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### Article

Kyungpook Mathematical Journal 2023; 63(3): 425-435

Published online September 30, 2023

### Miyachi's Theorem for the k-Hankel Transform on ℝd

Mohamed Amine Boubatra

Teacher Equcation College of Setif, P. O. Box 556, El Eulma, 19600, Setif, Algeria Laboratory of applied mathematics, University of Ferhat Abbes of Setif 1, Algeria
e-mail : boubatra.amine@yahoo.fr; m.boubatra@ens-setif.dz

Received: July 19, 2021; Revised: June 6, 2022; Accepted: June 8, 2022

The classical Hardy Theorem on states that a function f and its Fourier transform cannot be simultaneously very small; this fact was generalized by Miyachi in terms of L1+L and log+-functions. In this paper, we consider the k-Hankel transform, which is a deformation of the Hankel transform by a parameter k>0 arising from Dunkl's theory. We study Miyachi's theorem for the k-Hankel transform on d.

Keywords: k-Hankel transform, Miyachi's theorem, Hardy's theorem

Let d be a real d-dimensional Euclidean space with scalar product x,y and norm x=x,x. Let Sd1 be the unit Euclidean sphere in d, Δ be the Laplace operator, dμ(x)=(2π)d/2dx be the normalized Lebesgue measure, Lp(d),1p<+ be the Lebesgue space with norm fp:=(d|f|pdμ)1/p, and S(d) be the Schwartz space.

The Euclidian Fourier transform is defined by

Ff(y)=(2π)d/2d f(x)eix,ydx.

We introduce the real parameters α,β such that α,β>0 and let f be a measurable function on satisfying |f(x)|λeαx2 and |F(y)|λeβξ2. The function f reduces to the null function if αβ14. A generalization of Hardy's theorem is estabilished by Miyachi in [18] where the following is shown.

If f is a measurable function on such that

eαx2fL1()+L()

and

log+|F(ξ)e ξ2 4α |λdξ<,

where α,λ are two positive constants, then f is a constant multiple of eαx2.

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

Tk,a:=x2aΔkxa,a>0,

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

Fk,a:=eiπ22k+da2aexpπi2aTk,a,

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 d. 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 d 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 d. In § 3, we provide keys lemmas used to prove our main result of Miyachi's theorem for the k-Hankel transform.

### 2. Background for the k-Hankel transform on ℝd

Let Rd0 be a root system, R+ be a positive subsystem of R, G(R)O(d) be a reflection group formed by reflections σa:aR, where σa is a reflection with respect to hyperplane a,x=0, and k:RR+ be a multiplicity function invariant under groups G. This is a G-invariant positive homogeneous of degree 2γk1, where

γk= αR+ kα.

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

υk(x)=x1 αR+x,α2k(α),dμk(x)=υk(x)dx.

Denote by λk=2γk+d1 the homogeneous dimension of the system.

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

Tjf(x)=jf(x)+ αR+k(α)f(x)f(σαx)x,αα,ej,1jd,

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

TkTl=TlTk,for 1k,ld.

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

Δkf=Δf+ αRk(α)f(x),αα,xf(x)f(σαx) α,x2,xd,

where and Δ are the classical gradient and Laplacian operators.

### 2.1. The k-Hankel transform

We define the kernel

Bk(x,y)=Γλk 2VkJ˜ λk 21(z)yy,

with z=2xy(1+xx,.). Here, Vk denotes the Dunkl intertwining operator defined by

Vkf(x)=df(y)dσx(y),xd,

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

Let us define the space:

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

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

tVkf(y)=d f(x)dνy(x),xd.

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

tVk(f)=FFk(f).

The operators Vk and tVk possess the following property : For all fD(d) and gE(d) we have

d tVk(f)(y)g(y)dy=df(x)Vk(g)(x)dμk(x).

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

d tVk(f)(y)dy=df(x)dμk(x),

Moreover, for all x,yd, the kernel Bk(x,.) possesses the following properties:

For all x,yd, we have

Bk(0,y)=1,  |Bk(x,y)|1.
|zνBk(x,z)|x|ν|exez,

where

zν=νz1ν1...zdνdand|ν|=ν1+ν2+...+νd.

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 fL1(d,μk)

Fkf(y)=ckdf(x)Bk(x,y)dμk(x),yd,

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

ck1=d e xdμk(x).

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(d) and its inverse is given by

Fk1=Fk.

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

FkfL2(d,μk)=fL2(d,μk).

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

The generalized translation operator fτykf, yd is defined on L2(d,μk) by

Fk(τykf)(ξ)=Bk(y,ξ)Fk(f)(ξ),ξd.

It plays the role of the arbitrary translation τykf(.)=f(.y) in d, since the Euclidean Fourier transform satisfies τykf^(x)=eix,yf^(x).

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

Ak(d)={fL1(d,μk)/FkfL1(d,μk)}.

Note that Ak(d)L1L(d,μk) and hence is a subspace of L2(d,μκ).

The operator τyk satisfies the following properties:

Proposition 2.1.2. Assume that fAk(d) and gL1L(d,μk). Then

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

• (ii) For every yd, the operator τyk satisfies

d τykf(x)g(x)dμk(x)=df(x)τykg(x)dμk(x).

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

Case 1. G=2 (see [1]).

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

τykf(x)=Γ(λk2)Γ(λk212)Vk11 f0 _x,y,u;.1u2λk232du(yy),

where _x,y,u;.=x+y2xy(1+xx,.)u.

According to the positivity of the intertwining operator (2.2) it follows that τykf(x)0 for all yd,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 fLrad1(d,μk) the subspace of radial functions in L1(d,μk), we have:

d τykf(x)dμk(x)=df(x)dμk(x).

• (iii) The generalized translation operator is well defined on L2(d,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,gL2(d,μk), we define the k-Hankel convolution product k, by

fkg(x)=ckdf(y)τxkg(y)dμk(y),xd.

Note that the generalized convolution k is well defined since τxkgL2(d,μk) and it may be rewrite

fkg(x)=ckd F kf(λ)Fkg(λ)Bk(x,λ)dμk(λ),xd.

d|fkg(x)|2dμk(x)=d|Fkf(λ)|2|Fkg(λ)|2dμk(λ),xd.

This convolution has considered by [2]. It satisfies

fkg=gkf;Fk(fkg)=FkfFkg.

### 3. Miyachi’s theorem for the k-Hankel transform

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

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

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

qtk(x)=cktλkext,fort>0

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 d such that

eαxfLp(d,μk)+Lq(d,μk)

and

d log+|Fk(f)(ξ)eβξ|λdξ<

for some constants α,β,λ>0 and 1p,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),PP(d), 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 gd be an entire function, for some positive constants C1 and C2 such that

|g(z)|C1eC2ezd log+ |g(y)|dy<gis a constant.

Proof. Using Fubini's theorem together with relation (3.3), there is s subset E of d1 with λ(Ec)=0 (here λ denote the Lebsegue measure). Such that for all sequence (xi)2idE,

d log+|g(x,(xi)1id)|dx<+.

Additionally, the function z1g(z1,(xi)2id) is an entire function and O(eC2(z1)2) on . Then by Miyachi's Lemma [18][lemma 4], the function g is bounded in . Moreover, by using Liouville theorem, we see that for all z1 and all sequence (xi)2idE

g((xi)1id)=g(0,(xi)2id).

For all (zi)1id, 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 hLr(d,μk), there is a constant K>0 such that

d e αrx| tVk(eαyh)|rdx1/rKd |h(x)|rdμk(x)1/r.

Proof. By means assertion (3.4), one can assert that eαyhL1(d,μk). Then by relations (2.2), (2.5) and (2.6) tVk(eayh) is defined a.e on d. Here two cases to be discussed:

Case 1. If r[1,[, then

d e αrx|tVk(eαyh)|rdxd e αrx d eαy |h(y)|dνx (y)rdxd e αrxd |h(y)|rdνx(y) d eαry |h(y)|dνx (y)r/r dx,

where r' is the conjugate exponent of r.

Hence

d e tydνx(y)=Ketxfort>0.

According to relation (2.6), we see that

d e αrx|tVk(eαyh)|rdxKd t V k (|h | r)(x)dx          =Kd |h(x)|rdμk(x),

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

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

eαx|tVk(eαyh)(x)|eαxtVk(eαy)(x)hk,        =Khk,<,

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

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

eαxfLp(d,μk)+Lq(d,μk).

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

|Fk(f)(z)|Kevα.

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 fL1(d,μk), and tVk(f)L1(d,μk), consequently, by (2.4), for all z=u+ivd,u,vd, we have

Fk(f)(z)=d t V k (f)(x)e ix,zdx.

Using Lemma 3.1.2, we can write

|Fk(f)(z)|d e αx|tVk(f)(x)|eαx+xvdx    d e αx|tVk(f)(x)|eαydx

with y=x(1v).

Relation (3.6) yields that there exists f1Lp(d,μk) and f2Lq(d,μk) for which

d e αx|tVk(f)(x)|eαydxK(f1k,p+f2k,q)<+,

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 d defined by

g(z)= i=1deziαFk(f)(z)

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

|g(z)|Keuα,  for all  ud.

Moreover, observe that

d log+|g(y)|dy=d log+|eyα Fk(f)(y)|dy      =d log+eβy|Fk(f)(y)|λλe1αβy dy.

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

d log+|g(y)|dy=d log+eβy|Fk(f)(y)|λ dy+dλe1αβydy.

Since αβ>14, relation (3.2) yields that

d log+|g(y)|dy<+.

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

Fk(f)(y)=Keyα.

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)=Keyα. 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)eδy, then f and Fk(f) satisfy (3.1) and (3.2) for all δ]β,α1[, 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.

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