HARDY-TYPE INEQUALITIES FOR INTEGRAL TRANSFORMS ASSOCIATED WITH JACOBI OPERATOR

We establish Hardy-type inequalities for the Riemann-Liouville and Weyl transforms associated with the Jacobi operator by using Hardy-type inequalities for a class of integral operators.

This paper is devoted to the study of the Riemann-Liouville and Weyl transforms on the spaces of measurable functions on [0, ∞[ such that (1.5) The main results of this work are Theorems 4.2 and 4.4 in Section 4.
To obtain those results we use the following integral operators: where (i) ν is a nonnegative locally integrable function on [0,∞[, (ii) dµ(t) is a nonnegative measure, locally finite on [0, ∞[, (iii) the following is a measurable function satisfying some properties [10,12,18]: Both operators T ϕ and T * ϕ are connected by the following duality relation: for all nonnegative measurable functions f and g we have In this paper, we give some conditions on the functions ϕ, ν and the measure dµ so that the operator T ϕ and its dual T * ϕ satisfy the following Hardy inequalities: for all real numbers p, q satisfying M. Dziri and L. T. Rachdi 331 there exists a positive constant C p,q such that for all nonnegative measurable functions f and g we have where p and q are the conjugate exponents, respectively, of p and q.
In [5], we have studied inequalities (1.10), in the case 1 < q < p < ∞.The inequalities obtained below for the operators T ϕ and T * ϕ will allow us to obtain the main results of this paper.
This paper is arranged as follows.
In Section 2, we consider a continuous nonincreasing function 11) for which there exists a positive constant D satisfying (1.12) Then we give necessary and sufficient conditions such that the operators T ϕ and T * ϕ satisfy the inequalities (1.10).
In Section 3, we suppose only that the function ϕ is nondecreasing and we give the sufficient conditions such that the precedent inequalities hold.
In Section 4, we use the results obtained below to study and to establish the Hardy inequalities for Riemann and Weyl operators associated with Jacobi differential operator ∆ α,β .

Hardy operator T ϕ and its dual T *
ϕ when the function ϕ is nonincreasing on ]0,1[ In this section, we consider a measurable positive and nonincreasing function ϕ defined on ]0,1[ for which we associate the operator T ϕ and its dual T * ϕ defined, respectively, for every nonnegative and measurable function f , by where ν is a measurable nonnegative function on ]0,∞[ such that and dµ(t) is a nonnegative measure on [0,∞[ satisfying 332 Hardy-type inequalities The main result of this section is Theorem 2.1.
Theorem 2.1.Let p and q be two real numbers such that Let ν be a nonnegative measurable function on ]0,+∞[ satisfying (2.2), and dµ(t) a nonnegative measure on ]0,+∞[ which satisfies the relation (2.3).Lastly, suppose that is a continuous nonincreasing function so that (i) there exists a positive constant D such that (ii) for all a > 0, Then the following assertions are equivalent.
(1) There exists a positive constant C p,q such that for every nonnegative measurable function f , (2) The functions are bounded on ]0,+∞[, where The proof of this theorem uses the idea of [10,13,14,18] and is left to the reader.
To obtain similar inequalities for the dual operator T * ϕ , we use the following duality lemma.
(1) There exists a positive constant C p,q such that for every nonnegative measurable function f

.12)
(2) There exists a positive constant C p,q such that for every nonnegative measurable function g q dµ(t)

.13)
A consequence of Theorem 2.1 and Lemma 2.2 is the following.

Theorem 2.3 (dual theorem). Under the hypothesis of Theorem 2.1, the following assumptions are equivalent.
(1) There exists a positive constant C p,q such that for every nonnegative measurable function g (2) Both functions are bounded on ]0,+∞[.

Integral operator T ϕ and its dual when the function ϕ is nondecreasing
In this section, we suppose only that the function is nondecreasing, we will give a sufficient condition, which permits to prove that the integral operators T ϕ and T * ϕ satisfy the Hardy inequalities [1,8,15,16].Theorem 3.1.Let p and q be two real numbers such that and p = p/(p − 1), q = q/(q − 1).Let ν be a nonnegative function on ]0,+∞[ satisfying (2.2), and dµ(t) a nonnegative measure on ]0,+∞[ which satisfies the relation (2.3).Finally, 334 Hardy-type inequalities be a measurable nondecreasing function.
If there exists β ∈ [0,1] such that the function is bounded on ]0,+∞[, then there exists a positive constant C p,q such that for every nonnegative measurable function f , Proof of Theorem 3.1.Let h be the function defined by By Hölder's inequality, we have If we replace h(y) by its value, then we obtain Since the function ϕ is nondecreasing and β ∈ [0,1], we have from the hypothesis, the function belongs to L 1 (]0,+∞[,dz) and then, from [16, Lemma 1], we deduce that therefore inequality (3.14) involves J(x) ≤ p + q q h(x) q . (3.17) From inequalities (3.7) and (3.17), we obtain so, we obtain Since q/ p ≥ 1, then, from Minkowski's inequality [17], we deduce that 336 Hardy-type inequalities So On the other hand, from the hypothesis, the function is bounded on ]0,+∞[, we denote which means that . (3.25) From inequalities (3.21) and (3.25) we obtain Since ϕ is nondecreasing, we have  From inequalities (3.29) and (3.31), we deduce that for every nonnegative measurable function f , we have where C p,q = p + q q 1/p p + q p 1/q M p,q (3.33) and M p,q is the constant given by (3.23).This completes the proof of Theorem 3.1.
From Lemma 2.2 and Theorem 3.1 we obtain the following result.
Theorem 3.2 (dual theorem).Under the hypothesis of Theorem 3.1 if there exists β ∈ [0,1] such that the function is bounded on ]0,+∞[, then there exists a positive constant C p,q such that for every nonnegative measurable function g

Riemann-Liouville and Weyl transforms associated with Jacobi operator
The Jacobi operator stated in the introduction has been studied by many authors [2,6,7,19,20].In particular, we know that, for every complex number λ, the differential equation admits a unique solution ϕ α,β λ (x) given by where F is the Gaussian hypergeometric function [9,11].Furthermore the function ϕ α,β λ has the following Mehler integral representation: where k α,β is the nonnegative kernel given by the relation (1.3).Many properties of harmonic analysis associated with the operator ∆ α,β have been studied and established (convolution product, Fourier-transform, inversion formula, Plancherel and Paley-Wiener theorems).
On the other hand, the following integral transforms are defined for the Jacobi operator.

Definition 4.1. (1)
The Riemann-Liouville transform associated with Jacobi operator is the integral transform defined, for every nonnegative measurable function f , by (2) The Weyl transform associated with Jacobi operator is defined, for every nonnegative measurable function f , by where k α,β is the kernel given by the relation (1.3).Those integral operators are linked by the following duality relation: for all nonnegative measurable functions f and g, As mentioned in the introduction, those integral transforms have been studied on spaces of regular functions.
Our purpose in this section is to study those operators on the spaces Theorem 4.2.For −1/2 < β ≤ α, α ≥ 1/2, and p > 2α + 2, there exists a positive constant C p,α,β such that (1) (2) for all where The proof of this theorem needs the following lemma.

.16)
Proof of Theorem 4.2.Let T ϕ and T * ϕ be the Hardy-type operators defined, respectively, by where the function ϕ is continuous and nonincreasing on ]0,1[.Furthermore for all a,b ∈ ]0,1[ we have then by using the inequality we deduce that

.23)
Since p > 2α + 2, then for all a > 0, (1) Now we will prove that the operator T ϕ defined latterly satisfies the sufficient condition of Theorem 2.1.Then we must show that the functions are bounded on ]0,∞[.We put then Now, we have and consequently Furthermore, we have then Thus, we deduce that From the relations (4.32) and (4.36), we obtain and from the relations (4.29), it follows that both functions F and G are bounded on ]0,∞[.Therefore from Theorem 2.1, there exists a positive constant D p,α,β such that, for every nonnegative measurable function g, we have Let T α,β be the operator defined, for every nonnegative measurable function f , by Then the operators T α,β and T ϕ are connected by the following relation: for every nonnegative measurable function f , we have where and from Lemma 4.3 we deduce that there exists a positive constant C p,α,β such that for every nonnegative measurable function f , we have (4.44) Hence the function is finite almost everywhere.Then the function is defined almost everywhere, and This completes the proof of Theorem 4.2(1).
(2) From Theorem 2.3, we deduce that there exists a positive constant D p,α,β such that for every nonnegative measurable function h, we have Let g be a nonnegative measurable function, by setting and using the inequality (4.49), we deduce that where (2) for every function g where

.57)
The proof of this theorem needs the following lemma.
Proof of Theorem 4.4.In this proof, we consider the functions ϕ, ν and the measure dµ defined, respectively, by We will prove that the operator T ϕ satisfies the sufficient conditions of Theorem 3.1, that is, there exists λ ∈ [0,1] such that the function is bounded on ]0,∞[.In fact we denote  We complete the proof of Theorem 4.4 by the same way as Theorem 4.2 and using Lemma 4.5.