Journal of Inequalities in Pure and
Applied Mathematics
HARDY TYPE INEQUALITIES FOR INTEGRAL TRANSFORMS
ASSOCIATED WITH A SINGULAR SECOND ORDER
DIFFERENTIAL OPERATOR
volume 7, issue 1, article 38,
2006.
M. DZIRI AND L.T. RACHDI
Institut Supérieur de Comptabilité et d’Administration des Entreprises
Campus Universitaire de la Manouba
la Manouba 2010, Tunisia.
EMail: moncef.dziri@iscae.rnu.tn
Department of Mathematics
Faculty of Sciences of Tunis
1060 Tunis, Tunisia.
EMail: lakhdartannech.rachdi@fst.rnu.tn
Received 04 November, 2004;
accepted 17 January, 2006.
Communicated by: S.S. Dragomir
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c 2000 Victoria University
ISSN (electronic): 1443-5756
211-04
Quit
Abstract
We consider a singular second order differential operator ∆ defined on ]0, ∞[.
We give nice estimates for the kernel which intervenes in the integral transform
of the eigenfunction of ∆. Using these results, we establish Hardy type inequalities for Riemann-Liouville and Weyl transforms associated with the operator ∆.
2000 Mathematics Subject Classification: 44Xxx, 44A15.
Key words: Hardy type inequalities, Integral transforms, Differential operator.
Contents
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
M. Dziri and L.T. Rachdi
1
2
3
4
5
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
The Eigenfunctions of the Operator ∆ . . . . . . . . . . . . . . . . . . . 9
The Kernel h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Hardy Type Operators Tϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
The Riemann - Liouville and Weyl Transforms Associated
with the Operator ∆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
References
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
1.
Introduction
In this paper we consider the differential operator on ]0, ∞[, defined by
∆=
d2
A0 (x) d
+
+ ρ2 ,
dx2
A(x) dx
where A is a real function defined on [0, ∞[ , satisfying
A(x) = x2α+1 B(x); α > −
1
2
and B is a positive, even C ∞ function on R such that B(0) = 1, and ρ ≥ 0. We
suppose that the function A satisfies the following assumptions
i) A(x) is increasing, and lim+∞ A(x) = +∞.
ii)
A0 (x)
A(x)
is decreasing and lim+∞
A0 (x)
A(x)
B 0 (x)
B(x)
= e−δx F (x),
M. Dziri and L.T. Rachdi
Title Page
Contents
= 2ρ.
iii) there exists a constant δ > 0, satisfying
 0
B (x)
2α+1
−δx

 B(x) = 2ρ − x + e F (x), for ρ > 0,


Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
for ρ = 0,
where F is C ∞ on ]0, ∞[, bounded together with its derivatives on the
interval [x0 , ∞[, x0 > 0.
JJ
J
II
I
Go Back
Close
Quit
Page 3 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
This operator plays an important role in harmonic analysis, for example,
many special functions (orthogonal polynomials,...) are eigenfunctions of operators of the same type as ∆.
The Bessel and Jacobi operators defined respectively by
∆α =
d2
2α + 1 d
+
;
2
dx
x dx
α>−
1
2
and
∆α,β
d
d2
= 2 + ((2α + 1) coth x + (2β + 1) tanh x)
+ (α + β + 1)2 ,
dx
dx
1
α≥β>− ,
2
are of the type ∆, with
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
M. Dziri and L.T. Rachdi
Title Page
A(x) = x
2α+1
;
ρ = 0,
respectively
A(x) = sinh2α+1 x cosh2β+1 x;
ρ = α + β + 1.
Also, the radial part of the Laplacian - Betrami operator on the Riemannian
symmetric space, is of type ∆.
The operator ∆ has been studied from many points of view ([1], [7], [13],
[14], [15], [16]).In particular, K. Trimèche has proved in [15] that the differential equation
∆u(x) = −λ2 u(x), λ ∈ C
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 4 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
has a unique solution on [0, ∞[, satisfying the conditions u(0) = 1, u0 (0) = 0.
We extend this solution on R by parity and we denote it by ϕλ . He has also
proved that the eigenfunction ϕλ has the following Mehler integral representation
Z x
ϕλ (x) =
k(x, t) cos λtdt,
0
where the kernel k(x, t) is defined by
1
1
1
k(x, t) = 2h(x, t) + Cα A− 2 (x)x 2 −α (x2 − t2 )α− 2 ,
with
1
h(x, t) =
Π
Z
0<t<x
∞
ψ(x, λ) cos(λt)dλ,
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
M. Dziri and L.T. Rachdi
0
2Γ(α + 1)
Cα = √
,
ΠΓ(α + 12 )
Title Page
Contents
and
∀ λ ∈ R, x ∈ R;
ψ(x, λ) = ϕλ (x) − x
where
α+ 12
A
− 12
(x)jα (λ x),
Jα (z)
zα
and Jα is the Bessel function of the first kind and order α ([8]).
The Riemann - Liouville and Weyl transforms associated with the operator
∆ are respectively defined, for all non-negative measurable functions f by
Z x
R(f )(x) =
k(x, t)f (t)dt
jα (z) = 2α Γ(α + 1)
0
JJ
J
II
I
Go Back
Close
Quit
Page 5 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
and
Z
W(f )(t) =
∞
k(x, t)f (x)A(x)dx.
t
These operators have been studied on regular spaces of functions. In particular,
in [15], the author has proved that the Riemann-Liouville transform R is an
isomorphism from E ∗ (R) (the space of even infinitely differentiable functions
on R) onto itself, and that the Weyl transform W is an isomorphism from D∗ (R)
(the space of even infinitely differentiable functions on R with compact support)
onto itself.
The Weyl transform has also been studied on Schwarz space S∗ (R) ([13]).
Our purpose in this work is to study the operators R and W on the spaces
p
L ([0, ∞[, A(x)dx) consisting of measurable functions f on [0, ∞[ such that
Z ∞
p1
p
< ∞; 1 < p < ∞.
||f ||p,A =
|f (x)| A(x) dx
0
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
M. Dziri and L.T. Rachdi
Title Page
Contents
The main results of this paper are the following Hardy type inequalities
• For ρ > 0 and p > max (2, 2α + 2), there exists a positive constant Cp,α
such that for all f ∈ Lp ([0, ∞[, A(x)dx),
||R(f )||p,A ≤ Cp,α ||f ||p,A
(1.1)
(1.2)
II
I
Go Back
Close
Quit
0
and for all g ∈ Lp ([0, ∞[, A(x)dx),
1
W(g)
A(x)
JJ
J
Page 6 of 40
≤ Cp,α ||g||p0 ,A ,
p0 ,A
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
where p0 =
p
.
p−1
• For ρ = 0 and p > 2α + 2 there exists a positive constant Cp,α such that
(1.1) and (1.2) hold.
In ([5], [6]) we have obtained (1.1) and (1.2) in the cases
A(x) = x2α+1 ,
α>−
1
2
respectively
A(x) = sinh2α+1 (x) cosh2β+1 (x);
1
α≥β>− .
2
This paper is arranged as follows. In the first section, we recall some properties of the eigenfunctions of the operator ∆. The second section deals with the
study of the behavior of the kernel h(x, t). In the third section, we introduce the
following integral operator
Z x t
Tϕ (f )(x) =
ϕ
f (t)ν(t)dt
x
0
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
M. Dziri and L.T. Rachdi
Title Page
Contents
JJ
J
II
I
Go Back
where
• ϕ is a measurable function defined on ]0, 1[,
• ν is a measurable non-negative function on ]0, ∞[ locally integrable.
Close
Quit
Page 7 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
Then we give the criteria in terms of the function ϕ to obtain the following
Hardy type inequalities for Tϕ ,
for all real numbers, 1 < p ≤ q < ∞, there exists a positive constant Cp,q
such that for all non-negative measurable functions f and g we have
Z
∞
q
(Tϕ (f (x))) µ(x)dx
0
1q
Z
≤ Cp,q
∞
p
(f (x)) ν(x)dx
p1
.
0
In the fourth section, we use the precedent results to establish the Hardy type
inequalities (1.1) and (1.2) for the operators R and W.
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
M. Dziri and L.T. Rachdi
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 8 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
2.
The Eigenfunctions of the Operator ∆
As mentioned in the introduction, the equation
(2.1)
∆u(x) = −λ2 u(x),
λ∈C
has a unique solution on [0, ∞[, satisfying the conditions u(0) = 1, u0 (0) = 0.
We extend this solution on R by parity and we denote it ϕλ . Equation (2.1)
possesses also two solutions φ∓λ linearly independent having the following behavior at infinity φ∓λ (x) ∼ e(∓λ−ρ)x . Then there exists a function c such that
ϕλ (x) = c(λ)φλ (x) + c(−λ)φ−λ (x).
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
In the case of the Bessel operator ∆α , the functions ϕλ , φλ and c are given
respectively by
M. Dziri and L.T. Rachdi
(2.2)
jα (λx) = 2α Γ(α + 1)
Jα (λx)
, λx 6= 0,
(λx)α
Title Page
Contents
Kα (iλx)
,
kα (iλx) = 2 Γ(α + 1)
(iλx)α
α
1 Π
λx 6= 0,
1
c(λ) = 2α Γ(α + 1)e−i(α+ 2 ) 2 λ−(α+ 2 ) ,
λ > 0,
where Jα and Kα are respectively the Bessel function of first kind and order α,
and the MacDonald function of order α.
In the case of the Jacobi operator ∆α,β , the functions ϕλ , φλ and c are respectively
1
1
α,β
2
ϕλ (x) = 2 F1
(ρ − iλ), (ρ + iλ), (α + 1), − sinh (x) , x ≥ 0, λ ∈ C,
2
2
JJ
J
II
I
Go Back
Close
Quit
Page 9 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
φα,β
λ (x)
(iλ−ρ)
= (2 sinh x)
and
c(λ) =
1
1
−2
(ρ − 2α − iλ), (ρ − iλ), 1 − iλ, (sinh x)
;
2 F1
2
2
x > 0, λ ∈ C − (−iN)
2ρ−iλ Γ(α + 1)Γ(iλ)
Γ 12 (ρ − iλ) Γ 12 (α − β + 1 + iλ)
where 2 F1 is the Gaussian hypergeometric function.
From ([1], [2], [15], [16]) we have the following properties:
i) We have:
• For ρ = 0 : ∀ x ≥ 0, ϕ0 (x) = 1,
• For ρ ≥ 0 : there exists a constant k > 0 such that
(2.3)
∀x ≥ 0,
e−ρx ≤ ϕ0 (x) ≤ k(1 + x)e−ρx .
M. Dziri and L.T. Rachdi
Title Page
Contents
ii) For λ ∈ R and x ≥ 0 we have
(2.4)
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
|ϕλ (x)| ≤ ϕ0 (x).
iii) For λ ∈ Csuch that |=λ| ≤ ρ and x ≥ 0 we have |ϕλ (x)| ≤ 1.
iv) We have the integral representation of Mehler type,
Z x
(2.5)
∀x > 0, ∀λ ∈ C,
ϕλ (x) =
k(x, t) cos(λt)dt,
JJ
J
II
I
Go Back
Close
Quit
0
where k(x, ·) is an even positive C ∞ function on ] − x, x[ with support in
[−x, x].
Page 10 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
v) For λ ∈ R, we have c(−λ) = c(λ).
vi) The function |c(λ)|−2 is continuous on [0, +∞[ and there exist positive
constants k, k1 , k2 such that
• If ρ ≥ 0 : ∀λ ∈ C, |λ| > k
k1 |λ|2α+1 ≤ |c(λ)|−2 ≤ k2 |λ|2α+1 ,
• If ρ > 0 : ∀λ ∈ C, |λ| ≤ k
k1 |λ|2 ≤ |c(λ)|−2 ≤ k2 |λ|2 ,
• If ρ = 0, α > 0 : ∀λ ∈ C, |λ| ≤ k
k1 |λ|2α+1 ≤ |c(λ)|−2 ≤ k2 |λ|2α+1 .
(2.6)
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
M. Dziri and L.T. Rachdi
Now, let us put
Title Page
1
v(x) = A 2 (x)u(x).
Contents
The equation (2.1) becomes
JJ
J
v 00 (x) − (G(x) − λ2 )v(x) = 0,
where
1
G(x) =
4
A0 (x)
A(x)
2
1
+
2
Let
1
4
A0 (x)
A(x)
2
0
− ρ2 .
−α
.
x2
Thus from hypothesis of the function A, we deduce the following results for
the function ξ.
ξ(x) = G(x) +
II
I
Go Back
Close
Quit
Page 11 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
Proposition 2.1.
1. The function ξ is continuous on ]0, ∞[.
2. There exist δ > 0 and a ∈ R such that the function ξ satisfies
ξ(x) =
a
+ exp(−δx)F1 (x),
x2
where F1 is C ∞ on ]0, ∞[, bounded together with all its derivatives on the
interval [x0 , ∞[ ,x0 > 0.
Proposition 2.2 ([15]). Let
(2.7)
1
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
1
ψ(x, λ) = ϕλ (x) − xα+ 2 A− 2 (x)jα (λx),
M. Dziri and L.T. Rachdi
where jα is defined by (2.2).
Then there exist positive constants C1 and C2 such that
Title Page
Contents
(2.8) ∀x > 0, ∀λ ∈ R∗ , |ψ(x, λ)| ≤ C1 A
with
˜ =
ξ(x)
Z
−1
2
3
−α− 2
˜
(x)ξ(x)λ
˜
ξ(x)
exp C2
λ
!
|ξ(r)|dr.
−1
2
II
I
Close
Quit
The kernel k(x, t) given by the relation (2.5) can be written
k(x, t) = 2h(x, t) + Cα A
JJ
J
Go Back
x
0
(2.9)
,
1
1
(x)x 2 −α (x2 − t2 )α− 2 ,
Page 12 of 40
0 < t < x,
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
where
(2.10)
1
h(x, t) =
Π
Z
∞
ψ(x, t) cos(λt)dλ,
0
2Γ(α + 1)
Cα = √
,
ΠΓ(α + 12 )
and ψ(x, λ) is the function defined by the relation (2.7).
Since the Riemann-Liouville and Weyl transforms associated with the operator ∆ are given by the kernel k, then, we need some properties of this function.
But from the relation (2.9) it suffices to study the kernel h.
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
M. Dziri and L.T. Rachdi
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 13 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
The Kernel h
3.
In this section we will study the behaviour of the kernel h.
Lemma 3.1. For any real a > 0 there exist positive constants C1 (a) ,C2 (a)
such that for all x ∈ [0, a],
C1 (a)x2α+1 ≤ A(x) ≤ C2 (a)x2α+1 .
From Proposition 1, and [16], we deduce the following lemma.
Lemma 3.2. There exist positive constants a1 , a2 , C1 and C2 such that for |λ| >
a1

α+ 12 − 12
A (x) (jα (λx) + O(λx))
for
|λx| ≤ a2
C(α)x






1
1
C(α)λ−(α+ 2 ) A− 2 (x) (C1 exp −iλx + C2 exp iλx)
ϕλ (x) =



× (1 + O(λ−1 ) + O((λx)−1 ))



for
|λx| > a2 ,
where
1
C(α) = Γ(α + 1)A (1) exp −
2
1
2
Z
1
B(t)dt .
0
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
M. Dziri and L.T. Rachdi
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Theorem 3.3. For any a > 0, there exists a positive constant C1 (α, a) such that
∀ 0 < t < x ≤ a;
|h(x, t)| ≤ C1 (α, a)x
α− 12
A
− 12
Page 14 of 40
(x).
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
Proof. By (2.10) we have for 0 < t < x,
Z
1 ∞
|h(t, x)| ≤
|ψ(x, λ)|dλ
Π 0
Z a1
Z
1
1 ∞
|ψ(x, λ)|dλ +
|ψ(x, λ)|dλ
=
Π 0
Π a1
(3.1)
= I1 (x) + I2 (x),
where a1 is the constant given by Lemma 3.2.
We put
1
1
fλ (x) = x 2 −α A 2 (x)|ψ(x, λ)|,
0 < x < a, λ ∈ R.
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
M. Dziri and L.T. Rachdi
From Proposition 2.2 the function
(x, λ) −→ fλ (x)
is continuous on [0, a] × [0, a1 ]. Then
Z
1
1
1 a1
|ψ(x, λ)|dλ ≤ Cα1 xα− 2 A− 2 (x),
(3.2)
I1 (x) =
Π 0
where
Cα1
a1
=
Π
Title Page
Contents
JJ
J
II
I
Go Back
sup
|fλ (x)|.
Close
(x,λ)∈[0,a]×[0,a1 ]
Quit
Let us study the second term
1
I2 (x) =
Π
Z
Page 15 of 40
∞
|ψ(x, λ)|dλ.
a1
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
i) Suppose − 12 < α ≤ 12 . From inequality (2.8) we get
˜
ξ(x)
λ
exp C2
|λ|
a1
!
˜
1
˜ exp C2 ξ(x) xα− 12 .
≤ C̃1 A− 2 (x)ξ(x)
a1
C1 − 1
˜
I2 (x) ≤
A 2 (x)ξ(x)
Π
Z
∞
−α− 32
!
dλ
Since ξ˜ is bounded on [0, ∞[, we deduce that
(3.3)
I2 (x) ≤ C2,α x
α− 21
A
− 12
(x).
This completes the proof in the case − 12 < α ≤ 12 .
ii) Suppose now that α > 12 .
• Let a1 , a2 be the constants given in Lemma 3.2. From this lemma we
deduce that there exists a positive constant C1 (α) such that
1
1
a2
(3.4)
∀x > , λ > a1 ; |ϕλ (x)| ≤ C1 (α)A− 2 (x)λ−(α+ 2 ) .
a1
On the other hand, the function
1
s −→ sα+ 2 jα (s)
is bounded on [0, ∞[.
Then from equality (2.7), we have, for x > aa12
Z
1 ∞
(3.5)
|ψ(x, λ)|dλ
Π a1
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
M. Dziri and L.T. Rachdi
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 16 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
≤
≤
≤
≤
Z
Z ∞
1 ∞
1 α+ 1 − 1
2
2
A (x)
|jα (λx)|dλ
|ϕλ (x)|dλ + x
Π a1
Π
a1
Z ∞
Z ∞
C1 (α) − 1
1 α− 1 − 1
−(α+ 12 )
2
2
2
A (x)
λ
dλ + x
A (x)
|jα (u)|du
Π
Π
a1
a2
(α− 12 )
Z ∞
C1 (α)
1
1 α− 1 − 1
− 12
A (x)
+ x 2 A 2 (x)
|jα (u)|du
a1
Π
α − 12 Π
a2
(α− 12 )
Z ∞
1
x
1 α− 1 − 1
C1 (α)
−2
2
2
A (x)
+ x
A (x)
|jα (u)|du
Π
a2
α − 12 Π
a2
1
1
≤ C2 (α)xα− 2 A− 2 (x),
M. Dziri and L.T. Rachdi
where
1
C1 (α)
1
(a2 )(−α+ 2 ) +
C2 (α) =
1
Π
α− 2 Π
• 0<x<
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
a2
.
a1
Z
∞
Title Page
|jα (u)|du.
a2
Contents
JJ
J
From Lemma 3.2 and the fact that
∀x ∈ R,
|jα (λx)| ≤ 1
Go Back
we deduce that there exists a positive constant M1 (α) such that
a2
a2
∀0<x< , 0≤λ≤
a1
x
|ψ(x, λ)| ≤ M1 (α)x
α+ 21
A
− 12
II
I
Close
(x).
Quit
Page 17 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
This involves
Z a2
a
x
1
M1 (α) α+ 1 − 1
2
|ψ(x, λ)|dλ ≤
x 2 A 2 (x)
− a1
Π a1
Π
x
1
1
a2
(3.6)
≤ M1 (α)xα− 2 A− 2 (x).
Π
Moreover
Z
Z ∞
1
C1 (α) − 1
1 ∞
λ−(α+ 2 ) dλ
|ψ(x, λ)|dλ ≤
A 2 (x)
a2
Π
Π ax2
x
Z ∞
1 α− 1 − 1
+ x 2 A 2 (x)
|jα (u)|du
Π
a2
(α− 12 )
1
C1 (α)
− 12
≤
A
(x)
a2
α − 12 Π
Z ∞
1 α− 1 − 1
2
2
+ x
A (x)
|jα (u)|du
Π
a2
(3.7)
≤ C2 (α)x
α− 12
A
− 12
(x).
From (3.6) and (3.7) we deduce that
Z
1
1
a2
1 ∞
(3.8)
∀0<x< ;
|ψ(x, λ)|dλ ≤ M2 (α)xα− 2 A− 2 (x)
a1
Π a1
where
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
M. Dziri and L.T. Rachdi
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 18 of 40
a2
M2 (α) = M1 (α) + C2 (α).
Π
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
From (3.5), (3.8) it follows that
∀ 0 < x < a;
1
1
I2 (x) ≤ M2 (α)xα− 2 A− 2 (x).
This completes the proof.
In order to provide some estimates for the kernel h for later use, we need the
following lemmas
Lemma 3.4.
i) For ρ > 0, we have
A(x) ∼ e2ρx ,
(x −→ +∞)
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
M. Dziri and L.T. Rachdi
Title Page
ii) For ρ = 0, we have
A(x) ∼ x2α+1 , (x −→ +∞).
Contents
JJ
J
This lemma can be deduced from hypothesis of the function A.
Lemma 3.5 ([2]). For ρ = 0 and α >
D1 (α) and D2 (α) satisfying
1
2
there exist two positive constants
II
I
Go Back
Close
Quit
Page 19 of 40
i)
|ϕλ (x)| ≤ D1 (α)x
α+ 12
A
− 12
(x),
x > 0, λ ≥ 0.
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
ii)
1
|ϕλ (x)| ≤ D2 (α)|c(λ)|A− 2 (x),
x > 1, λx > 1
where
λ −→ c(λ)
is the spectral function given by (2.6).
Using previous results we will give the behavior of the function h for large
values of the variable x
Theorem 3.6. For ρ = 0, α > 12 , and a > 0 there exists a positive constant
Cα,a such that
0 < t < x,
1
1
|h(x, t)| ≤ Cα,a xα− 2 A− 2 (x).
x > a,
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
M. Dziri and L.T. Rachdi
Proof. We have
1
h(x, t) =
Π
Z
Title Page
∞
|ψ(x, λ)| cos(λt)dλ,
0
then
Z
(3.9)
∞
1
|ψ(x, λ)|dλ
Π 0
Z
Z
1 ∞
1 1
=
|ψ(x, λ)|dλ +
|ψ(x, λ)|dλ.
Π 0
Π 1
|h(x, t)| ≤
From Proposition 2.2 and the fact that α > 12 we get
Z
Z ∞
3
1 ∞
C1 − 1
˜
˜
2
|ψ(x, λ)|dλ ≤
A (x)ξ(x) exp C2 (ξ(x)
λ−α− 2 dλ.
Π 1
Π
1
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 20 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
Since the function ξ˜ is bounded on [0, ∞[, we deduce that there exists dα > 0
verifying
Z
1
1
1 ∞
(3.10)
|ψ(x, λ)|dλ ≤ dα xα− 2 A− 2 (x).
Π 1
On the other hand, we have
Z
Z
Z 1
1 1
1 1
1 α+ 1 − 1
|ψ(x, λ)|dλ ≤
|ϕλ (x)|dλ + x 2 A 2 (x)
|jα (λx)|dλ.
Π 0
Π 0
Π
0
However,
1
Π
Z
0
1
1
|ϕλ (x)|dλ =
Π
Z
0
1
x
1
|ϕλ (x)|dλ +
Π
Z
1
|ϕλ (x)|dλ
1
x
from Lemma 3.5 i) we have
Z 1
C1 α− 1 − 1
1 x
|ϕλ (x)|dλ ≤
x 2 A 2 (x).
(3.11)
Π
Π 0
Furthermore from Lemma 3.5 ii) and the relation (2.6) it follows that there exists
d2 (α) > 0 such that
Z
Z 1
1
d2 (α) − 1
1 1
|ϕλ (x)|dλ ≤
A 2 (x)
λ−(α+ 2 ) dλ
1
Π
Π x1
Z x∞
1
d2 (α) − 1
≤
A 2 (x)
λ−(α+ 2 ) dλ
1
Π
x
(3.12)
≤
d2 (α) α− 1 − 1
x 2 A 2 (x).
Π(α − 12 )
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
M. Dziri and L.T. Rachdi
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 21 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
The theorem follows from the relations (3.9), (3.10), (3.11) and (3.12).
Theorem 3.7. For ρ > 0 and a > 1 there exists a positive constant Cα,a such
that
1
∀ 0 < t < x; x ≥ a; |h(x, t)| ≤ C2 (α, a)xγ A− 2 (x),
where γ = max 1, α + 21 .
Proof. This theorem can be obtained in the same manner as Theorem 3.6, using
the properties (2.3) and (2.4).
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
M. Dziri and L.T. Rachdi
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 22 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
Hardy Type Operators Tϕ
4.
In this section, we will define a class of integral operators and we recall some
of their properties which we use in the next section to obtain the main results of
this paper.
Let
ϕ : ]0, 1[ −→ ]0, ∞[
be a measurable function, then we associate the integral operator Tϕ defined for
all non-negative measurable functions f by
Z x t
f (t)ν(t)dt
∀x > 0;
Tϕ (f )(x) =
ϕ
x
0
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
M. Dziri and L.T. Rachdi
where
• ν is a measurable non negative function on ]0, ∞[ such that
Z a
(4.1)
∀a > 0,
ν(t)dt < ∞
0
and
• µ is a non-negative function on ]0, ∞[ satisfying
Z b
(4.2)
∀ 0 < a < b,
µ(t)dt < ∞.
a
These operators have been studied by many authors. In particular, in [5], see
also ([6], [10], [11]), we have proved the following results.
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 23 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
Theorem 4.1. Let p, q be two real numbers such that
1 < p ≤ q < ∞.
Let ν and µ be two measurable non-negative functions on ]0, ∞[, satisfying
(4.1) and (4.2). Lastly, suppose that the function
ϕ : ]0, 1[ −→
]0, ∞[
is continuous non increasing and satisfies
∀x, y ∈]0, 1[,
ϕ(xy) ≤ D(ϕ(x) + ϕ(y))
where D is a positive constant. Then the following assertions are equivalent
1. There exists a positive constant Cp,q such that for all non-negative measurable functions f :
Z ∞
1q
Z ∞
p1
p
q
(Tϕ (f )(x)) µ(x)dx
≤ Cp,q
(f (x)) ν(x)dx .
0
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
M. Dziri and L.T. Rachdi
Title Page
Contents
0
2. The functions
∞
Z
F (r) =
r
1q Z r 0
10
p
x p
µ(x)dx
ϕ
ν(x)dx
r
0
JJ
J
II
I
Go Back
Close
and
Z
G(r) =
r
∞
1q Z r
10
r q
p
µ(x)dx
ϕ
ν(x)dx
x
0
are bounded on ]0, ∞[, where p0 =
p
.
p−1
Quit
Page 24 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
Theorem 4.2. Let p and q be two real numbers such that
1<p≤q<∞
and µ, ν two measurable non-negative functions on ]0, ∞[, satisfying the hypothesis of Theorem 4.1.
Let
ϕ : ]0, 1[ −→ ]0, ∞[
be a measurable non-decreasing function.
If there exists β ∈ [0, 1] such that the function
∞
Z
r −→
r
1q Z r 0
10
r βq
p
x p (1−β)
ϕ
µ(x)dx
ϕ
ν(x)dx
x
r
0
is bounded on ]0, ∞[, then there exists a positive constant Cp,q such that for all
non-negative measurable functions f , we have
∞
Z
(Tϕ (f (x)))q µ(x)dx
0
where p0 =
1q
Z
≤ Cp,q
∞
(f (x))p ν(x)dx
p1
0
p
.
p−1
The last result that we need is:
Corollary 4.3. With the hypothesis of Theorem 4.1 and ϕ = 1, the following
assertions are equivalent:
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
M. Dziri and L.T. Rachdi
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 25 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
1. there exists a positive constant Cp,q such that for all non-negative measurable functions f we have
Z
∞
(H(f )(x))q µ(x)dx
1q
Z
≤ Cp,q
0
∞
(f (x))p ν(x)dx
p1
,
0
2. The function
Z
I(r) =
1q Z
∞
µ(x)dx
10
r
p
ν(x)dx
r
0
is bounded on ]0, ∞[,
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
M. Dziri and L.T. Rachdi
where H is the Hardy operator defined by
Z
∀x > 0,
H(f )(x) =
Title Page
x
f (t)ν(t)dt.
0
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 26 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
5.
The Riemann - Liouville and Weyl Transforms
Associated with the Operator ∆
This section deals with the proof of the Hardy type inequalities (1.1) and (1.2)
mentioned in the introduction.
We denote by
• Lp ([0, ∞[, A(x)dx) ; 1 < p < ∞, the space of measurable functions on
[0, ∞[, satisfying
Z
||f ||p,A =
p1
∞
p
(f (x)) A(x)dx
< ∞.
0
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
M. Dziri and L.T. Rachdi
• R0 the operator defined for all non-negative measurable functions f by
Z x
∀x > 0, R0 (f )(x) =
h(x, t)f (t)dt,
0
where h is the kernel studied in the third section.
• R1 the operator defined for all non-negative measurable functions f by
Z x
1
2Γ(α + 1) α− 1 − 1
∀x > 0, R1 (f )(x) = √
(x2 −t2 )α− 2 f (t)dt.
x 2 A 2 (x)
1
ΠΓ α + 2
0
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 27 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
Definition 5.1.
1. The Riemann-Liouville transform associated with the operator ∆ is defined for all non-negative measurable functions f on ]0, ∞[ by
Z x
R(f )(x) =
k(x, t)f (t)dt.
0
2. The Weyl transform associated with operator ∆ is defined for all nonnegative measurable functions f by
Z ∞
W(f )(t) =
k(x, t)f (x)A(x)dx
t
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
M. Dziri and L.T. Rachdi
where k is the kernel given by the relation (2.5).
Title Page
Proposition 5.1.
1. For ρ > 0, α > − 12 and p > max(2, 2α+2) there exists a positive constant
C1 (α, p) such that for all f ∈ Lp ([0, ∞[, A(x)dx),
||R0 (f )||p,A ≤ C1 (α, p)||f ||p,A .
Contents
JJ
J
II
I
Go Back
1
2
2. For ρ = 0, α > and p > 2α + 2, there exists a positive constant
C2 (α, p) such that for all f ∈ Lp ([0, ∞[, A(x)dx)
||R0 (f )||p,A ≤ C2 (α, p)||f ||p,A .
Close
Quit
Page 28 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
1. Suppose that ρ > 0 and p > max(2, 2α + 2). Let
Proof.
0
ν(x) = A1−p (x)
and
p
1
p
µ(x) = C1 (α, a)xp(α− 2 ) A1− 2 (x)1]0,a] (x) + C2 (α, a)xpγ A1− 2 (x)1[a,∞[ (x),
with a > 1, C1 (α, a), C2 (α, a) and γ are the constants given in Theorem
3.3 and Theorem 3.7.
Then
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
0
ν(x) ≤ m1 (α, p)x(2α+1)(1−p )
and
M. Dziri and L.T. Rachdi
µ(x) ≤ m2 (α, p)x2α+1−p .
These inequalities imply that
Title Page
Z
∀b > 0;
b
Contents
ν(x)dx < ∞,
0
Z
JJ
J
b2
∀ 0 < b1 < b2 ;
µ(x)dx < ∞
b1
Go Back
and
Z
I(r) =
p1 Z
∞
µ(x)dx
≤
p
ν(x)dx
r
Close
10
r
II
I
Quit
0
Z
m2 (α, p)
x
r
p1 ∞
2α+1−p
dx
Z
m1 (α, p)
r
x
0
(2α+1)(1−p0 )
10
p
Page 29 of 40
dx
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
1
1
(m2 (α, p)) p (m1 (α, p)) p0
≤
1
1
(p − 2α − 2) p ((2α + 1)(1 − p0 ) + 1) p0
1
1
(m2 (α, p)) p × ((p − 1)m1 (α, p)) p0
=
.
p − 2α − 2
From Corollary 4.3, there exists a positive constant Cp,α such that for all
non-negative measurable functions g we have
Z
p1
∞
p
Z
≤ Cp,α
(H(g)(x)) µ(x)dx
(5.1)
∞
(g(x)) ν(x)dx
0
0
with
Z
Title Page
Now let us put
µ(x)
A(x)
p1 Z
then we have
H(g)(x) =
where
,
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
g(t)ν(t)dt.
0
T (f )(x) =
p1
M. Dziri and L.T. Rachdi
x
H(g)(x) =
p
µ(x)
A(x)
Contents
x
f (t)dt,
0
− p1
T (f )(x),
JJ
J
II
I
Go Back
Close
0
g(x) = f (x)Ap −1 (x).
Quit
Page 30 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
From inequality (5.1), we deduce that for all non-negative measurable
functions f , we have
p1
∞
Z
p
Z
≤ Cp,α
(T (f )(x)) A(x)dx
(5.2)
p1
∞
p
(f (x)) A(x)dx
0
.
0
On the other hand from Theorems 3.3 and 3.7 we deduce that the function
Z x
R0 (f )(x) =
h(x, t)f (t)dt
0
is well defined and we have
|R0 (f )(x)| ≤ T (|f |)(x).
(5.3)
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
M. Dziri and L.T. Rachdi
Thus, the relations (5.2) and (5.3) imply that
Title Page
Z
∞
|R0 (f )(x)|p A(x)dx
p1
Z
∞
≤ Cp,α
|f (x)|p A(x)dx
p1
,
0
0
which proves 1).
2. Suppose that ρ = 0 and α > 12 . From Theorems 3.3 and 3.6 we have
∀ 0 < t < x;
|h(t, x)| ≤ Cx
α− 12
A
− 12
(x).
Contents
JJ
J
II
I
Go Back
Close
Quit
Therefore if we take
1
p
µ(x) = x(α− 2 )p A1− 2 (x)
Page 31 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
and
0
ν(x) = A1−p (x),
we obtain the result in the same manner as 1).
Proposition 5.2. Suppose that − 12 < α ≤
positive constant a such
∀ 0 < t < x,
x > a,
1
,
2
ρ = 0 and that there exists a
h(x, t) = 0.
Then for all p > 2α + 2, we can find a positive constant Cα,a satisfying
∀f ∈ Lp ([0, ∞[, A(x)dx);
||R0 (f )||p,A ≤ Cα,a ||f ||p,A .
Proof. The hypothesis and Theorem 3.3 imply that there exists a positive constant a such that
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
M. Dziri and L.T. Rachdi
Title Page
Contents
∀ 0 < t < x;
|h(t, x)| ≤ C(α, a)x
α− 12
A
− 12
(x)1]0,a] (x).
Therefore, if we take
p
1
µ(x) = C(α, a)xp(α− 2 ) A1− 2 (x)1]0,a] (x)
JJ
J
II
I
Go Back
Close
and
0
ν(x) = A1−p (x)
then, we obtain the result using a similar procedure to that in Proposition 1,
2).
Quit
Page 32 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
Now, let us study the operator R1 defined for all measurable non-negative
functions f by
Z x
1
1
−α − 12
2
R1 (f )(x) = Cα x
A (x)
(x2 − t2 )α− 2 f (t)dt,
0
where
2Γ(α + 1)
Cα = √
.
ΠΓ α + 12
Proposition 5.3.
1. For α > − 21 , ρ > 0 and p > max(2, 2α + 2), there exists a positive
constant Cp,α such that for all f ∈ Lp ([0, +∞[, A(x)dx), we have
||R1 (f )||p,A ≤ Cp,α ||f ||p,A .
− 12 ,
2. For α >
ρ = 0 and p > 2α + 2 there exists a positive constant Cp,α
such that for all f ∈ Lp ([0, +∞[, A(x)dx), we have
||R1 (f )||p,A ≤ Cp,α ||f ||p,A .
Proof. Let Tϕ the Hardy type operator defined for all non-negative measurable
functions f by
Z x t
f (t)ν(t)dt,
Tϕ (f )(x) =
ϕ
x
0
where
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
M. Dziri and L.T. Rachdi
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 33 of 40
1
ϕ(x) = (1 − x2 )α− 2
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
and
0
ν(x) = A1−p (x).
Then for all non-negative measurable functions f , we have
(5.4)
1
1
R1 (f )(x) = Cα x− 2 +α A− 2 (x)Tϕ (g)(x),
where
0
g(x) = f (x)Ap −1 (x).
Let
1
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
p
µ(x) = xp(α− 2 ) A1− 2 (x),
then, according to the hypothesis satisfied by the function A, it follows that
there exist positive constants C1 , C2 such that for all α > − 21 and ρ > 0 we
have
(5.5)
(5.6)
∀x > 0;
∀x > 0;
0 ≤ µ(x) ≤ C1 x
2α+1−p
M. Dziri and L.T. Rachdi
Title Page
Contents
0
0 ≤ ν(x) ≤ C2 x(2α+1)(1−p ) .
JJ
J
II
I
Go Back
Thus from the relations (5.5) and (5.6) we deduce that for α ≥ 12 , ρ > 0 and
p > 2α + 2, we have
Close
Quit
• the function ϕ is continuous and non-increasing on ]0, 1[.
Page 34 of 40
• the functions ϕ, ν and µ satisfy the hypothesis of Theorem 4.1.
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
• the functions
Z
∞
F (r) =
r
p1 Z r 0
10
p
x p
µ(x)dx
ϕ
ν(x)dx
r
0
and
Z
G(r) =
r
∞
p1 Z r
10
r p
p
µ(t)dt
ϕ
ν(t)dt
x
0
are bounded on [0, ∞[.
Hence from Theorem 4.1, there exists Cp,α > 0 such that for all measurable
non-negative functions f we have
Z
∞
(Tϕ (f (x)))p µ(x)dx
p1
∞
Z
(f (x))p ν(x)dx
≤ Cp,α
0
.
0
0
∞
(R1 (f (x)))p A(x)dx
p1
Z
≤ Cp,α
∞
(f (x))p A(x)dx
0
which proves the Proposition 1, 1) in the case α ≥ 12 .
For − 12 < α < 12 and p > 2 we have
• the function ϕ is continuous and non-decreasing on ]0, 1[.
M. Dziri and L.T. Rachdi
p1
Title Page
Contents
This inequality together with the relation (5.4) lead to
Z
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
p1
JJ
J
II
I
Go Back
Close
Quit
Page 35 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
• if we pick
1 − p( 21 + α)
1
, min 1, 1
β ∈ max 0,
p( 12 − α)
p( 2 − α)
and using inequalities (5.5) and (5.6), we deduce that the function
Z ∞ p1 Z r 10
p
r βp
x (1−β)p0
H(r) =
ϕ
µ(x)dx
ϕ
ν(x)dx
x
r
r
0
is bounded on ]0, ∞[.
Consequently, the result follows from Theorem 4.2 and relation (5.4).
2) can be obtained in the same fashion as 1).
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
M. Dziri and L.T. Rachdi
Now we will give the main results of this paper.
Title Page
Theorem 5.4.
1. For α > − 12 , ρ > 0 and p > max(2, 2α + 2), there exists a positive
constant Cp,α such that for all f ∈ Lp ([0, ∞[, A(x)dx),
||R(f )||p,A ≤ Cp,α ||f ||p,A .
2. For α > − 12 , ρ > 0 and p > max(2, 2α + 2), there exists a positive
0
constant Cp,α such that for all g ∈ Lp ([0, ∞[, A(x)dx),
1
W(g)
A(x)
where p0 =
p
.
p−1
≤ Cp,α ||g||p0 ,A
p0 ,A
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 36 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
Proof. 1) follows from Proposition 1, 1) and Proposition 1, 1), and the fact that
R(f ) = R0 (f ) + R1 (f ).
2) follows from 1) and the relations
Z
||g||p0 ,A = max
(5.7)
||f ||p,A ≤1
∞
f (x)g(x)A(x)dx,
0
for all measurable non-negative functions f and g
Z ∞
Z ∞
(5.8)
R(f )(x)g(x)A(x)dx =
W(g)(x)f (x)dx.
0
0
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
M. Dziri and L.T. Rachdi
Theorem 5.5.
Title Page
1
,
2
1. For α >
ρ = 0 and p > 2α + 2 there exists a positive constant Cp,α
such that for all f ∈ Lp ([0, ∞[, A(x)dx)
||R(f )||p,A ≤ Cp,α ||f ||p,A .
2. For α > 12 , ρ = 0 and p > 2α + 2 there exists a positive constant Cp,α
0
such that for all g ∈ Lp ([0, ∞[, A(x)dx)
1
W(g)
A(x)
where p0 =
p
.
p−1
≤ Cp,α kgkp0 ,A
p0 ,A
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 37 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
3. For − 12 < α ≤ 12 , ρ = 0, p > 2α + 2 and under the hypothesis of
Proposition 5.2, the previous results hold.
Proof. This theorem is obtained by using Propositions 1, 2), 5.2 and 1, 2).
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
M. Dziri and L.T. Rachdi
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 38 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
References
[1] A. ACHOUR AND K. TRIMÈCHE, La g- fonction de Littlewood-Paley
associèe à un operateur differentiel singulier sur (0, ∞), Ann Inst Fourier
(Grenoble), 33 (1983), 203–206.
[2] W.R. BLOOM AND ZENGFU XU, The Hardy - Littlewood maximal function for Chébli -Trimèche hypergroups, Contemporary Mathematics, 183
(1995).
[3] W.R. BLOOM AND ZENGFU XU, Fourier transforms of Shwartz functions on Chébli - Trimèche hypergroups, Mh. Math., 125 (1998), 89–109.
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
[4] W.C. CONNET AND A.L. SHWARTS, The Littlewood Paley theory for
Jacobi expansions, Trans. Amer. Math. Soc., 251 (1979).
M. Dziri and L.T. Rachdi
[5] M. DZIRI AND L.T. RACHDI, Inégalités de Hardy Littelwood pour une
classe d’operateurs integraux, Diplôme d’études Approfondies, Faculté
des Sciences de Tunis, Departement de Mathematiques, Juillet 2001.
[6] M. DZIRI AND L.T. RACHDI, Hardy type inequalities for integral transforms associated with Jacobi operator, International Journal Of Mathematics And Mathematical Sciences, 3 (2005), 329–348.
[7] A. FITOUHI AND M.M. HAMZA, A uniform expansion for the eigenfunction of a singular second order differential operator, SIAM J. Math.
Anal., 21 (1990), 1619–1632.
[8] M.N. LEBEDEV, Special Function and their Applications, Dover Publications, Inc. New-York .
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 39 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au
[9] M.N. LAZHARI, L.T. RACHDI AND K. TRIMÈCHE, Asymptotic expansion and generalized Schlafli integral representation for the eigenfunction
of a singular second-order differential operator, J. Math. Anal. Appl., 217
(1998), 269–292.
[10] F.J. MARTIN-REYES AND E. SAWYER, Weighted inequalities for Riemann -Liouville fractional integrals of order one greater, Proc. Amer.
Math. Soc., 106(3) (1989).
[11] B. MUCKENHOUPT, Hardy’s inequalities with weights, Studia Math., 34
(1972), 31–38.
[12] M.M. NESSIBI, L.T. RACHDI AND K. TRIMÈCHE, The local central
limit theorem on the Chébli -Trimèche hypergroups and the Euclidiean
hypergroup Rn , J. Math. Sciences, 9(2) (1998), 109–123.
[13] K. TRIMÈCHE, Inversion of the Lions transmutation operators using
generalized wavelets, Applied and Computational Harmonic Analysis, 4
(1997), 97–112.
[14] K. TRIMÈCHE, Generalized Transmutation and translation associated
with partial differential operators, Contemp. Math., 1983 (1995), 347–
372.
[15] K. TRIMÈCHE, Transformation intégrale de Weyl et théoréme de Paley
Wiener associés un opérateur différentiel singulier sur (0, ∞), J. Math.
Pures et Appl., 60 (1981), 51–98.
[16] Z. XU, Harmonic Analysis on Chébli-Trimèche Hypergroups, PhD Thesis,
Murdoch Uni. Australia, 1994.
Hardy Type Inequalities For
Integral Transforms Associated
With A Singular Second Order
Differential Operator
M. Dziri and L.T. Rachdi
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 40 of 40
J. Ineq. Pure and Appl. Math. 7(1) Art. 38, 2006
http://jipam.vu.edu.au