Bootstrapping Weighted Inequalities for Hankel Transform

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.


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.

Journal of Function Spaces and Applications
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 0, ∞ . In particular, expressions of the form 0∞, ∞/∞, 0/0, 0 0 are all taken to be 0, while ∞ 0 1.

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 α > −1/2, we define where J α denotes the Bessel function of order α. We denote that by L For f ∈ L 1 α , H α f is bounded and continuous for x ≥ 0, see 9, page 336 .
is an isomorphism from S * R onto itself, and H −1 Example 2.4. If we take T δ 0 where δ 0 is the Dirac measure at zero, then

2.12
Since j α 0 1, 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 f, g ∈ L 1 α , the Hankel convolution f * α g of f and g is defined by 2.14 where the T x is the Hankel translation given by T 0 f y f y , ∀y ≥ 0.

2.15
From properties of the kernel D α x, y, z , we deduce the following properties. i ii If f ∈ L p α , p ≥ 1; then for all x ≥ 0 the function T x f belongs to L p α , p ≥ 1 and we have 2.17 iii Let f ∈ S * R ; then for all x ≥ 0 the function T x f belongs to S * R and we have 6. Let f ∈ S * R and T ∈ S * R . The generalized convolution of f and T is defined by the following: where T s is the Hankel translation, which is given by relation 2.15 .
Proposition 2.7. Let σ and ν ∈ S * R ; then for all f ∈ S * R , we have Proof. For σ ∈ S * R and f ∈ S * R , we have that σ * α f belongs to ξ * R the space of even infinitely differentiable function on R and increase slowly.
Thus σ * α f ∈ S * R , and we have

2.22
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 S * R is a dense subset of L 1 α and it is easy to verify that f → H α f * α σ , f → H α f H α σ , f → H α fH α ν , and f → H α f * α ν are all continuous maps from L 1 α to L ∞ , thus, the identities in 2.21 extend to be valid for all f ∈ L 1 α .

Weighted Inequalities for Hankel Transforms
Let L α denote the nonnegative, extended real-valued function on the measure space 0, ∞ , dμ α . We say that a map T : L α → L α has a formal adjoint T * :

3.2
Note that by our convention, C h 1 when p q, even if This result is a special case n 1 and r 1 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 C 1 when p q. Proposition 3.2. Suppose 1 < q ≤ p < ∞, T : L α → L α has a formal adjoint T * , and v, h ∈ L α with 0 < h < ∞. Set

3.4
Then Observe that if p > q, then the formulas for the constants C given in these propositions may take useful alternative forms. In the first,

3.7
In the second,

3.9
Let σ be finite complex-valued Borel measure on 0, ∞ and |σ| denote its absolute value. Then 3.10 The fact that |j α x | ≤ 1 allows us to get that the Hankel transform of σ defined by is continuous for 0, ∞ and that Furthermore, the bounded function H α σ has a Hankel transform in the distributional sense and Moreover, if f is real measurable function on 0, ∞ and σ a finite complex-valued Borel measure on 0, ∞ , we formally set

Journal of Function Spaces and Applications
where T x is the Hankel translation given by relation 2.15 . If f ∈ L 1 α , then f * α σ is defined in L 1 α , and if f ∈ L ∞ , the integral 3.15 is defined in L ∞ . For σ a finite complex-valued Borel measure on 0, ∞ , define the positive operator Thus, if f ∈ L 1 α ∪ L ∞ , the convolution operator f * α σ is well defined and we have For σ a finite complex-valued Borel measure on 0, ∞ , the positive operator K σ defined by relation 3.16 is autoadjoint. That is, K σ has a formal adjoint operator K * σ and Proof. If f, g ∈ L α , then by using properties of the kernel D α and by applying Fubini Tonelli argument we have

3.19
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 f ∈ L 1 α ∩ L p α v is known to be valid for some fixed p 0 , q 0 , u 0 , and v 0 . For each appropriate function g, define f g * α σ /H α ν , where σ and ν are finite, complexe-valued Borel measures on 0, ∞ .
Using Proposition 2.7, we get H α g H α f * α ν /H α σ . For p 1 ≥ p 0 , q 1 ≤ q 0 and arbitrary functions h σ and h ν we apply Propositions 3.1 and 3.2 to give formulas for u 1 and v 1 so that

3.21
Journal of Function Spaces and Applications 9 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 H α g and g that results from this composition is just the above inequality with new indices p 1 and q 1 and new weights u 1 and v 1 . This is our "output" inequality.

3.24
Also set

3.25
If H α ν is bounded away from zero, then the Hankel inequality holds for all g ∈ L 1 α . Here C 1 C ν C 0 C σ .
Proof. Let g ∈ L 1 α and set f g * α σ /H α ν . Since H α ν is bounded away from zero, 1/H α ν ∈ L ∞ so f ∈ L 1 α . Taking the Hankel transform of both sides of the equation g * α σ fH α ν and using identities 2. give

3.33
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.