Dini Lipschitz functions for the Dunkl transform in the Space (Formula presented.)

DOI 10.1007/s12215-015-0195-9

Dini Lipschitz functions for the Dunkl transform in the Space L ² ( R ^d , w k ( x ) d x )

Mohamed El Hamma

· Radouan Daher

Received: 15 December 2014 / Accepted: 3 March 2015

© The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract Using a generalized spherical mean operator, we obtain an analog of Theorem 5.2 in Younis (J Math Sci 9(2),301–312 1986) for the Dunkl transform for functions satisfying the d-Dunkl Dini Lipschitz condition in the space L

(

, w

(x )d x), where w

is a weight function invariant under the action of an associated reflection group.

Keywords Dunkl transform · Dunkl kernel · Generalized spherical mean operator Mathematics Subject Classification 42B37

1 Introduction and preliminaries

Younis Theorem 5.2 [13] characterized the set of functions in L

(

) satisfying the Cauchy Lipschitz condition by means of an asymptotic estimate growth of the norm of their Fourier transforms, namely, we have the following

Theorem 1.1 [13] Let f ∈ L

(

). Then the following are equivalents 1. f (x + h) − f (x )

= O

h^α (log¹_h)^β

as h −→ 0, 0 < α < 1, β ≥ 0, 2.

|x|≥r

|

( f )(x)|

d x = O

(logr)^2β

as r −→ +∞, where

stands for the Fourier transform of f .

Dedicated to Professor François Rouvière for his 69’s birthday.

B

1 Department of Mathematics, Faculty of Sciences Aïn Chock, University of Hassan II,

In this paper, we obtain an analog of Theorem 1.1 for the Dunkl transform on

. For this purpose, we use a generalized spherical mean operator. We point out that similar results have been established in the Bessel transform [4].

We consider the Dunkl operators D

; 1 ≤ i ≤ d, on

, which are the differential- difference operators introduced by Dunkl in [6]. These operators are very important in pure mathematics and in physics. The theory of Dunkl operators provides generalizations of var- ious multivariable analytic structures, among others we cite the exponential function, the Fourier transform and the translation operator. For more details about these operators see [5–7]. The Dunkl Kernel E

has been introduced by Dunkl in [8]. This Kernel is used to define the Dunkl transform.

Let R be a root system in

, W the corresponding reflection group, R

a positive sub- system of R (see [5,7,9–11]) and k a non-negative and W -invariant function defined on R.

The Dunkl operators is defined for f ∈ C

(

) by D

f (x ) = ∂f

∂x

(x) +

α∈R+

k(α)α

f (x) − f (σ

(x))

α, x , x ∈

(1 ≤ j ≤ d) Here , is the usual euclidean scalar product on

with the associated norm |.| and σ

the reflection with respect to the hyperplane H

orthogonal to α, and α

= α, e

, (e

, e

, . . . , e

) being the canonical basis of

.

The weight function w

defined by w

(x ) =

ζ∈R+

|ζ, x |

, x ∈

,

where w

is W-invariant and homogeneous of degree 2 γ where γ = γ ( R ) =

ζ∈R+

k (ζ ) ≥ 0 .

The Dunkl Kernel E

on

×

has been introduced by Dunkl in [8]. For y ∈

the function x → E

(x, y) is the unique solution on

of

D

u(x , y) = y

u(x , y) for 1 ≤ j ≤ d u ( 0 , y ) = 1 for all y ∈

E

is called the Dunkl Kernel.

Proposition 1.2 [5] Let z , w ∈

and λ ∈

. Then 1. E

(z, 0) = 1.

2. E

(z, w) = E

(w, z).

3. E

(λ z , w) = E

( z , λw) .

4. For all ν = (ν

, . . . , ν

) ∈

, x ∈

, z ∈

, we have

|∂

E

(x , z)| ≤ |x|

ex p(|x ||Re(z)|), where

∂

= ∂

∂z

. . . ∂ z

, |ν| = ν

+ · · · + ν

. In particular

|∂

E

(i x, z)| ≤ |x |

,

for all x , z ∈

.

The Dunkl transform is defined for f ∈ L

(

) = L

(

, w

( x ) d x ) by f (ξ) = c

R^d

f (x)E

(−iξ, x )w

(x )d x,

‘where the constant c

is given by c

=

R^d

e

2

2

w

(z)d z.

The Dunkl transform shares several properties with its counterpart in the classical case, we mention here in particular that Parseval Theorem holds in L

= L

(

) = L

(

, w

(x)d x), when both f and f are in L

(

), we have the inversion formula

f (x ) =

R^d

f (ξ)E

(i x, ξ)w

(ξ)dξ, x ∈

.

In L

(

) , consider the generalized spherical mean operator defined by M

f (x) = 1

d

S^d−1

τ

( f )(hy)dη

(y), (x ∈

, h > 0)

where τ

Dunkl translation operator (see [11,12]), η is the normalized surface measure on the unit sphere

in

and set d η

( y ) = w

( x ) d η( y ) , η

is a W -invariant measure on S

and d

= η

( S

) .

We see that M

f ∈ L

(

) whenever f ∈ L

(

) and M

f

k

≤ f

k

, for all h > 0.

For p ≥ −

, we introduce the normalized Bessel function of the first kind j

defined by j

( z ) = ( p + 1 )

(−1)

(

)

n!(n + p + 1) , z ∈

. (1)

Lemma 1.3 [1] The following inequalities are fulfilled 1. | j

(x)| ≤ 1,

2. 1 − j

(x ) = O(1), x ≥ 1.

3. 1 − j

(x ) = O(x

); 0 ≤ x ≤ 1.

From lemma 1.3, we have

|1 − j

(x)| ≤ C

x , ∀x ∈

(2)

Lemma 1.4 The following inequality is true

| 1 − j

( x )| ≥ c ,

with |x | ≥ 1, where c > 0 is a certain constant which depends only on p.

Proof (Analog of lemma 2.9 in [3])

Moreover, from (1) we see that

z→0

lim

j

2−1

(z) − 1

z

= 0 (3)

Proposition 1.5 Let f ∈ L

(

) . Then ( M

f )(ξ) = j

2−1

( h |ξ|) f (ξ).

i.e

M

f (x) =

R^d

j

2−1

(h|ξ |) f (ξ)E

(i x, ξ)w

(ξ)dξ and

f (x) =

R^d

f (ξ)E

(i x, ξ)w

(ξ)dξ.

We have

M

f (x ) − f (x ) =

R^d

( j

2−1

(h|ξ |) − 1) f (ξ)E

(i x, ξ)w

(ξ)dξ. (4) Invoking Parseval’s identity (4) gives

M

f (x) − f (x)

k

=

R^d

| j

2−1

(h|ξ |) − 1|

| f (ξ)|

w

(ξ)dξ.

2 Dini Lipschitz condition

Definition 2.1 Let f (x) ∈ L

(

), and let M

f (x ) − f (x )

k

≤ C h

(log

)

, γ ≥ 0 i.e

M

f ( x ) − f ( x )

k

= O

h

(log

)

,

for all x in

and for all sufficiently small h, C being a positive constant. Then we say that f satisfies a d-Dunkl Dini Lipschitz of order α , or f belongs to Li p (α, γ ) .

Definition 2.2 If however

M

f (x ) − f (x )

k h^α

(log¹_h)^γ

→ 0 as h → 0 i.e

M

f ( x ) − f ( x )

k

= o

h

(log

)

as h → 0 , γ ≥ 0 then f is said to be belong to the little d-Dunkl Dini Lipschitz class li p (α, γ ) . Remark It follows immediately from these definitions that

li p (α, γ ) ⊂ Li p (α, γ ).

Theorem 2.3 Let α > 1. If f ∈ Li p (α, γ ) , then f ∈ li p ( 1 , γ ) . Proof For x ∈

and h small, f ∈ Li p(α, γ ) we have

M

f ( x ) − f ( x )

k

≤ C h

( log

)

.

Then

log 1 h

M

f (x ) − f (x )

k

≤ C h

Therefore

(log

)

h M

f (x ) − f (x )

k

≤ C h

, which tends to zero with h → 0. Thus

(log

)

h M

f (x ) − f (x )

k

→ 0 as h → 0

Then f ∈ li p ( 1 , γ ) .

Theorem 2.4 If α < β , then Li p (α, 0 ) ⊃ Li p (β, 0 ) and li p (α, 0 ) ⊃ li p (β, 0 ) . Proof We have 0 ≤ h ≤ 1 and α < β, then h

≤ h

.

Then the proof of this theorem.

Theorem 2.5 Let f, g ∈ L

(

) such that M

( f g)(x ) = M

f (x)M

g(x). If f, g ∈ Li p (α, γ ) , then f g ∈ Li p (α, γ ) .

Proof Since f, g ∈ Li p(α, γ ), we have for all x in

M

f ( x ) − f ( x )

k

≤ C

h

( log

)

and

M

g ( x ) − g ( x )

k

≤ C

h

(log

)

It is clear that

M

( f g)(x ) − f (x )g(x)

k

= M

( f g)(x ) − f (x )M

g(x ) + f (x )M

g(x ) − f (x )g(x )

k

= M

f (x)M

g(x) − f (x)M

g(x) + f (x )M

g(x ) − f (x )g(x)

k

= M

g(x)(M

f (x ) − f (x )) + f (x )(M

g(x ) − g(x ))

k

≤ M

g(x )

k

M

f (x ) − f (x )

k

+ f (x)

k

M

g(x ) − g(x )

k

≤ K

C

h

( log

)

+ K

C

h

( log

)

≤ M h

(log

)

,

where M = max(K C , K C ). Then f g ∈ Li p(α, γ )

3 New results on Dini Lipschitz class

Theorem 3.1 Let α > 2. If f belong to the d-Dunkl Dini Lipschitz class, i.e f ∈ Li p (α, γ ), α > 2 , γ ≥ 0 .

Then f is equal to the null function in

. Proof Assume that f ∈ Li p(α, γ ). Then

M

f (x ) − f (x )

k

≤ C

h

(log

)

.

We have to recall that the Dunkl transform of f (x) satisfies the Parseval’s identity f

k

=

f

k

. So

M

f (x ) − f (x )

k

= M

f − f

k

i.e (1 − j

2−1

(h|ξ|)) f (ξ)

L²_k

≤ C

h

(log

)

. it follows that

R^d

| 1 − j

2−1

( h |ξ|)|

| f (ξ)|

w

(ξ) d ξ ≤ C

h

(log

)

.

Then

R^d

| 1 − j

2−1

( h |ξ |)|

| f (ξ)|

w

(ξ) d ξ

h

≤ C

h

( log

)

. Since α > 2 we have

h→0

lim h

( log

)

= 0 Then

h→0

lim

R^d

|1 − j

2−1

(h|ξ |)|

|ξ|

h

|ξ|

| f (ξ)|

w

(ξ) d ξ = 0 . and also from the formula (3) and Fatou’s theorem, we obtain |ξ |

f (ξ)

k

= 0. Thus

|ξ|

f (ξ) = 0 for all ξ ∈

, then f (x ) is the null function.

Analog of the theorem 3.1, we obtain this theorem Theorem 3.2 Let f ∈ L

(

) . If f belong to li p ( 2 , 0 ) , i.e

M

f ( x ) − f ( x )

k

= o ( h

) as h → 0 . Then f is equal to null function in

.

Now, we give another the main result of this paper analog of theorem 1.1.

Theorem 3.3 Let α ∈ ( 0 , 1 ) , γ ≥ 0 and f ∈ L

(

) . Then the following are equivalents 1. f ∈ Li p(α, γ )

2.

|ξ|≥s

| f (ξ)|

w

(ξ)dξ = O

(logs)^2γ

as s → +∞

Proof 1) ⇒ 2) Assume that f ∈ Li p(α, γ ). Then we have M

f (x) − f (x)

k

= O

h

( log

)

as h −→ 0, Proposition 1.5 and Parseval’s identity give

M

f ( x ) − f ( x )

k

=

R^d

| 1 − j

2−1

( h |ξ|)|

| f (ξ)|

w

(ξ) d ξ.

If |ξ| ∈ [

,

] then h |ξ| ≥ 1 and lemma 1.4 implies that 1 ≤ 1

c

|1 − j

2−1

(h|ξ |)|

. Then

1

h≤|ξ|≤²_h

| f (ξ)|

w

(ξ)dξ ≤ 1 c

1

h≤|ξ|≤²_h

|1 − j

2−1

(h|ξ|)|

| f (ξ)|

w

(ξ)dξ

≤ 1 c

R^d

|1 − j

2−1

(h|ξ |)|

| f (ξ)|

w

(ξ)dξ

= O

h

(log

)

. We obtain

s≤|ξ|≤2s

| f (ξ)|

w

(ξ) d ξ ≤ C s

( log s )

. where C is a positive constant.

So that

|ξ|≥s

| f (ξ)|

w

(ξ)d ξ =

s≤|ξ|≤2s

+

2s≤|ξ|≤4s

+

4s≤|ξ|≤8s

+ · · ·

| f (ξ)|

w

(ξ)dξ

≤ C

s

(log s)

+ (2s)

(log 2s)

+ (4s)

(log 4s)

+ · · ·

≤ C s

(log s)

(1 + 2

+ (2

)

+ (2

)

· · · )

≤ C K

s

( log s )

, where K

= C (1 − 2

)

since 2

< 1.

This proves that

| f (ξ)|

w

(ξ)dξ = O

s

(log s)

as s −→ +∞

2 ) ⇒ 1 ) Suppose now that

|ξ|≥s

| f (ξ)|

w

(ξ)dξ = O

s

( log s )

as s −→ +∞.

We have to show that

0

r

| 1 − j

2−1

( hr )|

φ( r ) dr = O

h

(log

)

, as h → 0 where

φ(r) =

S^d−1

| f (r y)|

w

(y)d y.

We write

0

r

|1 − j

2−1

(hr )|

φ(r)dr = I

+ I

, where

I

=

0

r

| 1 − j

2−1

( hr )|

φ( r ) dr , and

I

=

1/h

r

|1 − j

2−1

(hr )|

φ(r)dr.

Firstly, from (1) in lemma 1.3 we see that I

≤ 4

1/h

r

φ( r ) dr = O

h

(log

)

as h −→ 0 . Set

ψ(r) =

r

x

φ(x )d x.

From formula 2, an integration by parts yields I

=

0

r

|1 − j

2−1

(hr)|

φ(r)dr

≤ −C

h

0

r

ψ

(r)dr

≤ −C

ψ(1/h) + 2C

h

0

rψ(r)dr

≤ 2C

h

0

r r

( log r )

dr

≤ C

h

( log

)

where C

is a positive constant, and this ends the proof

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