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Hardy-Type Inequalities for Integral Transforms Associated with Jacobi Operator

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Hardy-Type Inequalities for Integral Transforms Associated with Jacobi Operator HARDY-TYPE INEQUALITIES FOR INTEGRAL TRANSFORMS ASSOCIATED WITH JACOBI OPERATOR M. DZIRI AND L. T. RACHDI Received 8 April 2004 and in revised form 24 December 2004 We establish Hardy-type inequalities for the Riemann-Liouville and Weyl transforms as- sociated with the Jacobi operator by using Hardy-type inequalities for a class of integral operators. 1. Introduction It is well known that the Jacobi second-order differential operator is defined on ]0,+∞[ by 1 d du 2 ∆α,βu(x) = Aα,β(x) + ρ u(x), (1.1) Aα,β(x) dx dx where 2ρ 2α+1 2β+1 (i) Aα,β(x) = 2 sinh (x)cosh (x), (ii) α,β ∈ R; α ≥ β>−1/2, (iii) ρ = α + β +1. The Riemann-Liouville and Weyl transforms associated with Jacobi operator ∆α,β are, respectively, defined, for every nonnegative measurable function f ,by x Rα,β( f )(x) = kα,β(x, y) f (y)dy, 0 ∞ (1.2) Wα,β( f )(y) = kα,β(x, y) f (x)Aα,β(x)dx, y where kα,β is the nonnegative kernel defined, for x>y>0, by −1 2 2−α+3/2Γ(α +1) cosh(2x) − cosh(2y) α / kα,β(x, y) = √ πΓ(α +1/2)coshα+β(x)sinh2α(x) (1.3) 1 cosh(x) − cosh(y) × F α + β,α − β;α + ; 2 2cosh(x) Copyright © 2005 Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences 2005:3 (2005) 329–348 DOI: 10.1155/IJMMS.2005.329 330 Hardy-type inequalities and F is the Gaussian hypergeometric function. Such integral transforms have many ap- plications to science and engineering [3, 4]. These operators have been studied on regular spaces of functions. In particular in [19], the author has proved that the Riemann-Liouville transform Rα,β is an isomorphism from ξ∗(R) (the space of even infinitely differentiable functions on R) on itself, and that the Weyl transform Wα,β is an isomorphism from D∗(R) (the space of even infinitely differentiable functions on R with compact support) on itself. The Weyl transform has also been studied on Shwartz space S∗(R)[20]. This paper is devoted to the study of the Riemann-Liouville and Weyl transforms on the spaces p L [0,∞[,Aα,β(x)dx 1 <p<∞ (1.4) of measurable functions on [0,∞[suchthat ∞ 1/p p f p,α,β = f (x) Aα,β(x)dx < ∞. (1.5) 0 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: x t Tϕ( f )(x) = ϕ f (t)ν(t)dt, 0 x ∞ (1.6) ∗ = x Tϕ (g)(x) ϕ g(t)dµ(t), x t 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]: ϕ : ]0,1[ −→ ]0,∞[. (1.7) ∗ Both operators Tϕ and Tϕ are connected by the following duality relation: for all non- negative measurable functions f and g we have ∞ ∞ = ∗ ν Tϕ( f )(x)g(x)dµ(x) f (y)Tϕ (g)(y) (y)dy. (1.8) 0 0 In this paper, we give some conditions on the functions ϕ, ν and the measure dµ so ∗ that the operator Tϕ and its dual Tϕ satisfy the following Hardy inequalities: for all real numbers p, q satisfying 1 <p≤ q<∞, (1.9) M. Dziri and L. T. Rachdi 331 there exists a positive constant Cp,q such that for all nonnegative measurable functions f and g we have ∞ 1/q ∞ 1/p q p Tϕ( f )(x) dµ(x) ≤ Cp,q f (x) ν(x)dx , 0 0 ∞ ∞ (1.10) 1/p 1/q ∗ p ν ≤ q Tϕ (g)(x) (x)dx Cp,q g(x) dµ(x) , 0 0 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 Tϕ and Tϕ 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 ϕ : ]0,1[ −→ ]0,∞[ (1.11) for which there exists a positive constant D satisfying ∀x, y ∈ ]0,1[, ϕ(xy) ≤ D ϕ(x)+ϕ(y) . (1.12) ffi ∗ Then we give necessary and su cient conditions such that the operators Tϕ and Tϕ 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. ∗ 2. Hardy operator Tϕ and its dual Tϕ 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 Tϕ and its dual Tϕ defined, respectively, for every nonnegative and measurable function f ,by x t ∀x>0, Tϕ( f )(x) = ϕ f (t)ν(t)dt, 0 x ∞ (2.1) ∀ ∗ = x x>0, Tϕ ( f )(x) ϕ f (t)dµ(t), x t where ν isameasurablenonnegativefunctionon]0,∞[suchthat a ∀a>0, ν(t)dt < ∞ (2.2) 0 and dµ(t) is a nonnegative measure on [0,∞[ satisfying b ∀0 <a<b, dµ(t) < ∞. (2.3) a 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 1 <p≤ q<+∞. (2.4) Let ν be a nonnegative measurable function on ]0,+∞[ satisfying (2.2), and dµ(t) a nonneg- ative measure on ]0,+∞[ which satisfies the relation (2.3). Lastly, suppose that ϕ : ]0,1[ −→ ]0,+∞[ (2.5) is a continuous nonincreasing function so that (i) there exists a positive constant D such that ∀x, y ∈ ]0,1[, ϕ(xy) ≤ D ϕ(x)+ϕ(y) , (2.6) (ii) for all a>0, a t ϕ ν(t)dt < +∞. (2.7) 0 a Then the following assertions are equivalent. (1) There exists a positive constant Cp,q such that for every nonnegative measurable func- tion f , ∞ 1/q ∞ 1/p q p Tϕ( f )(t) dµ(t) ≤ Cp,q f (t) ν(t)dt . (2.8) 0 0 (2) The functions ∞ 1/q r t p 1/p r −→ dµ(t) ϕ ν(t)dt , r 0 r (2.9) ∞ r q 1/q r 1/p r −→ ϕ dµ(t) ν(t)dt r t 0 are bounded on ]0,+∞[,where p p = . (2.10) p − 1 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 Tϕ , we use the following duality lemma. Lemma 2.2 [12, 18]. Let p, q, p , q be real numbers such that 1 1 1 1 1 <p≤ q<+∞, + = 1, + = 1 (2.11) p p q q M. Dziri and L. T. Rachdi 333 let µ be a σ-finite measure on ]0,+∞[ and ν a nonnegative locally integrable function on ]0,+∞[. Then the following statements are equivalent. (1) There exists a positive constant Cp,q such that for every nonnegative measurable func- tion f ∞ 1/q ∞ 1/p q p Tϕ( f )(t) dµ(t) ≤ Cp,q f (t) ν(t)dt . (2.12) 0 0 (2) There exists a positive constant Cp,q such that for every nonnegative measurable func- tion g ∞ ∞ 1/p 1/q ∗ p ν ≤ q Tϕ (g)(t) (t)dt Cp,q g(t) dµ(t) . (2.13) 0 0 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 assump- tions are equivalent. (1) There exists a positive constant Cp,q such that for every nonnegative measurable func- tion g ∞ 1/q ∞ 1/p ∗ qν ≤ p Tϕ (g)(x) (x)dx Cp,q g(x) dµ(x) . (2.14) 0 0 (2) Both functions ∞ 1/p r t q 1/q r −→ dµ(x) ϕ ν(t)dt , r 0 r (2.15) ∞ r p 1/p r 1/q r −→ ϕ dµ(x) ν(t)dt r x 0 are bounded on ]0,+∞[. 3. Integral operator Tϕ and its dual when the function ϕ is nondecreasing In this section, we suppose only that the function ϕ : ]0,1[ −→ ]0,+∞[ (3.1) is nondecreasing, we will give a sufficient condition, which permits to prove that the ∗ integral operators Tϕ and Tϕ satisfy the Hardy inequalities [1, 8, 15, 16]. Theorem 3.1. Let p and q be two real numbers such that 1 <p≤ q<∞ (3.2) and p = p/(p − 1), q = q/(q − 1).Letν 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 let ϕ : ]0,1[ −→ ]0,+∞[ (3.3) be a measurable nondecreasing function. If there exists β ∈ [0,1] such that the function − ∞ r βq 1/q r x p (1 β) 1/p r −→ ϕ dµ(x) ϕ ν(x)dx (3.4) r x 0 r is bounded on ]0,+∞[, then there exists a positive constant Cp,q such that for every nonneg- ative measurable function f , ∞ 1/q ∞ 1/p q p Tϕ( f )(x) dµ(x) ≤ Cp,q f (x) ν(x)dx .
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