New results for the Hankel two-wavelet multipliers

ABSTRACT The aim of this paper is to improve and generalize some results stated in Ghobber (A variant of the Hankel multiplier. Banach J Math Anal. 2018;12(1):144–166). We also give new results about the boundedness and compactness of Hankel two-wavelet multipliers on the space for .

One of the aims of the Fourier transform is the study of the theory of wavelet multipliers. This theory was initiated by He and Wong [11], developed in the paper [12] by Du and Wong, and detailed in the book [13] by Wong. Recently, Ghobber [3], with the aid of the harmonic analysis associated with the Bessel operator, has defined and studied the Hankel wavelet multipliers. In the same paper [3], Ghobber has given a trace formula for the Hankel wavelet multiplier as a bounded linear operator in the trace class from L 2 (R + , dν α (r)) into L 2 (R + , dν α (r)) in terms of the symbol and the two admissible wavelets.
The aim of this paper is to improve, generalize the results given in [3], and to give other new results on the L p boundedness and compactness of Hankel twowavelet multipliers.
The remainder of this paper will be arranged as follows.
In Section 2, we state some basic notions and results from harmonic analysis associated with the Bessel operator that will be needed throughout this paper.
The aim of Section 3 is to survey and revisit some results for the Hankel two-wavelet multipliers. More precisely, we give an explicit formula for the function that occurs in the lower bound for the trace norm as that of class Hankel two-wavelet multipliers. A result concerning the trace of products of Hankel two-wavelet multipliers is proved. In this section, we also introduce the generalized Landau-Pollak-Slepian operator which allows us to give an example of a Hankel two-wavelet multiplier. Our result generalize and improve Ghober's result concerning the Landau-Pollak-Slepian operator from the Hankel one-wavelet multiplier.
Section 4 is devoted to study the L p boundedness and compactness of these two-wavelet multipliers when suitable conditions on the symbols and the two admissible wavelets are satisfied.

Preliminaries
This section deals with some basic notions and results in harmonic analysis associated with the Bessel operator that will be needed in the sequel. For more details, we refer the interested reader to [14] for instance.
Throughout this paper, we fix α ≥ − 1 2 . We denote by: the Schwartz space of even rapidly decreasing functions on R. D * (R) the space of even C ∞ -functions on R which are of compact support. C p * (R) the space of even functions of class C p on R. For p ∈ [1, ∞], p denotes as in all that follows, the conjugate exponent of p.
For p = 2, we provide the space L 2 α (R + ) with the inner product Let α be the Bessel operator defined by Then, the following problem admits a unique solution j α (λ.), where j α is the function defined by and J α is the Bessel function of the first kind and index α (see [15,16]). The function j α has the following Poisson representation formula: and for all n ∈ N, we have The Hankel translation operator τ x is defined on It is well known that the Hankel translation operator satisfies the following product formula The Hankel convolution product of f,g in L 1 The Young inequality for the Hankel convolution product '' * α " reads as follows: if p,q and r ∈ [1, ∞] are such that 1/p + 1/q = 1 + 1/r, then for all f in L p α (R + ) and g in L q α (R + ), the function f * α g belongs to the space L r α (R + ), with the following inequality The following properties hold.

Schatten-von Neumann classes
Notation. We denote by • l p (N), 1 ≤ p < ∞, the set of all infinite sequences of real (or complex) numbers x := (x j ) j∈N , such that For p = 2, we provide this space l 2 (N) with the scalar product

Remark 3.1:
We mention that the space S 2 is the space of Hilbert-Schmidt operators, and S 1 is the space of trace class operators.

Definition 3.2:
The trace of an operator A in S 1 is defined by where (v n ) n is any orthonormal basis of L 2 α (R + ).

Remark 3.2:
If A is positive, then Moreover, a compact operator A on the Hilbert space belongs to the space of trace class S 1 . With this, we have for any orthonormal basis (v n ) n of L 2 α (R + ).

Revisiting Hankel two-wavelet multipliers
The aim of this subsection is to survey and revisit some results for the Hankel two-wavelet multipliers.

Definition 3.4:
Let u, v, σ be measurable functions on R + , we define the Hankel two-wavelet multiplier oper- In accordance with the different choices of the symbols σ and the different continuities required, we need to impose different conditions on u and v. We then obtain an operator on L p α (R + ). It is often more convenient to interpret the definition In the sequel of this section, u and v denote two any Below, we recall fourth results that have been proved by Ghobber [3].

Proposition 3.3 ([3]): Let σ be in L
and the following trace formula (3.10) In the remainder of this subsection we establish some new results for the Hankel two-wavelet multipliers.

Proposition 3.5: Let p ∈ [1, ∞). The adjoint of linear operator
Thus we get (3.13) whereσ is given bỹ (3.14) Proof: First, it is easy to see thatσ belongs to L 1 where s j , j = 1, 2 . . . refer to the positive singular values of P u,v (σ ) corresponding to φ j . We then get According to formula (3.15), we obtain Thanks to Fubini's theorem, we then deduce Now, using Plancherel's formula given by (2.12) we can write The proof is complete.

Remark 3.4:
If u = v and if σ is a real-valued and nonnegative functions in L 1 is a positive operator. So, by (3.3) and Proposition 3.3 we have Now we state a result concerning the trace of products of Hankel two-wavelet multipliers. Corollary 3.1: Let σ 1 and σ 2 be any real-valued and non-negative functions in L 1 α (R + ). We assume that u = v is a function in L 2 α (R + ) such that ||u|| L 2 α (R + ) = 1. Then, the Hankel two-wavelet multipliers P u,v (σ 1 ), P u,v (σ 2 ) are positive trace class operators and for any natural number n.
Proof: By Theorem 1 in the paper [17], we know that if A and B are in the trace class S 1 and are positive operators, then ∀n ∈ N, tr(AB) n ≤ (tr(A)) n (tr(B)) n .
So, if we take A = P u,v (σ 1 ), B = P u,v (σ 2 ) and we invoke the previous remark, we obtain the desired result, so completing the proof.

The generalized Landau-Pollak-Slepian operator
The purpose of this subsection is to give an example of a Hankel two-wavelet multiplier which improve and generalize Ghobber's result concerning the Landau-Pollak-Slepian operator from the Hankel onewavelet multiplier.
Let R, R 1 and R 2 be positive numbers. We define the linear operators by , where χ [0,s) stands for the characteristic function of the interval [0, s).
Adapting the proof of Proposition 20.1 in the book [13], we prove the following.

Proposition 3.6: The linear operators
are self-adjoint projections.
The bounded linear operator P R 2 Q R P R 1 : it is called the generalized Landau-Pollak-Slepian operator. We can show that the generalized Landau-Pollak-Slepian operator is in fact a Hankel two-wavelet multiplier.

Theorem 3.2: Let u and v be the functions on
.
Then the generalized Landau-Pollak-Slepian operator is unitary equivalent to a scalar multiple of the Hankel twowavelet multiplier In fact .

(3.18)
Proof: It is easy to see that u and v belong to By simple calculations we find for all f, g in S * (R) and hence the proof is complete.
The next result gives a formula for the trace of the generalized Landau-Pollak-Slepian operator.   P u,v (σ )

Boundedness for symbols in L
We wish to establish that P u,v (σ ) is a bounded operator on L p α (R + ). Let us start with the following propositions.

is a bounded linear operator and we have
Proof: Let f in L ∞ α (R + ). As above, from the relations (3.6), (2.10) and (2.3) we obtain Remark 4.1: Proposition 4.2 is also a corollary of Proposition 4.1, since the adjoint of Using an interpolation of Propositions 4.1 and 4.2, we get the following result.

Theorem 4.1: Let u and v be functions in L
. Then for all σ in L 1 α (R + ), there exists a unique bounded linear operator With a Schur technique, we can obtain an L p αboundedness result as in the previous theorem, but the estimate for the norm P u,v (σ ) B(L p α (R + )) is cruder.

. Then there exists a unique bounded linear operator
(4.4) Proof: Let N be the function defined on R + × R + by We have By simple calculations, it is not hard to see that Thus by Schur Lemma (cf. [18]), we can conclude that is a bounded linear operator for any 1 ≤ p ≤ ∞, and we have

Remark 4.2:
The previous theorem tells us that the unique bounded linear operator on L p α (R + ), 1 ≤ p ≤ ∞, obtained by interpolation in Theorem 4.1 is in fact the integral operator on L p α (R + ) with kernel N given by (4.5).
We can give another version of the L p α -boundedness. Firstly we generalize and we improve Proposition 4.2.

Proposition 4.3: Let σ be in L
Then the Hankel two-wavelet multiplier is a bounded linear operator, and we have Proof: For any f ∈ L p α (R + ), consider the linear functional From the relation (3.7) Using the relation (2.9), (2.3) and Hölder's inequality, we get . Thus, the operator I f is a continuous linear functional on L p α (R + ), and the operator norm .
, by the Riesz representation theorem, we have which concludes the proof.
Combining Proposition 4.1 and Proposition 4.3, we have the following theorem.
We can now state and prove the main result of this subsection.
Proof: Consider the linear functional Due to Proposition 4.1 and Proposition 3.1 we obtain (4.11) and (4.12) Therefore, by (4.11), (4.12) and the multi-linear interpolation theory, see Section 10.1 in [19], we get a unique bounded linear operator such that where By the definition of I, we have As the adjoint of P u,v (σ ) is P v,u (σ ), so P u,v (σ ) is a bounded linear map on L r α (R + ) with its operator norm 14) where Using an interpolation of (4.13) and (4.14), we have that, for any p ∈ [r, r ],
Proof: From the previous proposition, we only need to show that the conclusion holds for p = ∞. In fact, the operator P u,v (σ ) : L ∞ α (R + ) −→ L ∞ α (R + ) is the adjoint of the operator P v,u (σ ) : L 1 α (R + ) −→ L 1 α (R + ), which is compact by the previous proposition. Thus by the duality property, P u,v (σ ) : L ∞ α (R + ) −→ L ∞ α (R + ) is compact. Finally, by an interpolation of the compactness on L 1 α (R + ) and on L ∞ α (R + ) such as that given in the book [20, p.202-203] by Bennett and Sharpley, we conclude the proof.
The following result is an analogue of Theorem 4.4 for compact operators.

Remark 4.3:
Our hope that this work motivates the researchers to study the two-wavelet multipliers for the Bessel-Struve transform on R d , [21].

Conclusions
In the present paper, we have successfully studied many spectral theorems associated with the Hankel twowavelet multipliers. The obtained results have a novelty and contribution to the literature, and they improve and generalize the results of Ghobber [3].