Besov-type spaces for the κ -Hankel wavelet transform on the real line

: In this paper, we shall introduce functions spaces as subspaces of L pκ ( R ) that we call Besov- κ -Hankel spaces and extend the concept of κ -Hankel wavelet transform in L pκ ( R ) space. Subsequently we will characterize the Besov- κ -Hankel space by using κ -Hankel wavelet coefficients.


Introduction
Besov spaces B p,q α (R) are subspaces of L p (R) , having functions of smoothness α and q gives a ner graduation to the smoothness. It is extension of classical Sobolev and Hölder spaces. It is also expressed as interpolation space lies in between two Sobolev spaces H x p and H y p ( ≤ p, q ≤ ∞) with α = ( − β)x + βy ; α, x, y ∈ R, β ∈ ( , ). The Besov spaces B p,q α (α ∈ R and ≤ p, q ≤ ∞) were recognized in about sixty th decade of ninteen th century [3,4]. They were generalized in mid of seventies by various authors in di erent directions with different ideas. The classical de nition of Besov spaces depends on the modulus of smoothness [11,13]. The Littlewood-Paley theory interlink Besov spaces with the fourier transform. Michael Frazier and Björn Jawerth [6] characterise Besov spaces with the help of Calderon's formula while Dang Vu Giang and Ferenc Moricz [7] characterise Besov spaces in terms of it's Riesz mean and Dirichlet integral. In 1996, Valerie Perrier and Claude Basdevant [10] characterise Besov spaces by the behavior of the continuous wavelet coe cients for α ∈ R + , α ∉ Z + . Jorge J. Betancor and L. Rodríguez-Mesa [5] pave the way for exploration of Besov-Hankel spaces and characterized by mean of the Bochner-Riesz mean and the partial Hankel integrals. Recently Salem Ben Saïd, Mohamed Amine Boubatra, Mohamed Si [2] come out with deformed Besov-Hankel spaces and characterised it in terms of the deformed Bochner-Riesz means and the deformed partial Hankel integral. Hatem Mejjaoli, khalifa Trimèche [8] presented κ-Hankel wavelet transform in the year 2020. Using the approach of Betancor et al. [5] and Ben Saïd et al. [2], we de ne Besov κ-Hankel space and by exploiting the technique of Perrier et al. [10] we characterise Besov κ-Hankel space with the help of continuous κ-Hankel wavelet transform. Present paper is organized in following manner: section 1 is introductory, in which we de ne the development from Besov space to Besov κ-Hankel space with the help of κ-Hankel wavelet and it's characterisation with time period. section 2 is preliminary, in which we recall some properties of κ-Hankel transform , Besov κ-Hankel space and continuous κ-Hankel wavelet transform. Section 3 is related the continuous κ-Hankel wavelet transform in L p κ (R) . In the section 4, we characterize Besov κ-Hankel norms in terms of continuous κ-Hankel wavelet transform.

Preliminary
In this paper, we denotes the weighted L p κ (R) space as The κ-Hankel transformation of the function f ∈ L κ (R) for order κ ≥ is de ned as see [8] ( and Bκ(µ, x) is the κ-Hankel kernel given as Here ω n denote the normalized Bessel function of index λ. Iff ∈ L κ (R), then the inverse of κ-Hankel transformations is given by Also, Parseval's formula of the κ-Hankel transformation for f , g ∈ L κ (R) L κ (R) is given by By denseness and continuity the Parseval's formula can be extended to all f , g ∈ L κ . Hence κ is isometry on L κ (R). If f , g ∈ L κ (R), then the convolution associated with the κ-Hankel transform is de ned as see [1] where the operatore τ κ x is κ-Hankel translation is given by From (1) and (3), we have and moreover, where Mκ is independent of x and y such that Mκ Now, we recall some properties of κ-Hankel convolution [1] which are useful throught the paper.

The Continuous κ-Hankel Wavelet Transform in L p κ (R)
In this section we extend the concept of κ-Hankel wavelet transform on L p κ (R).
Theorem 3.1. Suppose that a function ψ ∈ L κ (R) satis es the admissibility condition whereψ denote the κ-Hankel transform of ψ then continuous κ-Hankel wavelet transform is a bounded linear operator Proof. Let Sp denote the space L κ (R + , dσκ(a) a κ ) × L p κ (R) associated to the norm If we take p = , then from Plancherel's theorem [8]: where C κ,ψ = ∞ ω − |ψ(ω)| dω < ∞, if ψ is real. From singular integral theorem, the operators on L κ (R + , dσκ(a) a κ ) holds inequality: where the constant Cp depends only on p and ψ(see [12]). Due to duality the inequality is also valid for < p < ∞. It follows that Conversely suppose that f ∈ L κ (R) ∩ L p κ (R). Since continuous κ-Hankel wavelet transform is isomerty for using Schwarz inequality and then Holder's inequality, we have where p + q = .
From equation (6), we get By Density theorem

. An inversion formula
Theorem 3.3. Let us consider ϕ ∈ L p κ (R) with < p < ∞ and ψ is a real wavelet. Then The equality holds in L p κ (R) sense and the integral of right hand side have to be taken in the sense of distributions.
Proof. The proof followed from theorem 3.2.

Characterization of Besov κ-Hankel Norms
In present section, By using the above results, we characterize the Besov κ-Hankel norms associated with the κ-Hankel wavelet transform.
Proof. By the de nition of continuous κ-Hankel wavelet transform and equation (4), we have Taking L p κ − norm of the wavelet coe cient Using Minkowski inequality of integrability for p ≠ ∞ Suppose that q < ∞ and integrating w.r.t. a, we get Again using Minkowski integrabilty inequality Next theorem is the converse of the above theorem. The κ-Hankel wavelet coe cients is su cient to characterize Besov κ-Hankel spaces.