ON WEIGHTED INEQUALITIES FOR CERTAIN
FRACTIONAL INTEGRAL OPERATORS
R. K. RAINA
Received 31 December 2005; Revised 8 May 2006; Accepted 9 May 2006
This paper considers the modified fractional integral operators involving the Gauss hypergeometric function and obtains weighted inequalities for these operators. Multidimensional fractional integral operators involving the H-function are also introduced.
Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1. Introduction and preliminaries
Tuan and Saigo [7] introduced the multidimensional modified fractional integrals of order α (Re(α) > 0) by
α
f (x) =
X+;n
1
Dn
Γ(α + 1)
(−1)n n
X−α ;n f (x) =
D
Γ(α + 1)
Rn+
min
Rn+
α
x
x1
,..., n − 1
t1
tn
x
x
1 − max 1 ,..., n
t1
tn
+
f (t)dt,
(1.1)
α
+
f (t)dt,
where Rn+ = {(t1 ,...,tn ) | ti > 0 (i = 1,...,n)}, ϕ+ (x) is a real-valued function defined in
terms of the function ϕ(x) by
⎧
⎪
⎨ϕ(x),
ϕ(x) > 0,
⎩0,
ϕ(x) ≤ 0,
ϕ+ (x) = ⎪
(1.2)
and Dn denotes the derivative operator ∂n /∂x1 ,...,∂xn .
The operators in (1.1) provide multidimensional generalizations to the well-known
one-dimensional Riemann-Liouville and Weyl fractional integral operators defined in [5]
(see also [1]). The paper [7] considers several formulas and interesting properties of (1.1).
By invoking the Gauss hypergeometric function 2 F1 (α,β;γ;x), the following generalizations of the multidimensional modified integral operators (1.1) of order α (Re(α) > 0)
Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2006, Article ID 79196, Pages 1–7
DOI 10.1155/IJMMS/2006/79196
2
Inequalities for certain integral operators
were studied in [6]:
α,β,γ
S+;n f (x) =
α,β,γ
S−;n
α
xn
x1
,...,
−1
t1
tn
Rn+
+
x
x1
·2 F1 α + β,α + η;1 + α;1 − min
,..., n
f (t)dt,
t1
tn
(1.3)
xn α
x1
1
− max
,...,
t1
tn +
Rn+
xn
x1
·2 F1 α + β, −η;1 + α;1 − max
,...,
f (t)dt.
t1
tn
(1.4)
1
Dn
Γ(α + 1)
(−1)n n
f (x) =
D
Γ(α + 1)
min
For β = −α, the operators (1.3) and (1.4) reduce to the modified integral operators deα f (x) and X α f (x) defined
fined in (1.1), respectively. In [8], the integral operators X+;n
−;n
n
on the space ᏹγ (R+ ) are shown to satisfy some L p − Lq weighted inequalities. The space
ᏹγ (Rn+ ) represents the space of functions f which are defined on Rn+ , and are entire functions of exponential type (see [7]). The present paper is devoted to finding inequalities for
the generalized multidimensional modified integral operators (1.3) and (1.4) by making
use of the inequality stated in [8] (which was established with the aid of Pitt’s inequality).
Multidimensional operators have also been studied in [3, 4].
2. Inequalities for operators (1.3) and (1.4)
If (᏷ f )(x) denotes the integral operator
(᏷ f )(x) =
Rn+
k(xy) f (y)d y,
(2.1)
then following [8], we have
1
k(xy) f (y)d y =
n
n
(2πi)
R+
(1/2)
k∗ (s) f ∗ (1 − s)x−s ds,
(2.2)
where the integral over (1/2) stands for the multiple integral
(1/2)
=
1/2+i∞
1/2−i∞
···
1/2+i∞
1/2−i∞
,
(2.3)
and k∗ (s) and f ∗ (1 − s) are the Mellin transforms of the functions k(y) and f (y), respectively. It is proved in [8] that if
∗ k (s) ≤ C |s|−α s ∈ 1 , α ≥ 0 ,
2
(2.4)
then there holds the inequality
|log y − t |−b y 1/2−1/r (᏷ f )(y)
Lr (Rn+ )
≤ C |log y − t |d y 1/2−1/q f (y)Lq (Rn+ ) ,
(2.5)
R. K. Raina 3
or equivalently,
|log y − t |−b y 1/2−1/r (᏷ f )(y)
Lr (Rn+ )
a−b+n(1/r −1/q) 1/2−1/q
≤ C |log y − t y
f (y)Lq (Rn+ ) ,
(2.6)
for all t ∈ Rn+ , provided that
⎡
⎤
2 α 1
b
α 1 α 1
1
⎢max r , n + r − 1 ≤ n ≤ min 0, n − q , n + q − 1 + r ,⎥
⎢
⎥
⎢
⎢
⎣
1 1
max 0,n + − 1
r q
≤ α ≤ n, 1 < q ≤ r < ∞.
⎥.
⎥
⎦
(2.7)
In the paper [6], it was established that if
Re(α) > 0,
Re h j <
1
2
n
( j = 1,...,n),
Re h j <
j =1
n
+ min Re(β),Re(η) ,
2
(2.8)
α,β,γ
then the operator xh S+;n x−h f (x) is a homeomorphism of the space ᏹ1/2 (Rn+ ) onto itself,
and
α,β,γ
xh S+;n x−h f (x)
1
=
(2πi)n
Γ β + n − nj=1 h j − nj=1 s j Γ η + n − nj=1 h j − nj=1 s j
f ∗ (s)x−s ds.
n
n
n
n
(1/2) Γ n −
j =1 h j −
j =1 s j Γ α + β + η + n −
j =1 h j −
j =1 s j
(2.9)
We note that
Γ β + n − nj=1 h j − nj=1 s j Γ η + n − nj=1 h j − nj=1 s j
= O |s|−α ,
Γ n − nj=1 h j − nj=1 s j Γ α + β + η + n − nj=1 h j − nj=1 s j
(2.10)
so we can apply the inequality (2.6) to the multidimensional operator defined by (2.9),
which leads to
−h
|log y − t |−b y 1/2−1/r+h Sα,β,γ
f (y)Lr (Rn+ )
+;n y
≤ C |log y − t |a−b+n(1/r −1/q) y 1/2−1/q f (y)Lq (Rn+ ) ,
(2.11)
valid for all t ∈ Rn+ , provided that the constraints (2.7) and (2.8) are satisfied. On the
other hand, (see [6]) if
Re(α) > 0,
Re h j >
1
2
( j = 1,...,n),
n
j =1
Re h j >
n
+ Re(β − η) − 1,
2
(2.12)
4
Inequalities for certain integral operators
α,β,γ
then the operator xh S−;n x−h f (x) is a homeomorphism of the space ᏹ1/2 (Rn+ ) onto itself,
and we obtain
α,β,γ
xh S−;n x−h f (x)
1
=
(2πi)n
Γ 1 − n + nj=1 h j + nj=1 s j Γ 1 − β + η − n + nj=1 h j + nj=1 s j
n
n
n
n
(1/2) Γ 1 − β − n+ j =1 h j +
j =1 s j Γ 1 + α + η − n+ j =1 h j +
j =1 s j
× f ∗ (s)x−s ds.
(2.13)
By noting the estimate that
Γ 1 − n + nj=1 h j + nj=1 s j Γ 1 − β + η − n + nj=1 h j + nj=1 s j
= O |s|−α ,
n
n
n
n
Γ 1 − β − n + j =1 h j + j =1 s j Γ 1 + α + η − n + j =1 h j + j =1 s j
(2.14)
we again apply the inequality (2.6) to the multidimensional operator defined by (2.13) to
get
−h
|log y − t |−b y 1/2−1/r+h Sα,β,γ
f (y)Lr (Rn+ )
−;n y
≤ C t |log y − t |a−b+n(1/r −1/q) y 1/2−1/q f (y)Lq (Rn+ ) ,
(2.15)
valid for all t ∈ Rn+ , provided that the constraints (2.7) and (2.12) are satisfied.
3. Classes of multidimensional operators
We introduce the following classes of multidimensional modified fractional integral operators involving the well-known H-function [2, Section 8.3] (see also [1, page 343])
defined by
(aP ,αP )
(a1 ,α1 ),...,(aP ,αP )
M,N
HM,N
P,Q,+;n (bQ ,βQ ) f (x) = HP,Q,+;n (b1 ,β1 ),...,(bQ ,βQ ) f (x)
= Dn
M,N
HP,Q
min
n
R+
x x1
(aP ,αP )
,..., n (b
f (t)dt,
Q ,βQ )
t1
tn
(3.1)
(aP ,αP )
(a1 ,α1 ),...,(aP ,αP )
M,N
HM,N
P,Q,−;n (bQ ,βQ ) f (x) = HP,Q,−;n (b1 ,β1 ),...,(bQ ,βQ ) f (x)
= (−1)n Dn
M,N
HP,Q
max
n
R+
x x1
(aP ,αP )
,..., n (b
f (t)dt,
Q ,βQ )
t1
tn
(3.2)
where we assume that the parameters of the H-function involved in (3.1) and (3.2) satisfy
the existence conditions as given in [2].
The special cases of the operators of interest in this paper are the operators which
emerge from (3.1) and (3.2) in the case when N = 0, P = M, Q = M, and the parameters α1 = α2 = · · · = αm = 1, and β1 = β2 = · · · = βm = 1. Thus, we have the following
multidimensional fractional integral operators (defined in terms of Meijer’s G-function)
R. K. Raina 5
(see [6]):
(am );(bm )
(a1 ,...,am );(b1 ,...,bm )
G+;n
f (x) = G+;n
f (x)
n
=D
Rn+
Gm,0
m,m
x (am )
x
min 1 ,..., n f (t)dt,
t1
tn (bm )
(a1 ,...,am );(b1 ,...,bm )
m );(bm )
G(a
f (x) = G+;n
f (x)
−;n
n
n
= (−1) D
Rn+
(3.3)
x (am )
x
max 1 ,..., n f (t)dt.
t1
tn (bm )
Gm,0
m,m
By setting the parameters
m = 2,
a1 = 1 − β,
a2 = 1 − η,
a1 = 1 − β,
a2 = 1 + α + η,
b1 = 1 − α − β − η,
b2 = 0,
(3.4)
b1 = 1 − β + η,
b2 = 0,
(3.5)
in (3.1), and
m = 2,
in (3.2), and noting the relation (see [1, equation (1.1.18), page 18] )
a
1 2
G2,0
2,2 σ b1 ,b2 =
,a
σ b2 (1 − σ)a1 +a2 −b1 −b2 −1
Γ(a1 + a2 − b1 − b2 )
·2 F1 a2 − b1 ,a1 − b1 ;a1 + a2 − b1 − b2 ;1 − σ
(3.6)
(σ < 1),
we observe the following relationships:
(1−β,1−η);(1−α−β−η,0)
G+;n
α,β,γ
f (x) = (−1)α S+;n f (x),
(1−β,1+α+η);(1−β+η,0)
G−;n
(3.7)
α,β,γ
f (x) = S−;n f (x),
in terms of the multidimensional modified fractional integral operators (1.3) and (1.4).
We state below two useful lemmas concerning the multidimensional Mellin transform
of the functions f (max[x1 ,... ,xn ]) and f (min[x1 ,... ,xn ]) (see [3, 6]).
Lemma 3.1. Let Re(s j ) > 0 ( j = 1,...,n) and let τ s·1−1 f (τ) ∈ L1 (R+ ), then
Rn+
!
"
xs−1 f max x1 ,...,xn dx =
|s|
s1
f ∗ |s| ,
(3.8)
where s1 denotes the product s1 ,...,sn , and |s| = s1 + · · · + sn .
Lemma 3.2. Let Re(s j ) < 0 ( j = 1,...,n) and let τ s·1−1 f (τ) ∈ L1 (R+ ), then
Rn+
!
"
xs−1 f min x1 ,...,xn dx = (−1)n−1
|s|
s1
f ∗ |s| .
(3.9)
6
Inequalities for certain integral operators
Making use of (3.1), we have
(aP ,αP ) −s
HM,N
(x) = Dn
P,Q,+;n (bQ ,βQ ) x
M,N
HP,Q
min
n
R+
n 1−s
=D x
Rn+
t
s −2
M,N
HP,Q
x (aP ,αP ) −s
x1
,..., n t dt
t1
tn (bQ ,βQ )
(3.10)
(aP ,αP ) min t1 ,...,tn dt.
(bQ ,βQ )
Applying now (3.9) of Lemma 3.2, and the following result giving the Mellin transform
of the H-function [1, equation (E.20), page 348], namely,
M,N
HP,Q
#N
#M (a ,α ) ∗
j =1 Γ b j + β j s
i=1 Γ(1 − ai − αi s
P P
x
(s) = #P
#Q
(bQ ,βQ )
j =M+1 Γ 1 − b j − β j s
i=N+1 Γ(ai + αi s)
× − min
1≤ j ≤M
1 − Re ai
< Re(s) < min
1≤i≤N
αi
Re b j
βj
,
(3.11)
we obtain
(aP ,αP ) −s
HM,N
(x)
P,Q,+;n (bQ ,βQ ) x
#M
Γ 1 + n − |s|
= #P
Γ n − |s|
j =1 Γ
b j + β j |s| − n
i=N+1 Γ ai + αi |s| − n
#N
#Q
i =1 Γ
1 − ai − αi |s| − n
j =M+1 Γ 1 − b j − β j |s| − n
x−s .
(3.12)
Similarly, by using the multidimensional operator (3.2), we obtain
(aP ,αP ) −s
HM,N
(x)
P,Q,−;n (bQ ,βQ ) x
=
(−1)n Γ 1 − n + |s|)
Γ − n + |s|)
#P
#M
j =1 Γ
b j + β j |s| − n
i=N+1 Γ ai + αi |s| − n
#N
#Q
i =1 Γ
1 − ai − αi |s| − n
j =M+1 Γ 1 − b j − β j |s| − n
x−s .
(3.13)
The result (3.13) on specializing the parameters in accordance with (3.5) yields the formula [7, equation (3.6), page 148] involving the multidimensional modified integral operator (1.4). Similarly, we can deduce a result from (3.12) which involves the multidimensional modified integral operator (1.3).
Acknowledgments
The author is thankful to the referees for suggestions. The present investigation was supported by All India Council for Technical Education (AICTE), Government of India, New
Delhi.
R. K. Raina 7
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, Modified fractional integral operators in L p space with power-logarithmic weight, Mathe[8]
matica Japonica 49 (1999), no. 1, 145–150.
R. K. Raina: Department of Mathematics, College of Technology and Engineering, MP University of
Agriculture and Technology, Udaipur 313 001, Rajasthan, India
E-mail address: rainark 7@hotmail.com