Journal of Function Spaces and ApplicationsVolume 2012 (2012), Article ID 315086, 12 pagesdoi:10.1155/2012/315086

Research Article

Bootstrapping Weighted Inequalities for Hankel Transform

M. Dziri

Department of Mathematics, Faculty of Sciences of Tunis, Tunisia

Received 30 October 2010; Accepted 12 April 2011

Academic Editor: Lars-Erik Persson

Copyright © 2012 M. Dziri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

A method of producing new inequalities for Hankel transform from known ones is given using the theory of positive integral operators. The new inequalities produced depend on six parameters, two real indices, two complex-valued measures, and two positive functions. The method may be iterated using the last inequality generated as input to the next stage.

1. Introduction

Weighted inequalities for the integral transforms with the general weights are of great importance in many branches of mathematics (functional analysis, integral equation, interpolation theory, etc.). They provide a tool to solve numerous problems related to the estimation of expressions with given integral transform. One of the most important problems is the characterization problem of the operator theory in function spaces such as the criteria of the continuity, compactness, and other qualitative estimations of a classical and nonclassical transformation. Many authors studied some generalizations of the Hardy inequalities and give some applications of these inequalities, [1–7].

In this work, we are interested in problems related to weighted inequalities for Hankel transform. More precisely, the main goal of this note is that from a given “input” Hankel inequality, a parametrized collection of “output” Hankel inequalities can be deduced. The idea is to exploit the close relationship of the Hankel transform to the operation of Hankel convolution and then to apply techniques from the theory of positive integral operators.

A single application of the theorem will produce new weighted Hankel inequalities from known ones. However, since the output inequalities are of the same form as the input inequality, it becomes possible to “bootstrap” the production of new inequalities by using the output at one stage as the input at the next. The implications of this sort of the iteration are not examined here.

Our investigation is inspired by the idea developed by Sinnamon [8], to the classical Fourier transform.

Throughout the paper, we adhere to conventions that are more common in the study of positive integral operators than in harmonic analysis generally. When integrals of non negative functions are involved, we will not concern ourselves with convergence; if the integral happens to take the value , then its appearance in formulas is to be interpreted according to arithmetic on . In particular, expressions of the form are all taken to be 0, while .

2. Hankel Convolution Structure and Hankel Transform

In the following, we give some basic definitions and some properties of Hankel transform analogous to those of the classical Fourier transform. For more details, see [9, 10].

For fixed , we define where denotes the Bessel function of order .

We denote that by , the space of all real-valued, measurable functions defined on with norm is finite, whereas that does not depend on denotes the space of those measurable functions defined on for which is finite.

Let be the area of the triangle with sides if such a triangle exists.

Set if exists and zero otherwise. We note that and that is symmetric in . Further we have the following basic formula: [11, page 411], from which it follows immediately, on setting , that Using (2.5), we may show that see [10, page 310].

For each in , it is clear, by (2.7), that the integral exists, so that we may define the Hankel transform of a function in by For is bounded and continuous for , see [9, page 336].

Proposition 2.1 (Haimo [9, page 338]). Let be such that and . Then

Proposition 2.2 (Trimèche [12]). The Hankel transform is an isomorphism from onto itself, and its inverse denoted , where is the space of even infinitely differentiable functions on , rapidly decreasing together with all their derivatives equipped with its usual topology.

Proposition 2.3 (Trimèche [12]). The Hankel transform on (the space of even tempered distribution on ), defined by is an isomorphism from onto itself, and .

Example 2.4. If we take where is the Dirac measure at zero, then Indeed, set Since , then we get This proves the result.

Haimo [9] and Hirschman [10] investigated a convolution operation and translation operation associated to the Hankel transformation. If , the Hankel convolution of and is defined by where the is the Hankel translation given by

From properties of the kernel , we deduce the following properties.(i)(ii)If ; then for all the function belongs to and we have (iii) Let ; then for all the function belongs to and we have

Proposition 2.5. (i) The Hankel convolution is commutative and associative. (ii) Assume that satisfies the Young conditions . Then extends to a continuous map from to and we have

Definition 2.6. Let and . The generalized convolution of and is defined by the following: where is the Hankel translation, which is given by relation (2.15).

Proposition 2.7. Let and ; then for all , we have

Proof. For and , we have that belongs to (the space of even infinitely differentiable function on ) and increase slowly.Thus , and we have This proves the relation (2.21) on the left.On the other hand, We complete the proof of the relation (2.21) on the right by the same way as the relation on the left.

Remark 2.8. Since the space is a dense subset of and it is easy to verify that ,, and are all continuous maps from to , thus, the identities in (2.21) extend to be valid for all .

3. Weighted Inequalities for Hankel Transforms

Let denote the nonnegative, extended real-valued function on the measure space . We say that a map has a formal adjoint provided for all .

Let and a map have formal adjoint. Fix a weight . For each , we define Note that by our convention, when , even if .

Proposition 3.1. If , almost everywhere, and , almost everywhere then for all .

This result is a special case ( and ) of Theorem  2.1 in [7].

The next result may be deduced from the last by duality argument. It is also a special case of Theorem  3.1 of [6]. Again note that when .

Proposition 3.2. Suppose has a formal adjoint , and with . Set Then for all .

Observe that if , then the formulas for the constants given in these propositions may take useful alternative forms. In the first, so In the second, so

Let be finite complex-valued Borel measure on and denote its absolute value. Then The fact that allows us to get that the Hankel transform of defined by is continuous for and that where Furthermore, the bounded function has a Hankel transform in the distributional sense and Moreover, if is real measurable function on and a finite complex-valued Borel measure on , we formally set where is the Hankel translation given by relation (2.15). If , then is defined in , and if , the integral (3.15) is defined in .

For a finite complex-valued Borel measure on , define the positive operator for all.

Thus, if , the convolution operator is well defined and we have

Proposition 3.3. For a finite complex-valued Borel measure on , the positive operator defined by relation (3.16) is autoadjoint. That is, has a formal adjoint operator and

Proof. If , then by using properties of the kernel and by applying Fubini Tonelli argument we have This completes the proof.

Before introducing any technical details, we give sketch of the argument behind the main following theorem. We suppose that the following Hankel inequality: for all is known to be valid for some fixed , and . For each appropriate function , define , where and are finite, complexe-valued Borel measures on .

Using Proposition 2.7, we get . For and arbitrary functions and we apply Propositions 3.1 and 3.2 to give formulas for and so that The arrow in the middle corresponds to the known “input” Hankel inequality, and the other arrows correspond to the Hankel convolution inequalities for the operators The inequality relating and that results from this composition is just the above inequality with new indices and and new weights and . This is our “output” inequality.

Theorem 3.4. Suppose is a positive constant, and are indices in , and and are nonnegative weight functions such that the Hankel inequality for all . Let and be finite complex-valued Borel measure and and positive functions on . For and satisfying and , set Also set If is bounded away from zero, then the Hankel inequality holds for all . Here .

Proof. Let and set . Since is bounded away from zero, so . Taking the Hankel transform of both sides of the equation and using identities (2.21) yields Proposition 3.2 shows that and the estimate gives Proposition 3.1 shows that and the trivial estimates give The three inequalities (3.30), (3.23), and (3.33) combine to yield (3.26) as required. This completes the proof.

When all indices are taken to 2 the theorem simplifies substantially.

Corollary 3.5. Suppose is a positive constant and and are nonnegative weight functions such that the Hankel inequality holds for all . Let and be finite complex-valued Borel measures and and positive function on . Set If is bounded away from zero, then the Hankel inequality holds for all .

A further simplification yields a following new weighted Hankel inequality.

Corollary 3.6. If is finite positive Borel measure and is a positive function on then for holds for all .

Proof. It is well known that the Hankel transform is continuous from into , where . This shows that the inequality (3.33) holds with and and . Thus if we take , and take to be the Dirac measure at zero. Then , which proves that reduce to the identity. Furthermore, since , then we get From the fact that is just convolution by and , it follows Then if we take , we get the result.

If we take and as Gaussian functions, that is, and , then we obtain the following new particular case of weighted Hardy-type inequality of Hankel transform.

Corollary 3.7. For and , we have

Proof. The result is obtained by using Propositions 2.1 and 2.7 and the fact that

Acknowledgment

The author is very grateful to Professor Gord Sinnamon for his help and critical comments.

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