Let ℝd be a real d-dimensional Euclidean space with scalar product 〈x,y〉 and norm ‖x‖=〈x,x〉. Let Sd−1 be the unit Euclidean sphere in ℝd, Δ be the Laplace operator, dμ(x)=(2π)−d/2dx be the normalized Lebesgue measure, Lp(ℝd),1≤p<+∞ be the Lebesgue space with norm ‖f‖p:=(∫ℝd|f|pdμ)1/p, and S(ℝd) be the Schwartz space.

The Euclidian Fourier transform is defined by

We introduce the real parameters α,β such that α,β>0 and let f be a measurable function on ℝ satisfying |f(x)|≤λe−αx2 and |F(y)|≤λe−βξ2. The function f reduces to the null function if αβ≥14. A generalization of Hardy's theorem is estabilished by Miyachi in [18] where the following is shown.

If f is a measurable function on ℝ such that

and

where α,λ are two positive constants, then f is a constant multiple of e−αx2.

A large family of theorems have been investigated in recent years, the most classical one is Titchmarsh's theorem [9, 12, 17], which says that a function and its classical Fourier transform on the real line cannot both be clearly localized. To be more precise, it is impossible for a non-zero function and its classical Fourier transform (CFT) to both be small. The notion of smallness have been given many defintions. See, for example, Hardy's work in [13], Cowling et al. in [7] and Miyachi in [18].

In harmonic analysis theory, an important role is played by the following infinitisimal generator operator

where Δk is the Dunkl Laplacian given by relation (2.1).

In the last decade, Ben Saïd et al. have generalized in [4] the classical situation by introducing a generalized integral transform Fk,a, which is defined by

where k is a parameter comes from the Dunkl differential-difference operators, and a arises from the interpolation of two minimal unitary representations of two different reductive groups, see [4, 3]. More recently, a convolution structure has been studied for this transform by the author jointly Negzaoui and Sifi in [5].

The transform Fk,a specialises to various well-known integral transforms:

• ▶ the classical Fourier transform, [14] (a=2,k=0).

• ▶ the classical Hankel transform, [15] (a=1,k=0).

• ▶ the Dunkl transform, [11] (a=2,k>0).

• ▶ the k-Hankel transform, [1] (a=1,k>0).

In this paper, we pin down the last case (k-Hankel transform Fk), we study Miyachi's theorem on ℝd. Analogous results have been studied by Chouchene et al. in [6] for the Dunkl transform, Loualid in [16] for the generalized Dunkl transform, by Daher in [8] for Jacobi-Dunkl transform, and Daher et al. in [10] for which a generalization of Miyachi's theorem on ℝd is established for the generalized Fourier transforms, the Chèbli-Triméche and the Dunkl transforms.

We briefly summarize the contents of this paper. In § 2, we collect some background materials for the harmonic analysis associated with the k-Hankel transform on ℝd. In § 3, we provide keys lemmas used to prove our main result of Miyachi's theorem for the k-Hankel transform.

Let R⊂ℝd∖0 be a root system, R+ be a positive subsystem of R, G(R)⊂O(d) be a reflection group formed by reflections σa:a∈R, where σa is a reflection with respect to hyperplane 〈a,x〉=0, and k:R↦R+ be a multiplicity function invariant under groups G. This is a G-invariant positive homogeneous of degree 2γk−1, where

Let's consider the weight and the Dunkl measure given respectively on ℝd by

Denote by λk=2γk+d−1 the homogeneous dimension of the system.

The Dunkl operators Tj,1≤j≤d on ℝd are the first-order differential-difference operators, introduced by Dunkl in [11] are given by

where ∂j denotes the usual partial derivatives and e1,...,ed the standard basis on ℝd. A fundamental property of these differential-difference operators is their commutativity:

The Dunkl Laplacian Δk=∑ j=1dTj2, is given explicitly for a regular function f, by

where ∇ and Δ are the classical gradient and Laplacian operators.

2.1. The k-Hankel transform

We define the kernel

with z=2‖x‖‖y‖(1+〈x‖x‖,.〉). Here, Vk denotes the Dunkl intertwining operator defined by

where σx is a probability measure on ℝd with support in the closed ball B(0,‖x‖) of center 0 and radius ‖x‖. The expression in (2.2) is Lebesgue integrable on ℝd, and J˜ν(z)=(z2)νJν(z), Jν being the Bessel function of first kind and index ν.

Let us define the space:

D(ℝd) is the space of test functions (that is infinitely differentiable functions f:ℝd↦ℂ with compact support contained in ℝd).

Let tVk denotes the dual operator of Vk on which is a topological automorphism of D(ℝd). It is defined by: There exists a positive probability measure νy on ℝd with support in the closed ball B(0,‖x‖) of center 0 and radius ‖x‖ such that

Relation (2.3) is also given in terms of the k-Hankel transform and the classical Fourier transform F by the following relation

The operators Vk and tVk possess the following property : For all f∈D(ℝd) and g∈E(ℝd) we have

If we take g=1 in (2.5), we obtain

Moreover, for all x,y∈ℝd, the kernel Bk(x,.) possesses the following properties:

For all x,y∈ℝd, we have

where

The kernel Bk plays an important role in the development of the k-Hankel transform, for more details, we refer the reader to [1, 2]. Relation (2.7) asserts that the k-Hankel transform is well defined for all f∈L1(ℝd,μk)

where c_k is the Macdonald-Mehta-Selberg integral given by

We collect some properties of the k-Hankel transform (for more details see [2]).

Proposition 2.1.1.

• (i) (Inversion formula) The k-Hankel transform Fk is a topological isomorphism of S(ℝd) and its inverse is given by

  Fk−1=Fk.

• (ii) (Plancherel Theorem) The k-Hankel transform extends to an isometry of L2(ℝd,μk). In particular, we have

  ‖Fkf‖L2(ℝd,μk)=‖f‖L2(ℝd,μk).

The definition of the k-Hankel transform permets us to define the generalized translation operator on L2(ℝd,μk).

The generalized translation operator f↦τykf, y∈ℝd is defined on L2(ℝd,μk) by

It plays the role of the arbitrary translation τykf(.)=f(.−y) in ℝd, since the Euclidean Fourier transform satisfies τykf^(x)=e−i〈x,y〉f^(x).

In the analysis of this translation a particular role is played by the space

Note that Ak(ℝd)⊂L1∩L∞(ℝd,μk) and hence is a subspace of L2(ℝd,μκ).

The operator τyk satisfies the following properties:

Proposition 2.1.2. Assume that f∈Ak(ℝd) and g∈L1∩L∞(ℝd,μk). Then

• (i) For every x,y∈ℝd, we have τykf(x)=τxkf(y).

• (ii) For every y∈ℝd, the operator τyk satisfies

  ∫ℝd τykf(x)g(x)dμk(x)=∫ℝdf(x)τykg(x)dμk(x).

A formula of τykf is known, at the moment, only in two cases.

Case 1. G=ℤ2 (see [1]).

Case 2. where a formula of τykf is known when f is a radial function in Ak(ℝd)(f(x)=fo(‖x‖)), G being any reflection group(see [2])

where ≺_x,y,u;.≽=‖x‖+‖y‖−2‖x‖‖y‖(1+〈x‖x‖,.〉)u.

According to the positivity of the intertwining operator (2.2) it follows that τykf(x)≥0 for all y∈ℝd,f(x)=f0(‖x‖)≥0.

Some properties of τykf (f being radial) follow from this formula. This is collected in the following proposition.

Proposition 2.1.3. (See [2])

• (i) For every f∈Lrad1(ℝd,μk) the subspace of radial functions in L1(ℝd,μk), we have:

  ∫ℝd τykf(x)dμk(x)=∫ℝdf(x)dμk(x).

• (ii) For 1≤p≤2, τyk:Lradp(ℝd,μk)↦Lradp(ℝd,μk), is a bounded operator.

• (iii) The generalized translation operator is well defined on L2(ℝd,μk) by the relation

  Fk(τxkf)(y)=Bk(x,y)Fkf(y).

2.2. The k-Hankel convolution product

The generalized translation operator can be used to define the k-Hankel convolution product.

Definition 2.2.1. For f,g∈L2(ℝd,μk), we define the k-Hankel convolution product ∗k, by

Note that the generalized convolution ∗k is well defined since τxkg∈L2(ℝd,μk) and it may be rewrite

Let f∈Lrad2(ℝ,μk) and g∈L2(ℝ,μk). Then

This convolution has considered by [2]. It satisfies

Our principal interest in this section is to prove a Miyachi's theorem associated with the k-Hankel transform.

Let us denote by P(ℝd) is the set of polynomials on ℝd.

For all x∈ℝd,  s>0 the k-Hankel heat kernel qtk is given by

the function qtk is a solution of the heat equation Hku(x,t)=0 and Hk=Tk,1, where Tk,1 is the infinitisimal generator operator defined by (1.1). For more details we refer the reader to [2, 4].

Now, we state our principal theorem of this section.

Theorem 3.1. Let f be a measurable function on ℝd such that

and

for some constants α,β,λ>0 and 1≤p,q≤+∞.

Case 1. If αβ>14, then f=0 a.e.

Case 2. If αβ=14, then f=Kqβk(.) with |K|≤λ.

Case 3. If αβ<14, then for all δ∈]β,1α[, if f takes the form f(x)=P(x)qδk(x),  P∈P(ℝd), the relations (3.1), (3.2) hold. To achieve the proof of Theorem 3.1 we need the following auxiliary lemmas.

3.1. Auxiliary lemmas

Lemma 3.1.1. Let g∈ℂd be an entire function, for some positive constants C1 and C2 such that

Proof. Using Fubini's theorem together with relation (3.3), there is s subset E of ℝd−1 with λ(Ec)=0 (here λ denote the Lebsegue measure). Such that for all sequence (xi)2≤i≤d∈E,

Additionally, the function z1↦g(z1,(xi)2≤i≤d) is an entire function and O(eC2(ℜz1)2) on ℂ. Then by Miyachi's Lemma [18][lemma 4], the function g is bounded in ℂ. Moreover, by using Liouville theorem, we see that for all z1∈ℂ and all sequence (xi)2≤i≤d∈E

For all (zi)1≤i≤d, the last equality has a sense because g is a continuous function. Then by induction we infer the result,

which furnishes the proof of Lemma 3.1.1.

Lemma 3.1.2. Let r∈[1,+∞[,  a>0. Then for h∈Lr(ℝd,μk), there is a constant K>0 such that

Proof. By means assertion (3.4), one can assert that e−α‖y‖h∈L1(ℝd,μk). Then by relations (2.2), (2.5) and (2.6) tVk(e−a‖y‖h) is defined a.e on ℝd. Here two cases to be discussed:

Case 1. If r∈[1,∞[, then

where r' is the conjugate exponent of r.

Hence

According to relation (2.6), we see that

which gives the result for the case r∈[1,∞[.

Case 2. If r=+∞, then by relation (3.5), we have

which furnishes the case r=+∞, and this infers the result.

Lemma 3.1.3. Let p,q∈[1,+∞[ and f a measurable function on ℝd, let α>0 such that

Then for all complex number z∈ℂd, Fk(f)(z), moreover it's entire, exists K>0 such that for all z∈ℂd,

Using relation (2.8) together with Hölder's inequality , we infer the relation (3.6).

For the relation (3.7), observe that relations (3.6) and (2.6) assert that f∈L1(ℝd,μk), and tVk(f)∈L1(ℝd,μk), consequently, by (2.4), for all z=u+iv∈ℂd,  u,v∈ℝd, we have

Using Lemma 3.1.2, we can write

with ‖y‖=‖x‖(1−‖v‖).

Relation (3.6) yields that there exists f1∈Lp(ℝd,μk) and f2∈Lq(ℝd,μk) for which

which furnishes the proof of Lemma 3.1.3.

Thanks to the tools collected above, we can now prove our theorem.

3.2. Proof of Theorem 3.1

Case 1. αβ>14. Let h be a function on ℝd defined by

g is an entire function belongs to ℂd, then according to relation (3.7), we write

Moreover, observe that

For all positive constants a,b>0 and using the fact that log+ab≤log+a+b, we get

Since αβ>14, relation (3.2) yields that

Relations (3.8) and (3.9) assert that the function g satisfies (3.3), consequently g is a constant, we have then

Since we have αβ>14, relation (3.2) makes sense as K=0, furthermore, the injectivity of the k-Hankel transform gives that f=0 a.e.

Case 2. αβ=14, as in the first case, we have that Fk(f)(y)=Ke−‖y‖α. So, (3.2) holds as |K|≤λ. Consequently, we get f=Kqβk(.) whenever |K|≤λ.

Now, it remains the third case when αβ<14. If f is given like the form f=Kqβk(.), then its k-Hankel transform takes the form Fk(f)(y)=P(y)e−δ‖y‖, then f and Fk(f) satisfy (3.1) and (3.2) for all δ∈]β,α−1[, which furnishes the proof of Theorem 3.1.

The author is very grateful to the referee for the useful comments and suggestions given for the paper.

No potential conflict of interest was reported by the author.