Titchmarsh's theorem for the dunkl transform in the space L2(Rd, wk(x)dx)

Titchmarsh’s Theorem for the Dunkl transform in the space L ² (R ^d , w k (x)dx)

Radouan Daher and

Mohamed El Hamma University of Hassan II, Morocco

Received : February 2013. Accepted : August 2013

Universidad Cat´ olica del Norte Antofagasta - Chile

Abstract

Using a generalized spherical mean operator, we obtain a general- ization of Titchmarsh’s theorem for the Dunkl transform for functions satisfying the (ψ, α, β)-Dunkl Lipschitz condition in L

(R

, w

(x)dx).

Keywords : Dunkl operator, Dunkl transform, generalized spher- ical mean operator.

Mathematics Subject Classification : 47B48; 33C52.

1. Intoduction and preliminaries

In [10], E. C. Titchmarsh characterized the set of functions in L ² (R) satis- fying the Cauchy Lipschitz condition for the Fourier transform, namely we have

Theorem 1.1. Let α ∈ (0, 1) and assume that f ∈ L ² (R). Then the following are equivalents

1. k f (x + h) − f(x) k L

( R ) = O(h ^α ) as h −→ 0, 2. ^R

_λ

_r | F(λ) | ² dλ = O(r

^2α ) as r −→ + ∞ , where F stands for the Fourier transform of f.

The main aim of this paper is to establish a generalization of Theorem 1.1 in the Dunkl transform setting by means of the generalized spherical mean operator.

In this paper we consider the Dunkl operators T _j , j = 1, 2, ..., d, which are the differential-difference operators introduced by C.F. Dunkl in [3].

These operators are very important in pure mathematics and in physics.

In the first we collect some notations and results on Dunkl operators and the Dunkl kernel (see [3], [4], [6]).

We consider R ^d with the Euclidean scalar product h ., . i and | x | = p h x, x i . For α ∈ R ^d \{ 0 } , let σ α be the reflection in the hyperplane H _α ⊂ R ^d orthogonal to α.

i.e.,

σ α (x) = x − 2 h α, x i

| x | ² α.

A finite set R ⊂ R ^d \{ 0 } is called a root system, if R ∩ R.α = { α, − α } and σ α R = R for all α ∈ R. For a given root system R the reflection σ _α , α ∈ R, generate a finite group W ⊂ O(d), the reflection group associ- ated with R. We fix β ∈ R ^d \ ∪ α

R H _α and define a positive root system R + = { α ∈ R/ h α, β i > 0 } .

A function k : R −→ C on a root system R is called a multiplicity func-

If one regards k as a function on the corresponding reflections, this means that k is constant on the conjugacy classes of reflections in W .

We consider the weight function w _k (x) = ^Y

α

R

|h α, x i| ^2k(α) ,

where w _k is W -invariant and homogeneous of degree 2γ where γ = ^X

α

R

k(α).

We let η be the normalized surface measure on the unit sphere S ^d

¹ in R ^d and set

dη k (y) = w k (y)dη(y).

Then η k is a W -invariant measure on S ^d

¹ , we let d k = η k (S ^d

¹ ).

Introduced by C.F. Dunkl in [3] the Dunkl operators T _j , 1 ≤ j ≤ d, on R ^d associated with the reflection group W and the multiplicity function k are the first-order differential-difference operators given by

T j f (x) = ∂f

∂x j

(x) + ^X

α

R

k(α)α j

f (x) − f (σ α (x))

h α, x i , f ∈ C ¹ (R ^d ), where α j = h α, e j i ; (e 1 , ...., e _d ) being the canonical basis of R ^d and C ¹ (R ^d ) is the space of functions of class C ¹ on R ^d .

The Dunkl kernel E _k on R ^d × R ^d has been introduced by C.F. Dunkl in [5]. For y ∈ R ^d the function x 7−→ E _k (x, y) can be viewed as the solution on R ^d of the following initial problem

( T _j u(x, y) = y _j u(x, y) f or 1 ≤ j ≤ d u(0, y) = 1 f or all y ∈ R ^d

This kernel has unique holomorphic extension to C ^d × C ^d .

M. R¨ osler has proved in [8] the following integral representation for the

Dunkl kernel

E k (x, z) = Z

R

e

^y,z

dµ x (y), x ∈ R ^d , z ∈ C ^d ,

where µ _x is a probability measure on R ^d with support in the closed ball B(0, | x | ) of center 0 and raduis | x | .

Proposition 1.2. [6]: Let z, w ∈ C ^d and λ ∈ C. Then 1. E _k (z, 0) = 1,

2. E _k (z, w) = E _k (w, z), 3. E _k (λz, w) = E _k (z, λw),

4. For all ν = (ν 1 , ..., ν d ) ∈ N ^d , x ∈ R ^d , z ∈ C ^d , we have

| D ^ν _z E _k (x, z) | ≤ | x |

^ν

exp( | x || Rez | ), where

D _z ^ν = ∂

^ν

∂z ₁ ^ν

...∂z _d ^ν

; | ν | = ν 1 + ... + ν _d . In particulier

| D ^ν _z E k (ix, z) | ≤ | x |

^ν

, for all x, z ∈ R ^d .

The Dunkl transform is defined for f ∈ L ¹ _k (R ^d ) = L ¹ (R ^d , w k (x)dx) by f b (ξ) = c

_k ¹

Z

R

f (x)E _k ( − iξ, x)w _k (x)dx, where the constant c _k is given by

c _k = Z

R

e

2

2

w _k (z)dz.

1. When both f and f ^b are in L ¹ _k (R ^d ), we have the inversion formula f (x) =

Z R

f b (ξ)E k (ix, ξ)w k (ξ)dξ, x ∈ R ^d ,

2. (Plancherel’s theorem) The Dunkl transform on S(R ^d ), the space of Schwartz functions, extends uniquely to an isometric isomorphism on L ² _k (R ^d ).

K. Trim` eche has introduced [11] the Dunkl translation operators τ x , x ∈ R ^d . For f ∈ L ² _k (R ^d ) and we have

(τ _x d (f ))(ξ) = E _k (ix, ξ) f(ξ), b and

τ _x (f )(y) = c

_k ¹ Z

R

f b (ξ)E _k (ix, ξ)E _k (iy, ξ)w _k (ξ)dξ.

The generalized spherical mean operator for f ∈ L ² _k (R ^d ) is defined by M _h f(x) = 1

d k

Z S

τ _x (hy)dη _k (y), x ∈ R ^d , h > 0.

From [7], we have M _h f ∈ L ² _k (R ^d ) whenever f ∈ L ² _k (R ^d ) and k M _h f k L

≤ k f k L

.

For p ≥ − ¹ ₂ , we introduce the normalized Bessel functuion j _p defined by

j _p (z) = Γ(p + 1) X

n=0

( − 1) ⁿ (z/2) ²ⁿ

n!Γ(n + p + 1) , z ∈ C, where Γ is the gamma-function.

Lemma 1.3. [1] The following inequalities are fulfilled 1. | j _p (x) | ≤ 1,

2. 1 − j p (x) = O(x ² ); 0 ≤ x ≤ 1.

Lemma 1.4. The following inequality is true

| 1 − j _p (x) | ≥ c, with | x | ≥ 1, where c > 0 is a certain constant.

(Analog of lemma 2.9 in [2])

Proposition 1.5. Let f ∈ L ² _k (R ^d ). Then (M d _h f)(ξ) = j _γ+

2−

1 (h | ξ | ) f(ξ). b (See [7])

For any function f(x) ∈ L ² _k (R ^d ) we define differences of the order m (m ∈ { 1, 2, ... } ) with a step h > 0.

∆ ^m _h f (x) = (M _h − I ) ^m f (x), (1.1)

here I is the unit operator.

Lemma 1.6. Let f ∈ L ² _k (R ^d ). Then k ∆ ^m _h f (x) k ² L

=

Z

R

| 1 − j _γ+

2−

1 (h | ξ | ) | ^2m | f ^b (ξ) | ² w _k (ξ)dξ.

From formula (1.1) and proposition 1.5, we have (∆ d ^m _h f )(ξ) = (j _γ+

2−

1 (h | ξ | ) − 1) ^m f ^b (ξ).

By Parseval’s identity, we obtain k ∆ ^m _h f (x) k ² _L

k

= Z

R

| 1 − j _γ+

2−

1 (h | ξ | ) | ^2m | f ^b (ξ) | ² w _k (ξ)dξ.

The lemma is proved 2. Main Results

In this section we give the main result of this paper. We need first to define

Definition 2.1. Let α > 0 and β > 0. A function f ∈ L ² _k (R ^d ) is said to be in the (ψ, α, β)-Dunkl Lipschitz class, denoted by Lip(ψ, α, β); if:

k ∆ ^m _h f (x) k L

= O(h ^α ψ(h ^β )) as h −→ 0; m ∈ { 1, 2, ... } ,

where ψ(t) is a continuous increasing function on [0, ∞ ), ψ(0) = 0 and ψ(ts) = ψ(t)ψ(s) for all t, s ∈ [0, ∞ ) and this function verify

Z 1/h 0

r ^2m

^2α

¹ ψ(r

^2β )dr = O(h ^2α

^2m ψ(h ^2β )) as h −→ 0.

Theorem 2.2. Let f ∈ L ² _k (R ^d ). Then the following are equivalents 1. f ∈ Lip(ψ, α, β),

2. ^R

_ξ

_s | f ^b (ξ) | ² w _k (ξ)dξ = O(s

^2α ψ(s

^2β )) as s −→ + ∞ . 1) = ⇒ 2) Assume that f ∈ Lip(ψ, α, β). Then we have

k ∆ ^m _h f (x) k L

= O(h ^α ψ(h ^β )) as h −→ 0, From lemma 1.6, we have

k ∆ ^m _h f (x) k ² L

= Z

R

| 1 − j _γ+

2−

1 (h | ξ | ) | ^2m | f(ξ) ^b | ² w k (ξ)dξ.

If | ξ | ∈ [ ¹ _h , ² _h ] then h | ξ | ≥ 1 and lemma 1.4 implies that 1 ≤ 1

c ^2m | 1 − j _γ+

2−

1 (h | ξ | ) | ^2m . Then

Z

1

h≤|

ξ

| f ^b (ξ) | ² w _k (ξ)dξ ≤ 1 c ^2m

Z

1

h≤|

ξ

| 1 − j _γ+

2−

1 (h | ξ | ) | ^2m | f ^b (ξ) | ² w _k (ξ)dξ

≤ 1 c ^2m

Z

R

| 1 − j _γ+

2−

1 (h | ξ | ) | ^2m | f(ξ) ^b | ² w k (ξ)dξ

≤ Ch ^2α (ψ(h ^β )) ² = Ch ^2α ψ(h ^2β ).

Therefore Z

s

ξ

2s | f(ξ) ^b | ² w _k (ξ)dξ ≤ Cs

^2α ψ(s

^2β ).

Furthermore, we have R

|

ξ

s | f ^b (ξ) | ² w _k (ξ)dξ

= ^³R _s

_ξ

_2s + ^R _2s

_ξ

_4s + ^R _4s

_ξ

_8s +... ^´ | f ^b (ξ) | ² w _k (ξ)dξ

≤ C ^³ s

^2α ψ(s

^2β ) + (2s)

^2α ψ((2s)

^2β ) + (2 ² s)

^2α ψ((2 ² s)

^2β ) + ... ^´

≤ C ^³ s

^2α ψ(s

^2β ) + 2

^2α s

^2α ψ(s

^2β ) + (2

^2α ) ² s

^2α ψ(s

^2β ) + .... ^´

≤ Cs

^2α ψ(s

^2β )(1 + 2

^2α + (2

^2α ) ² + (2

^2α ) ³ + ...)

≤ K α s

^2α ψ(s

^2β ),

where K α = C(1 − 2

^2α )

¹ . This proves that

Z

|

ξ

s | f ^b (ξ) | ² w _k (ξ)dξ = O(s

^2α ψ(s

^2β )) as s −→ + ∞ 2) = ⇒ 1) Suppose now that

Z

|

ξ

s | f ^b (ξ) | ² w _k (ξ)dξ = O(s

^2α ψ(s

^2β )) as s −→ + ∞ . We have to show that

Z

0

r ^2γ+d

¹ | 1 − j _γ+

2−

1 (hr) | ^2m φ(r)dr = O(h ^2α ψ(h ^2β )), where we have set

φ(r) = Z

S

| f ^b (ry) | ² w _k (y)dy.

We write I 1 =

Z 1/h 0

r ^2γ+d

¹ | 1 − j _γ+

2−

1 (hr) | ^2m φ(r)dr, and

I ₂ = Z

1/h

r ^2γ+d

¹ | 1 − j _γ+

2−

1 (hr) | ^2m φ(r)dr.

Firstly, from (1) in lemma 1.3 we see that I ₂ ≤ 4 ^m

Z

1/h

r ^2γ+d

¹ φ(r)dr = O(h ^2α ψ(h ^2β )) as h −→ 0.

g(r) = Z

r

x ^2γ+d

¹ φ(x)dx.

From (2) in lemma 1.3, an integration by parts yields

I 1 ≤ − C 1 h ^2m Z 1/h

0

r ^2m g

(r)dr

≤ − C 1 g(1/h) + 2mC 1 h ^2m Z 1/h

0

r ^2m

¹ g(r)dr

≤ 2mC ₁ h ^2m Z 1/h

0

r ^2m

¹ r

^2α ψ(r

^2β )dr

≤ 2mC ₁ h ^2m Z 1/h

0

r ^2m

^2α

¹ ψ(r

^2β )dr

≤ C 2 h ^2m h ^2α

^2m ψ(h ^2β )

≤ C ₂ h ^2α ψ(h ^2β ),

where C 1 and C 2 are positive constants, and this ends the proof

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Radouan Daher

Department of Mathematics, Faculty of Sciences A¨ ın Chock,

University of Hassan II, Casablanca, Morocco e-mail: rjdaher024@gmail.com

and

Mohamed El Hamma

Department of Mathematics, Faculty of Sciences A¨ ın Chock, University of Hassan II, Casablanca,

Morocco

e-mail: m elhamma@yahoo.fr