Jacobi-dunkl dini lipschitz functions in the space Lp(R, Aα,β(x)dx)

Jacobi-Dunkl Dini Lipschitz Functions In The Space L ^p ( R ; A ; (x)dx)

Salah El Ouadih ^y , Radouan Daher ^z

Received 06 July 2015

Abstract

In this paper, we obtain an analog of Younis Theorem 5.2 in [7] for the Jacobi- Dunkl transform on the real line for functions satisfying the ( ; )-Jacobi-Dunkl Lipschitz condition in the space L

( R ; A

(x)dx) where

and 6 =

.

1 Introduction

Younis Theorem 5.2 [7] characterized the set of functions in L ² ( R ) satisfying the Dini Lipschitz condition by means of an asymptotic estimate growth of the norm of their Fourier transforms, namely we have the following Theorem 1.1.

THEOREM 1.1 ([7, Theorem 5.2]). Let f 2 L ² ( R ). Then the following statements are equivalents

(a) k f (x + h) f (x) k = O _(log ^h

h

) as h ! 0; where 0 < < 1 and 0.

(b) R

j j r j f b ( ) j ² d = O _(log ^r _r)

as r ! 1 , where f b stands for the Fourier trans- form of f .

In this paper, we obtain an analog of Theorem 1.1 for the Jacobi-Dunkl transform on the real line. For this purpose, we use a generalized translation operator. In this section, we recapitulate from [1, 2, 3, 5] some results related to the harmonic analysis associated with Jacobi-Dunkl operator _; . The Jacobi-Dunkl function with parameters ( ; ); ₂ ¹ , and 6 = ₂ ¹ , de…ned by the formula

8 x 2 R ; ^; (x) =

( ' ^; (x) ^{i d} _dx ' ^; (x) , if 2 Cnf 0 g ;

1 , if = 0:

Mathematics Sub ject Classi…cations: 42B37, 42B10, 43A90, 42C15.

y

Departement of Mathematics, Faculty of Sciences Aïn Chock, University Hassan II, Casablanca, Morocco

z

Departement of Mathematics, Faculty of Sciences Aïn Chock, University Hassan II, Casablanca, Morocco

88

where ² = ² + ² , = + + 1, F is the Gausse hypergeometric function (see [1,6]), and ' ^; is the Jacobi function given by

' ^; (x) = F + i

2 ; i

2 ; + 1; (sinh(x)) ² :

Then ^; is the unique C ¹ -solution on R of the di¤erentiel-di¤erence equation (

; U = i U 2 C ; U (0) = 1;

where ; is the Jacobi-Dunkl operator given by

; U (x) = d U (x)

dx + [(2 + 1) coth x + (2 + 1) tanh x] U (x) U ( x)

2 :

The operator _; is a particular case of the operator D given by D U (x) = d U (x)

dx + A ⁰ (x) A(x)

U (x) U ( x)

2 ;

where A(x) = j x j ^{2 +1} B(x), and B a function of class C ¹ on R , even and positive. The operator ; corresponds to the function

A(x) = A _; (x) = 2 (sinh j x j ) ^{2 +1} (cosh j x j ) ^{2 +1} : Using the relation

d

dx ' ^; (x) =

2 + ²

4( + 1) sinh(2x)' ^{+1; +1} (x);

the function ^; can be written in the form above (see [2])

; (x) = ' ^; (x) + i

4( + 1) sinh(2x)' ^{+1; +1} (x); x 2 R :

Denote L ^p _; ( R ) = L ^p _; ( R ; A _; (x)dx), 1 < p 2, the space of measurable functions f on R such that

k f k p; ; = Z

R j f (x) j ^p A _; (x)dx

1=p

< + 1 :

Using the eigenfunctions ^; of the operator _; called the Jacobi-Dunkl kernels, we de…ne the Jacobi-Dunkl transform of a function f 2 L ^p _; ( R ) by

F ; f ( ) = Z

R

f (x) ^; (x)A _; (x)dx; 2 R ; and the inversion formula

f (x) = Z

R F ^; f ( ) ^; (x)d ( );

where

d ( ) = j j

8 p ₂

2 j C ; ( p ₂

2 ) j 1 _{Rn ] ; [} ( )d : Here,

C ; ( ) = 2 ⁱ ( + 1) (i )

( ¹ ₂ ( + i )) ( ¹ ₂ ( + 1 + i )) ; 2 Cn (i N );

and 1 _Rn ] ; [ is the characteristic function of Rn ] ; [. The Jacobi-Dunkl transform is a unitary isomorphism from L ² _; ( R ) onto L ² ( R ; d ( )), i.e.,

k f k ^{2; ;} = kF ^; (f ) k L

( R ;d ( )) : (1) Plancherel’s theorem (1) and the Marcinkiewics interpolation theorem (see [8]) we get for f 2 L ^p _; ( R ) with 1 < p 2 and q such that ¹ _p + ¹ _q = 1,

kF ^; (f ) k L

( R ;d ( )) K k f k ^{p; ;} ; (2) where K is a positive constant (see [9]).

The operator of Jacobi-Dunkl translation is de…ned by T _x f (y) =

Z

R

f(z)d _x;y ^; (z); 8 x; y 2 R ;

where _x;y ^; (z); x; y 2 R are the signed measures given by

d _x;y ^; (z) = 8 >

> <

> >

:

K ; (x; y; z)A ; (z)dz , if x; y 2 R ;

x , if y = 0;

y , if x = 0:

Here, x is the Dirac measure at x. And,

K ; (x; y; z) = M ; (sinh j x j sinh j y j sinh j z j ) ² 1 I

Z

0

(x; y; z) (g (x; y; z)) ₊ ¹ sin ² d ;

I x;y = [ j x j j y j ; jj x j j y jj ] [ [ jj x j j y jj ; j x j + j y j ];

(x; y; z) = 1 _x;y;z + _z;x;y + _z;y;x ;

8 z 2 R ; 2 [0; ]; _x;y;z =

( _{cosh x+cosh y cosh z cos}

sinh xsinh y , if xy 6 = 0;

0 , if xy = 0;

g (x; y; z) = 1 cosh ² x cosh ² y cosh ² z + 2 cosh x cosh y cosh z cos ;

t + =

( t , if t > 0;

0 , if t 0;

and

M ; = 8 <

:

2

( +1)

p ( ) ( +

) , if > ;

0 , if = :

In [2], we have

F ^; (T h f)( ) = ^; (h) F ^; (f )( ) and F ^; ( ; f )( ) = i F ^; (f )( ): (3) For ₂ ¹ , we introduce the Bessel normalized function of the …rst kind de…ned by

j (x) = ( + 1) X 1 n=0

( 1) ⁿ ( ^x ₂ ) ²ⁿ

n! (n + + 1) ; x 2 R : Moreover, we see that

x lim ! 0

j (x) 1

x ² 6 = 0; (4)

by consequence, there exists C ₁ > 0 and > 0 satisfying

j j (x) 1 j C 1 j x j ² for j x j : (5) By Lemma 9 in [4], we obtain the following Lemma 1.2.

LEMMA 1.2. Let ₂ ¹ ; 6 = ₂ ¹ . Then for j v j , there exists a positive constant C 2 such that

1 ' _+i ^; (x) C ₂ j 1 j ( x) j :

Denote by L ^m _p ( ; ); 1 < p 2; m = 0; 1; 2; :::, the class of functions f 2 L ^p _; ( R ) that have generalized derivatives satisfying ^m _; f 2 L ^p _; ( R ), i.e.,

L ^m _p ( _; ) = n

f 2 L ^p _; ( R ) : ^m _; f 2 L ^p _; ( R ) o

; where ⁰ _; f = f and ^m _; f = _; ( ^m _; ¹ f ) for m = 1; 2; :::.

2 Dini Lipschitz Condition

Denote N _h by

N _h = T _h + T _h 2I;

where I is the unit operator in the space L ^p _; ( R ).

DEFINITION 2.1. Let f 2 L ^m _p ( _; ), and de…ne k N _h ^m _; f (x) k p; ; C h

(log _h ¹ ) ; 0 and m = 0; 1; 2; :::;

i.e.,

k N _h ^m _; f (x) k p; ; = O h (log ¹ _h ) ;

for all x in R and for all su¢ ciently small h; C being a positive constant. Then we say that f satis…es a Jacobi-Dunkl Dini Lipschitz of order , or f belongs to Lip( ; ; p).

DEFINITION 2.2. If

k N _h ^m _; f (x) k p; ; h

(log

)

! 0; as h ! 0;

i.e.,

k N _h ^m _; f (x) k p; ; = O h (log ¹ _h ) ;

then f is said to belong to the little Jacobi-Dunkl Dini Lipschitz class lip( ; ; p).

REMARK. It follows immediately from these de…nitions that lip( ; ; p) Lip( ; ; p):

THEOREM 2.3. Let > 1 . If f 2 Lip( ; ; p), then f 2 lip(1; ; p).

PROOF. For x 2 R , h small and f 2 Lip( ; ; p); we have k N h m

; f (x) k ^{p; ;} C h (log ¹ _h ) : Then

(log 1

h ) k N _h ^m _; f (x) k p; ; Ch : Therefore,

(log ¹ _h )

h k N h m

; f (x) k ^{p; ;} Ch ¹ ; which tends to zero with h ! 0. Thus

(log ¹ _h )

h k N h m

; f (x) k ^{p; ;} ! 0; h ! 0:

Then f 2 lip(1; ; p).

THEOREM 2.4. If < , then Lip( ; 0; p) Lip( ; 0; p) and lip( ; 0; p) lip( ; 0; p):

PROOF. We have 0 h 1 and < , then h h . Then the proof of the

theorem is immediate.

3 New Results on Dini Lipschitz Class

LEMMA 3.1. For f 2 L ^m _p ( ; ), then Z

R

2 ^{q qm} j ' ^; (h) 1 j ^q jF ; f ( ) j ^q d ( )

1 q

K k N _h ^m _; f k p; ; ;

where ¹ _p + ¹ _q = 1 and m = 0; 1; 2; ::::

PROOF. From (3), we have

F ; ( ^m _; f )( ) = i ^{m m} F ; (f )( ); m = 0; 1; 2; :::: (6) By using formulas (3) and (6), we conclude that

F ; (N _h ^m _; f )( ) = i ^m ( ^{( ; )} (h) + ^{( ; )} ( h) 2) ^m F ; (f )( );

Since

( ; )

(h) = ' ^; (h) + i

4( + 1) sinh(2h)' ^{+1; +1} (h);

( ; )

( h) = ' ^; ( h) i

4( + 1) sinh(2h)' ^{+1; +1} ( h);

and ' ^; is even, we see that F ^; (N h m

; f )( ) = 2i ^m (' ^; (h) 1) ^m F ^; (f )( ):

Now by formula (2), we have the result.

THEOREM 3.2. Let > 2. If f belongs to the Jacobi-Dunkl Dini Lipschitz class, i.e.,

f 2 Lip( ; ; p); > 2; 0:

Then f is null almost everywhere on R .

PROOF. Assume that f 2 Lip( ; ; p). Then we have k N h m

; f(x) k ^{p; ;} C h

(log _h ¹ ) ; 0:

From Lemma 3.1, we get Z

R

qm j ' ^; (h) 1 j ^q jF ^; f ( ) j ^q d ( ) K ^q C ^q 2 ^q

h ^q (log _h ¹ ) ^q : In view of Lemma 1.3, we conclude that

Z

R j 1 j ( h) j ^{q qm} jF ; f ( ) j ^q d ( ) K ^q C ^q 2 ^q C ₂ ^q

h ^q

(log _h ¹ ) ^q :

Then R

R j 1 j ( h) j ^{q qm} jF ^; f ( ) j ^q d ( ) h ^2q

K ^q C ^q 2 ^q C ₂ ^q

h ^{q 2q} (log ¹ _h ) ^q : Since > 2; we have

h lim ! 0

h ^{q 2q} (log ¹ _h ) ^q = 0:

Thus

h lim ! 0

Z

R

j 1 j ( h) j

2 h ²

q

2q qm

jF ^; f ( ) j ^q d ( ) = 0;

and also from the formula (4) and Fatou theorem, we obtain Z

R 2q qm

jF ^; f ( ) j ^q d ( ) = 0:

Hence ^{2 m} F ^; f ( ) = 0 for all 2 R , and so f (x) is the null function.

Analog of the Theorem 3.2, we obtain this theorem.

THEOREM 3.3. Let f 2 L ^m _p ( _; ). If f belong to lip(2; 0; p), i.e., k N _h ^m _; f (x) k p; ; = O(h ² ); as h ! 0:

Then f is null almost everywhere on R .

THEOREM 3.4. Let f belong to Lip( ; ; p). Then Z

j j r

qm jF ; f ( ) j ^q d ( ) = O r ^q

(log r) ^q ; as r ! 1 ; where ¹ _p + ¹ _q = 1 and m = 0; 1; 2; :::.

PROOF. Let f 2 Lip( ; ; p). Then k N h m

; f (x) k ^{p; ;} = O h

(log _h ¹ ) ; as h ! 0:

From Lemma 3.1, we have Z

R

qm j ' ^; (h) 1 j ^q jF ^; f ( ) j ^q d ( ) K ^q

2 ^q k N h m

; f (x) k ^q p; ; : By (5) and Lemma 1.2, we get

Z

2h

j j

qm j 1 ' ^; (h) j ^q jF ^; (f )( ) j ^q d ( )

C ₁ ^q C ₂ ^q Z

2h

j j

qm j h j ^2q jF ; (f )( ) j ^q d ( ):

From _2h j j h ; we have

2h

2 2 2

h

2 2 ;

which implies that

2 h ²

2

4

2 h ² : Take h ₃ , then we have ² h ² C ₃ = C ₃ ( ). So

Z

2h

j j

qm j 1 ' ^; (h) j ^q jF ; (f )( ) j ^q d ( )

C ₁ ^q C ₂ ^q C ₃ ^q Z

2h

j j

qm jF ^; (f )( ) j ^q d ( ):

Note that there exists then a positive constant C ₄ such that Z

2h

j j

qm jF ^; (f )( ) j ^q d ( ) C 4

Z

R

qm j 1 ' ^; (h) j ^q jF ^; (f )( ) j ^q d ( )

= O h ^q

(log _h ¹ ) ^q : So we obtain Z

r j j 2r

qm jF ^; (f )( ) j ^q d ( ) C 5

r ^q (log r) ^q ; where C 5 is a positive constant. Now, we have

Z

j j r

qm jF ; (f )( ) j ^q d ( ) = X 1 i=0

Z

2

r j j 2

r

qm jF ; (f )( ) j ^q d ( )

C 5

r ^q

(log r) ^q + (2r) ^q

(log 2r) ^q + (4r) ^q (log 4r) ^q + C ₅ r ^q

(log r) ^q + (2r) ^q

(log r) ^q + (4r) ^q (log r) ^q + C ₅ r ^q

(log r) ^q 1 + 2 ^q + (2 ^q ) ² + (2 ^q ) ³ + K r ^q

(log r) ^q ;

where K = C 5 (1 2 ^q ) ¹ since 2 ^q < 1. Consequently Z

j j r

qm jF ^; f ( ) j ^q d ( ) = O r ^q

(log r) ^q ; as r ! 1 :

Thus, the proof is …nished.

COROLLARY 3.5. Let f 2 L ^m _p ( _; ). If k N _h ^m _; f k p; ; = O h

(log _h ¹ ) ; as h ! 0;

then Z

j j r jF ^; (f )( ) j ^q d ( ) = O r ^{qm q}

(log r) ^q ; as r ! 1 : where ¹ _p + ¹ _q = 1 and m = 0; 1; 2; :::.

DEFINITION 3.6. A function f 2 L ^m _p ( _; ) is said to be in the ( ; )-Jacobi-Dunkl Dini Lipschitz class, denoted by Lip( ; ; p), if

k N _h ^m _; f (x) k p; ; = O (h)

(log ¹ _h ) ; as h ! 0; 0;

where

(1) (t) a continuous increasing function on [0; 1 ), (2) (0) = 0,

(3) (ts) = (t) (s) for all t; s 2 [0; 1 ).

THEOREM 3.7. Let f belong to Lip( ; ; p). Then Z

j j r

qm jF ^; f ( ) j ^q d ( ) = O (r ^q )

(log r) ^q ; as r ! 1 ; where ¹ _p + ¹ _q = 1 and m = 0; 1; 2; :::.

PROOF. Let f 2 Lip( ; ; p). Then

k N _h ^m _; f (x) k p; ; = O (h)

(log _h ¹ ) ; as h ! 0:

From Lemma 3.1, we have Z

R

qm j ' ^; (h) 1 j ^q jF ; f ( ) j ^q d ( ) K ^q

2 ^q k N _h ^m _; f (x) k ^q p; ; : By (5) and Lemma 1.2, we get

Z

2h

j j

qm j 1 ' ^; (h) j ^q jF ^; (f )( ) j ^q d ( )

C ₁ ^q C ₂ ^q Z

2h

j j

j h j ^{2q qm} jF ; (f )( ) j ^q d ( ):

From _2h j j h we have

2h

2 2 2

h

2 2 ;

which implies that

2 h ²

2

4

2 h ² : Take h ₃ , then we have ² h ² C ₃ = C ₃ ( ). So

Z

2h

j j

qm j 1 ' ^; (h) j ^q jF ; (f )( ) j ^q d ( )

C ₁ ^q C ₂ ^q C ₃ ^q Z

2h

j j

qm jF ^; (f )( ) j ^q d ( ):

Note that there exists then a positive constant C ₄ such that Z

2h

j j

qm jF ^; (f )( ) j ^q d ( ) C 4

Z

R

qm j 1 ' ^; (h) j ^q jF ^; (f )( ) j ^q d ( )

= O ( (h)) ^q

(log _h ¹ ) ^q = O (h ^q ) (log ¹ _h ) ^q : So we obtain Z

r j j 2r

qm jF ^; (f )( ) j ^q d ( ) C 6

(r ^q ) (log r) ^q ; where C 6 is a positive constant. Now, we have

Z

j j r

qm jF ; (f )( ) j ^q d ( ) = X 1 i=0

Z

2

r j j 2

r

qm jF ; (f )( ) j ^q d ( )

C 6

(r ^q )

(log r) ^q + ((2r) ^q )

(log 2r) ^q + ((4r) ^q ) (log 4r) ^q + C ₆ (r ^q )

(log r) ^q + ((2r) ^q )

(log r) ^q + ((4r) ^q ) (log r) ^q + C ₆ (r ^q )

(log r) ^q 1 + (2 ^q ) + ² (2 ^q ) + ³ (2 ^q ) + C (r ^q )

(log r) ^q ;

where C = C 6 (1 (2 ^q )) ¹ since (2 ^q ) < 1. Consequently, Z

j j r

qm jF ^; f ( ) j ^q d ( ) = O (r ^q )

(log r) ^q ; as r ! 1 ;

and this ends the proof.

COROLLARY 3.8. Let f 2 L ^m _p ( _; ). If k N h m

; f k ^{p; ;} = O h

(log _h ¹ ) ; as h ! 0;

then Z

j j r jF ; (f )( ) j ^q d ( ) = O r ^qm (r ^q )

(log r) ^q ; as r ! 1 ; where ¹ _p + ¹ _q = 1 and m = 0; 1; 2; :::.

Acknowledgemnts. The authors would like to thank the referee for his valuable comments and suggestions.

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