Bootstrapping Weighted Inequalities for Hankel Transform

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 [0,∞]. In particular, expressions of the form 0∞,∞/∞,0/0,0⁰ are all taken to be 0, while ∞⁰=1.

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 𝛼>−1/2, we define𝑑𝜇_𝛼(𝑥)=𝑥^2𝛼+1𝑗2^𝛼Γ(𝛼+1)𝑑𝑥,_𝛼(𝑥)=2^𝛼Γ(𝛼+1)𝑥^−𝛼𝐽_𝛼(𝑥),(2.1) where 𝐽_𝛼 denotes the Bessel function of order 𝛼.

We denote that by 𝐿^𝑝𝛼,1≤𝑝<∞, the space of all real-valued, measurable functions 𝑓 defined on (0,∞) with norm‖𝑓‖_𝑝,𝛼=^∞0||𝑓(𝑥)||^𝑝𝑑𝜇_𝛼(𝑥)^1/𝑝(2.2) is finite, whereas 𝐿^∞𝛼=𝐿^∞ that does not depend on 𝛼 denotes the space of those measurable functions defined on (0,∞) for which‖𝑓‖_∞=esssup_0<𝑥<∞||𝑓(𝑥)||(2.3) is finite.

Let Δ(𝑥,𝑦,𝑧) be the area of the triangle with sides 𝑥,𝑦,𝑧 if such a triangle exists.

Set𝐷_𝛼(𝑥,𝑦,𝑧)=2^(𝛼−1)(Γ(𝛼+1))²Γ(𝛼+1/2)𝜋^1/2(𝑥𝑦𝑧)^−2𝛼𝐴(𝑥,𝑦,𝑧)^2𝛼−1(2.4) if Δ exists and zero otherwise. We note that 𝐷_𝛼(𝑥,𝑦,𝑧)>0 and that 𝐷_𝛼(𝑥,𝑦,𝑧) is symmetric in 𝑥,𝑦,𝑧. Further we have the following basic formula:𝑗_𝛼(𝑥𝑡)𝑗_𝛼(𝑦𝑡)=^∞0𝐷_𝛼(𝑥,𝑦,𝑧)𝑗_𝛼(𝑧𝑡)𝑑𝜇_𝛼(𝑧),(2.5) [11, page 411], from which it follows immediately, on setting 𝑡=0, that^∞0𝐷_𝛼(𝑥,𝑦,𝑧)𝑑𝜇_𝛼(𝑧)=1.(2.6) Using (2.5), we may show that||𝑗_𝛼(𝑥)||≤1,∀𝑥≥0,(2.7) see [10, page 310].

For each 𝑓 in 𝐿^1𝛼, it is clear, by (2.7), that the integral ∫^∞0𝑗_𝛼(𝑥𝑡)𝑓(𝑡)𝑑𝜇_𝛼(𝑡) exists, so that we may define the Hankel transform ℋ_𝛼(𝑓) of a function 𝑓 in 𝐿^1𝛼 byℋ_𝛼(𝑓)(𝑥)=^∞0𝑗_𝛼(𝑥𝑡)𝑓(𝑡)𝑑𝜇_𝛼(𝑡).(2.8) For 𝑓∈𝐿^1𝛼,ℋ_𝛼(𝑓) is bounded and continuous for 𝑥≥0, see [9, page 336].

Proposition 2.1 (Haimo [9, page 338]). Let 𝑓 be such that 𝑓 and ℋ_𝛼(𝑓)∈𝐿^1𝛼. Then 𝑓(𝑥)=^∞0ℋ_𝛼(𝑓)(𝑡)𝑗_𝛼(𝑥𝑡)𝑑𝜇_𝛼(𝑡)𝑎.𝑒.(2.9)

Proposition 2.2 (Trimèche [12]). The Hankel transform ℋ_𝛼 is an isomorphism from 𝒮_∗(ℝ) onto itself, and its inverse denoted ℋ^𝛼−1=ℋ_𝛼, where 𝒮_∗(ℝ) is the space of even infinitely differentiable functions on ℝ, rapidly decreasing together with all their derivatives equipped with its usual topology.

Proposition 2.3 (Trimèche [12]). The Hankel transform on 𝒮^∗(ℝ) (the space of even tempered distribution on ℝ), defined by ⟨ℋ_𝛼(𝑇),𝜑⟩=⟨𝑇,ℋ_𝛼(𝜑)⟩,𝑇∈𝒮^∗(ℝ),𝜑∈𝒮_∗(ℝ),(2.10) is an isomorphism from 𝒮^∗(ℝ) onto itself, and ℋ^𝛼−1=ℋ_𝛼.

Example 2.4. If we take 𝑇=𝛿₀ where 𝛿₀ is the Dirac measure at zero, then ℋ_𝛼𝛿₀=1.(2.11) Indeed, set 𝜑∈𝒮_∗(ℝ)ℋ_𝛼𝛿₀,𝜑=⟨𝛿₀,ℋ_𝛼(𝜑)⟩=ℋ_𝛼(𝜑)(0)=^∞0𝑗_𝛼(0)𝜑(𝑡)𝑑𝜇_𝛼(𝑡).(2.12) Since 𝑗_𝛼(0)=1, then we get ℋ_𝛼𝛿₀,𝜑=^∞0𝜑(𝑡)𝑑𝜇_𝛼(𝑡)=⟨1,𝜑⟩.(2.13) 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^1𝛼, the Hankel convolution 𝑓∗_𝛼𝑔 of 𝑓 and 𝑔 is defined by∀𝑥≥0,𝑓∗_𝛼𝑔(𝑥)=^∞0𝒯_𝑥(𝑓)(𝑦)𝑔(𝑦)𝑑𝜇_𝛼(𝑦),(2.14) where the 𝒯_𝑥 is the Hankel translation given by𝒯𝒯_𝑥(𝑓)(𝑦)=^∞0𝐷_𝛼(𝑥,𝑦,𝑧)𝑓(𝑧)𝑑𝜇_𝛼(𝑧),for𝑥>0,₀(𝑓)(𝑦)=𝑓(𝑦),∀𝑦≥0.(2.15)

From properties of the kernel 𝐷_𝛼(𝑥,𝑦,𝑧), we deduce the following properties.(i)𝒯_𝑥(𝑓)(𝑦)=𝒯_𝑦(𝑓)(𝑥).(2.16)(ii)If 𝑓∈𝐿^𝑝𝛼,𝑝≥1; then for all 𝑥≥0 the function 𝒯_𝑥𝑓 belongs to 𝐿^𝑝𝛼,𝑝≥1 and we have ‖‖𝒯_𝑥𝑓‖‖_𝑝,𝛼≤‖𝑓‖_𝑝,𝛼.(2.17)(iii) Let 𝑓∈𝒮_∗(ℝ); then for all 𝑥≥0 the function 𝒯_𝑥𝑓 belongs to 𝒮_∗(ℝ) and we haveℋ_𝛼𝒯_𝑥𝑓(𝑦)=𝑗_𝛼(𝑥𝑦)ℋ_𝛼(𝑓)(𝑦).(2.18)

Proposition 2.5. (i) The Hankel convolution ∗_𝛼 is commutative and associative. (ii) Assume that 𝑝,𝑞,𝑟∈[1,∞] satisfies the Young conditions 1/𝑝+1/𝑞=1+1/𝑟. Then (𝑓,𝑔)→𝑓∗_𝛼𝑔 extends to a continuous map from 𝐿^𝑝𝛼×𝐿^𝑞𝛼 to 𝐿^𝑟𝛼 and we have ‖‖𝑓∗_𝛼𝑔‖‖_𝑟,𝛼≤‖𝑓‖_𝑝,𝛼‖𝑔‖_𝑞,𝛼.(2.19)

Definition 2.6. Let 𝑓∈𝒮_∗(ℝ) and 𝑇∈𝒮^∗(ℝ). The generalized convolution of 𝑓 and 𝑇 is defined by the following: 𝑇∗_𝛼𝑓(𝑠)=⟨𝑇,𝒯_𝑠𝑓⟩,(2.20) 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 ℋ_𝛼𝑓∗_𝛼𝜎=ℋ_𝛼(𝑓)⋅ℋ_𝛼(𝜎),ℋ_𝛼𝑓⋅ℋ_𝛼(𝜈)=ℋ_𝛼(𝑓)∗_𝛼𝜈.(2.21)

Proof. For 𝜎∈𝒮^∗(ℝ) and 𝑓∈𝒮_∗(ℝ), we have that 𝜎∗_𝛼𝑓 belongs to 𝜉_∗(ℝ) (the space of even infinitely differentiable function on ℝ) and increase slowly.
Thus 𝜎∗_𝛼𝑓∈𝒮^∗(ℝ), and we have ℋ_𝛼𝜎∗_𝛼𝑓,𝜑==⟨𝜎∗_𝛼𝑓,ℋ_𝛼(𝜑)⟩=^∞0𝜎∗_𝛼𝑓(𝑥)ℋ_𝛼(𝜑)(𝑥)𝑑𝜇_𝛼(𝑥)_∞0=𝜎𝜎_𝑦,𝒯_𝑥𝑓(𝑦)ℋ_𝛼(𝜑)(𝑥)𝑑𝜇_𝛼(𝑥)_𝑦=𝜎,^∞0𝒯_𝑥𝑓(𝑦)ℋ_𝛼(𝜑)(𝑥)𝑑𝜇_𝛼(𝑥)_𝑦=𝜎,^∞0𝒯_𝑦𝑓(𝑥)ℋ_𝛼(𝜑)(𝑥)𝑑𝜇_𝛼(𝑥)_𝑦=𝜎,^∞0ℋ_𝛼𝒯_𝑦𝑓(𝑥)𝜑(𝑥)𝑑𝜇_𝛼(𝑥)_𝑦=𝜎,ℋ,^∞0_𝛼𝑗_𝛼(𝑥𝑦)ℋ_𝛼𝑓(𝑥)𝜑(𝑥)𝑑𝜇_𝛼(𝑥)𝜑ℋ_𝛼(𝑓)=⟨ℋ_𝛼(𝜎),ℋ_𝛼(𝑓)𝜑⟩=⟨ℋ_𝛼(𝜎)ℋ_𝛼(𝑓),𝜑⟩.(2.22) This proves the relation (2.21) on the left.
On the other hand, ℋ_𝛼𝑓ℋ_𝛼(𝜎),𝜑=⟨ℋ_𝛼(𝜎),𝑓ℋ_𝛼(𝜑)⟩=𝜎,ℋ_𝛼𝑓ℋ_𝛼(𝜑)=⟨𝜎,ℋ_𝛼(𝑓)∗_𝛼𝜑⟩.(2.23) 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^1𝛼 and it is easy to verify that 𝑓→ℋ_𝛼(𝑓∗_𝛼𝜎),𝑓→ℋ_𝛼(𝑓)ℋ_𝛼(𝜎),𝑓→ℋ_𝛼(𝑓ℋ_𝛼(𝜈)), and 𝑓→ℋ_𝛼(𝑓)∗_𝛼𝜈 are all continuous maps from 𝐿^1𝛼 to 𝐿^∞, thus, the identities in (2.21) extend to be valid for all 𝑓∈𝐿^1𝛼.

3. 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^∗∶𝐿^+𝛼→𝐿^+𝛼 provided^∞0𝑇(𝑓)(𝑥)𝑔(𝑥)𝑑𝜇_𝛼(𝑥)=^∞0𝑓(𝑥)𝑇^∗(𝑔)(𝑥)𝑑𝜇_𝛼(𝑥)(3.1) for all 𝑓,𝑔∈𝐿^+𝛼.

Let 1<𝑞≤𝑝<∞ and a map 𝑇∶𝐿^+𝛼→𝐿^+𝛼 have formal adjoint. Fix a weight 𝑢∈𝐿^+𝛼. For each ℎ∈𝐿^+𝛼, we define𝑣_ℎ=ℎ^1−𝑝𝑇^∗𝑢(𝑇ℎ)^𝑞−1,𝐶=^∞0ℎ^𝑝𝑣_ℎ𝑑𝜇_𝛼^{1/𝑞−1/𝑝}.(3.2) Note that by our convention, 𝐶_ℎ=1 when 𝑝=𝑞, even if ∫^∞0(ℎ(𝑥))^𝑝𝑣_ℎ(𝑥)𝑑𝜇_𝛼(𝑥)=∞.

Proposition 3.1. If 0<ℎ<∞𝜇_𝛼, almost everywhere, and 0<𝑇(ℎ)<∞𝑢𝜇_𝛼, almost everywhere then ^∞0(𝑇(𝑓)(𝑥))^𝑞𝑢(𝑥)𝑑𝜇_𝛼(𝑥)^1/𝑞≤𝐶^∞0(𝑓(𝑥))^𝑝𝑣_ℎ(𝑥)𝑑𝜇_𝛼(𝑥)^1/𝑝(3.3) for all 𝑓∈𝐿^+𝛼.

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 𝑢=ℎ𝑇𝑣^1−𝑝′𝑇^∗ℎ^𝑝′−1^1−𝑞,𝐶=^∞0ℎ^𝑞′𝑢^1−𝑞′^{1/𝑞−1/𝑝}.(3.4) Then ^∞0(𝑇(𝑓)(𝑥))^𝑞𝑢(𝑥)𝑑𝜇_𝛼(𝑥)^1/𝑞≤𝐶^∞0(𝑓(𝑥))^𝑝𝑣(𝑥)𝑑𝜇_𝛼(𝑥)^1/𝑝(3.5) for all 𝑓∈𝐿^+𝛼.

Observe that if 𝑝>𝑞, then the formulas for the constants 𝐶 given in these propositions may take useful alternative forms. In the first,^∞0ℎ^𝑝𝑣𝑑𝜇_𝛼=^∞0ℎ𝑇^∗𝑢(𝑇ℎ)^𝑞−1𝑑𝜇_𝛼=^∞0(𝑇ℎ)𝑢(𝑇ℎ)^𝑞−1=^∞0(𝑇ℎ)^𝑞𝑢𝑑𝜇_𝛼(3.6) so𝐶=^∞0(𝑇ℎ)^𝑞𝑢𝑑𝜇_𝛼^{1/𝑞−1/𝑝}.(3.7) In the second,^∞0ℎ^𝑞𝑢^1−𝑞=^∞0ℎ𝑇𝑣^1−𝑝′𝑇^∗ℎ^𝑝′−1𝑑𝜇^𝛼=^∞0𝑇^∗ℎ𝑣^1−𝑝𝑇^∗ℎ^𝑝−1(3.8) so𝐶=^∞0𝑇^∗ℎ^𝑝′𝑣^1−𝑝′𝑑𝜇_𝛼^{1/𝑞−1/𝑝}.(3.9)

Let 𝜎 be finite complex-valued Borel measure on (0,∞) and |𝜎| denote its absolute value. Then^∞0𝑑|𝜎|(𝑥)<∞.(3.10) The fact that |𝑗_𝛼(𝑥)|≤1 allows us to get that the Hankel transform of 𝜎 defined byℋ_𝛼(𝜎)(𝑡)=^∞0𝑗_𝛼(𝑥𝑡)𝑑𝜎(𝑥)(3.11) is continuous for [0,∞[ and that‖‖ℋ_𝛼(𝜎)‖‖_∞≤|||||𝜎|||||,(3.12) where|||||𝜎|||||=^∞0𝑑|𝜎|(𝑥)<∞.(3.13) Furthermore, the bounded function ℋ_𝛼(𝜎) has a Hankel transform in the distributional sense andℋ_𝛼ℋ_𝛼(𝜎)=𝜎.(3.14) Moreover, if 𝑓 is real measurable function on (0,∞) and 𝜎 a finite complex-valued Borel measure on (0,∞), we formally set𝑓∗_𝛼𝜎(𝑥)=^∞0𝒯_𝑥𝑓(𝑦)𝑑𝜎(𝑦),(3.15) where 𝒯_𝑥 is the Hankel translation given by relation (2.15). If 𝑓∈𝐿^1𝛼, then 𝑓∗_𝛼𝜎 is defined in 𝐿^1𝛼, 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 operator𝐾_𝜎(𝑓)=𝑓∗_𝛼|𝜎|(3.16) for all𝑓∈𝐿^+𝛼.

Thus, if 𝑓∈𝐿^1𝛼∪𝐿^∞, the convolution operator 𝑓∗_𝛼𝜎 is well defined and we have||𝑓∗_𝛼𝜎||≤𝐾_𝜎||𝑓||.(3.17)

Proposition 3.3. 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 𝐾^∗𝜎=𝐾_𝜎.(3.18)

Proof. If 𝑓,𝑔∈𝐿^+𝛼, then by using properties of the kernel 𝐷_𝛼 and by applying Fubini Tonelli argument we have ^∞0𝐾_𝜎(𝑓)(𝑥)𝑔(𝑥)𝑑𝜇_𝛼(𝑥)=^∞0^∞0=𝒯_𝑥𝑓(𝑦)𝑑|𝜎|(𝑦)𝑔(𝑥)𝑑𝜇_𝛼(𝑥)_∞0^∞0=𝑔(𝑥)^∞0𝐷_𝛼(𝑥,𝑦,𝑧)𝑓(𝑧)𝑑𝜇_𝛼(𝑧)𝑑𝜇_𝛼(𝑥)𝑑|𝜎|(𝑦)_∞0𝑓(𝑧)^∞0𝒯_𝑧𝑔(𝑦)𝑑|𝜎|(𝑦)𝑑𝜇(𝑧)=^∞0𝑓(𝑧)𝒦_𝜎𝑔(𝑧)𝑑𝜇_𝛼(𝑧).(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:^∞0ℋ_𝛼(𝑓)^𝑞𝑢𝑑𝜇_𝛼^1/𝑞≤^∞0||𝑓||^𝑝𝑣𝑑𝜇_𝛼^1/𝑝(3.20) for all 𝑓∈𝐿^1𝛼∩𝐿^𝑝𝛼(𝑣) is known to be valid for some fixed 𝑝₀,𝑞₀,𝑢₀, and 𝑣₀. For each appropriate function 𝑔, define 𝑓=(𝑔∗_𝛼𝜎)/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)∗_𝛼𝜈)/ℋ_𝛼(𝜎). For 𝑝₁≥𝑝₀,𝑞₁≤𝑞₀ and arbitrary functions ℎ_𝜎 and ℎ_𝜈 we apply Propositions 3.1 and 3.2 to give formulas for 𝑢₁ and 𝑣₁ so that𝐿^𝑝𝛼𝑣₁𝑔⟶⟼𝐿𝑓^𝑝0_𝛼⟼𝑣₀⟶ℋ_𝛼𝐿^𝑞0_𝛼𝑢₀⟶𝐿^𝑞1_𝛼𝑢₁,(𝑓)⟼ℋ_𝛼(𝑔).(3.21) The arrow in the middle corresponds to the known “input” Hankel inequality, and the other arrows correspond to the Hankel convolution inequalities for the operators𝑔⟶𝑔∗^𝛼𝜎ℋ_𝛼(𝜈),ℋ^𝛼(𝑓)⟶ℋ^𝛼(𝑓)∗_𝛼𝜈ℋ_𝛼(𝜎).(3.22) The inequality relating ℋ_𝛼(𝑔) and 𝑔 that results from this composition is just the above inequality with new indices 𝑝₁ and 𝑞₁ and new weights 𝑢₁ and 𝑣₁. This is our “output” inequality.

Theorem 3.4. Suppose 𝐶₀ is a positive constant, 𝑝₀ and 𝑞₀ are indices in (1,∞), and 𝑢₀ and 𝑣₀ are nonnegative weight functions such that the Hankel inequality ^∞0ℋ_𝛼(𝑓)(𝑥)^𝑞0𝑢₀(𝑥)𝑑𝜇_𝛼(𝑥)^1/𝑞0≤𝐶₀^∞0||𝑓(𝑥)||^𝑝0𝑣₀(𝑥)𝑑𝜇_𝛼(𝑥)^1/𝑝0(3.23) for all 𝑓∈𝐿^1𝛼. Let 𝜎 and 𝜈 be finite complex-valued Borel measure and ℎ_𝜎 and ℎ_𝜈 positive functions on [0,∞[. For 𝑝₁ and 𝑞₁ satisfying 1<𝑝₀≤𝑝₁<∞ and 1<𝑞₁≤𝑞₀<∞, set 𝑤_𝜈𝑤_𝜎=||ℋ(𝜈)||^−𝑝0𝑣₀,𝑣₁=ℎ^1−𝑝1_𝜎𝐾_𝜎𝑤^𝜎𝐾_𝜎ℎ_𝜎^𝑝0−1,=ℎ_𝜈𝐾^𝜈𝑢^1−𝑞′0𝐾_𝜈ℎ_𝜈^𝑞′0−1^1−𝑞1,𝑢₁=||ℋ(𝜎)||^𝑞1𝑤_𝜈.(3.24) Also set 𝐶_𝜎=^∞0ℎ^𝑝1_𝜎𝑣₁𝑑𝜇_𝛼^{1/𝑝0−1/𝑝1},𝐶_𝜈=^∞0ℎ^𝑞′1_𝜈𝑤^{1−𝑞1𝜈}𝑑𝜇_𝛼^{1/𝑞1−1/𝑞0}.(3.25) If ℋ_𝛼(𝜈) is bounded away from zero, then the Hankel inequality ^∞0ℋ_𝛼(𝑔)(𝑥)^𝑞1𝑢₁(𝑥)𝑑𝜇_𝛼(𝑥)^1/𝑞1≤𝐶₁^∞0||𝑔(𝑥)||^𝑝1𝑣₁(𝑥)𝑑𝜇_𝛼(𝑥)^1/𝑝1(3.26) holds for all 𝑔∈𝐿^1𝛼. Here 𝐶₁=𝐶_𝜈𝐶₀𝐶_𝜎.

Proof. Let 𝑔∈𝐿^1𝛼 and set 𝑓=(𝑔∗_𝛼𝜎)/ℋ_𝛼(𝜈). Since ℋ_𝛼(𝜈) is bounded away from zero, 1/ℋ_𝛼(𝜈)∈𝐿^∞ so 𝑓∈𝐿^1𝛼. Taking the Hankel transform of both sides of the equation 𝑔∗_𝛼𝜎=𝑓ℋ_𝛼(𝜈) and using identities (2.21) yields ℋ_𝛼(𝑔)ℋ_𝛼(𝜎)=ℋ_𝛼(𝑓)∗_𝛼𝜈.(3.27) Proposition 3.2 shows that ^∞0𝒦_𝜈ℋ_𝛼(𝑓)^𝑞1𝑤_𝜈𝑑𝜇_𝛼^1/𝑞1≤𝐶_𝜈^∞0||ℋ_𝛼(𝑓)||^𝑞0𝑢₀𝑑𝜇_𝛼^1/𝑞0(3.28) and the estimate ||ℋ_𝛼(𝑔)ℋ_𝛼(𝜎)||=||ℋ_𝛼(𝑓)∗_𝛼𝜈||≤𝒦_𝜈||ℋ_𝛼(𝑓)||(3.29) gives ^∞0||ℋ_𝛼(𝑔)||^𝑞1𝑢₁𝑑𝜇_𝛼^1/𝑞1≤𝐶_𝜈^∞0||ℋ_𝛼(𝑓)||^𝑞0𝑢₀𝑑𝜇_𝛼^1/𝑞0.(3.30) Proposition 3.1 shows that ^∞0𝒦_𝜎||𝑔||^𝑝0𝑤_𝜎𝑑𝜇_𝛼^1/𝑝0≤𝐶_𝜎^∞0||𝑔||^𝑝1𝑣₁𝑑𝜇_𝛼^1/𝑝1(3.31) and the trivial estimates ||𝑓ℋ_𝛼(𝜈)||=||𝑔∗_𝛼𝜎||≤𝒦_𝜎||𝑔||(3.32) give ^∞0||𝑓||^𝑝0𝑣₀𝑑𝜇_𝛼^1/𝑝0≤𝐶_𝜎^∞0||𝑔||^𝑝1𝑣₁𝑑𝜇_𝛼^1/𝑝1.(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.

Corollary 3.5. Suppose 𝐶 is a positive constant and 𝑢₀ and 𝑣₀ are nonnegative weight functions such that the Hankel inequality ^∞0ℋ_𝛼(𝑓)(𝑥)²𝑢₀(𝑥)𝑑𝜇_𝛼(𝑥)≤𝐶^∞0||𝑓(𝑥)||²𝑣₀(𝑥)𝑑𝜇_𝛼(𝑥)(3.34) holds for all 𝑓∈𝐿^1𝛼. Let 𝜎 and 𝜈 be finite complex-valued Borel measures and ℎ_𝜎 and ℎ_𝜈 positive function on [0,∞[. Set 𝑣₁=1ℎ_𝜎𝐾_𝜎𝑣⁰𝐾_𝜎ℎ_𝜎||ℋ_𝛼(𝜈)||²,𝑢¹=||ℋ^𝛼(𝜎)||²ℎ_𝜈𝐾_𝜈𝐾_𝜈ℎ_𝜈/𝑢₀.(3.35) If ℋ_𝛼(𝜈) is bounded away from zero, then the Hankel inequality ^∞0ℋ_𝛼(𝑔)(𝑥)²𝑢₁(𝑥)𝑑𝜇_𝛼(𝑥)≤𝐶^∞0||𝑔(𝑥)||²𝑣₁(𝑥)𝑑𝜇_𝛼(𝑥)(3.36) holds for all 𝑔∈𝐿^1𝛼.

A further simplification yields a following new weighted Hankel inequality.

Corollary 3.6. If 𝜎 is finite positive Borel measure and ℎ is a positive function on [0,∞[then for 𝛼>−1/2^∞0||ℋ_𝛼(𝑔)(𝑥)||²||ℋ_𝛼(𝜎)(𝑥)||²𝑑𝜇_𝛼(𝑥)≤𝐶^∞0||𝑔(𝑥)||²𝜎∗_𝛼𝜎∗_𝛼ℎ(𝑥)ℎ(𝑥)𝑑𝜇^𝛼(𝑥)(3.37) holds for all 𝑔∈𝐿^1𝛼.

Proof. It is well known that the Hankel transform is continuous from 𝐿^𝑝𝛼 into 𝐿^𝛼𝑝, where 1/𝑝+1/𝑝′=1,1≤𝑝≤2. This shows that the inequality (3.33) holds with 𝑝=2 and 𝑢₀=1 and 𝑣₀=1. Thus if we take ℎ_𝜈=1,ℎ_𝜎=ℎ, and take 𝜈 to be the Dirac measure at zero. Then 𝒦_𝜈=𝑓∗_𝛼𝛿₀=𝑓, which proves that 𝒦_𝜈 reduce to the identity. Furthermore, since ℋ_𝛼(𝜈)=1, then we get 𝑢₁=||ℋ^𝛼(𝜎)||²ℎ_𝜈𝐾_𝜈𝐾_𝜈ℎ_𝜈/𝑢₀=||ℋ^𝛼(𝜎)||²ℎ_𝜈ℎ_𝜈/𝑢₀=||ℋ_𝛼(𝜎)||².(3.38) From the fact that 𝐾_𝜎 is just convolution by 𝜎 and ℋ_𝛼(𝜈)=1, it follows 𝑣₁=1ℎ_𝜎𝐾_𝜎𝑣⁰𝐾_𝜎ℎ_𝜎||ℋ_𝛼(𝜈)||²=ℎ1_𝜎𝒦_𝜎𝑣₀𝒦_𝜎ℎ_𝜎.(3.39) Then if we take ℎ_𝜎=ℎ, we get the result.

If we take 𝜎 and ℎ as Gaussian functions, that is, 𝜎_𝑡=𝐶₁𝑒^−𝑡𝑥2 and ℎ_𝑠=𝐶₂𝑒^−𝑠𝑥2, then we obtain the following new particular case of weighted Hardy-type inequality of Hankel transform.

Corollary 3.7. For 𝛼>−1/2 and 𝑠,𝑡>0, we have ^∞0||ℋ_𝛼(𝑔)(𝑥)||²𝑒^{−(1/2𝑡)𝑥2}𝑑𝜇_𝛼(𝑥)≤𝐶_{𝛼,𝑡,𝑠}^∞0||𝑔(𝑥)||²𝑒^{(2𝑠2/(2𝑠+𝑡))𝑥2}𝑑𝜇_𝛼(𝑥).(3.40)

Proof. The result is obtained by using Propositions 2.1 and 2.7 and the fact that ℋ_𝛼𝜎_𝑡(𝑥)=𝐶₁𝑒^{−𝑥2/4𝑡}(2𝑡)^𝛼+1,ℋ_𝛼ℎ_𝑠(𝑥)=𝐶₂𝑒^{−𝑥2/4𝑠}(2𝑠)^𝛼+1.(3.41)

Acknowledgment

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