Ekincioglu, the Boundedness of High Order Riesz-Bessel Transformations Generated by Generalized Shift Operator in weigted Lp with General
Weights, Acta Appl. Math. 109, (2010), 591-598.
Department of Mathematics,
Dumlupınar University, Kütahya Turkey
ekinci@dumlupinar.edu.tr
Department of Mathematics,
Dumlupınar University, Kütahya Turkey
ckeskin@dumlupinar.edu.tr