psedidifferential-difference operators associated with dunkl operators

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Psedidifferential-difference operators associated withdunkl operatorsAzza Dachraoui aa Department of Mathematics, Faculty of Sciences of Tunis , Campus Universitaire , Tunis,1060, TunisiaPublished online: 03 Apr 2007.

To cite this article: Azza Dachraoui (2001) Psedidifferential-difference operators associated with dunkl operators, IntegralTransforms and Special Functions, 12:2, 161-178

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PSEUDODIFFERENTIAL-DIFFERENCE OPERATORS ASSOCIATED WITH DUNKL OPERATORS

Azza DACHRAOUI

Department of Mathematics, Faculty of Sciences of Tunis, Campus Universitaire, 1060 Tunis, Tunisia

(Received July 12, 2000)

In this paper, we consider the Dunkl operator A, of index (a + 1/2), a > -1/2, associated with the reflexion group ZZ on R. We introduce pseudodifferential-difference operators and Sobolev type spaces, associated with A, , and using the harmonic analysis associated with A, , we study some of their properties.

KEY WORDS: Dunkl operator, Dunkl transform, pseudodifferential-difference operator, Sobolev type spaces

MSC (2000): 42A38, 35805, 46335

1. INTRODUCTION

Many authors have generalized the classical notion of pseudodifferential operators to harmonic analysis associated with differential operators and have studied properties of these new pseudodifferential operators ([I], [2], [5 ] ) .

In this paper, we consider the differential-difference operator A, on R, given

This operator is the Dunkl operator of index ( a + 1/2), a 1 -1/2 associated with the reflexion group Z2 on W. Using the harmonic analysis associated with A, built by C.F. Dunkl and M.F.E. de Jeu ([6] ,[4]), we introduce pseudodifferential- difference operators and Sobolev type spaces associated with A,, and we study some of their properties.

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162 A. DACHRAOUI

The content of this paper is as follows. In the first section, we recall some properties of the eigenfunctions Qf; of Dunkl operator A,, cr 2 -1/2, and we give an expression of the derivatives d2nSz(x) /dx2nl n E N, by using the functions 9:+k(x), 0 5 k 5 n. In the second section, we give the main results of the harmonic analysis associated with A,. In the third section, we define two classes of symbols SF and Sm, m E W, with S"' C S r . For p E Sr, we associate a pseudodifferential-difference operator I f p and we prove that H, is continuous from S(R) (the space of Cm-functions on R, rapidly decreasing together with their derivatives) into itself. In the last section, we introduce Sobolav type spaces Gz', s E R, r E [I , +m], a 2 -1/2, and we establish that the pseudodifferential- difference operator H, associated with p E Sn', is continuous from G:'~" into G:OO, and from G:'~" into GO,", T 2 1.

2. EIGENFUNCTIONS OF DUNKL OPERATORS A,

In this section, we recall and establish some basic results of the Harmonic Analysis related to Dunkl operators. More details can be found in [6],[7] .

Dunkl operators are differential-difference operators A,, cr 2 -112, on W, defined by

For a 2 -112, and X E C, the equation

has a unique solution ST given by

where j, is the normalized Bessel function of order cr given by

j,(z) = I'(a + 1)

In particular, we have

v X , z E @ ,

v X , z E C ,

Using the following Laplace integral formula for j,

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PSEUDODIFFERENTIAL-DIFFERENCE OPERATORS 163

where A, z E C , a > -112, and formula ( 3 ) , we deduce the integral representation

for all A, z E C, a > -112. Derivation under integral sign in formula ( 6 ) gives the following inequality

( ) A n x E R , A , n € N . (7)

Formula (3) allows us to establish the following Proposition.

Proposition 2.1. For all n E N , a > -112, we have

d2n+l ii) - dx2n+l Q?(x ) = (iu2"+' k=o f: czn,k b : + k ( x ) - z+::i{2 j a + k + l ( ~ ~ ) ] , (9)

where C2n,k are given by

Proof. From formulas ( 2 ) and (3), we deduce that

d a + 112 d~ j = + l ( ~ x ) ] .

But, we have for all A, x in W

d ( ~ A ) ~ x - [ j a ( A x ) ] = - d x 2(a + j a + i 0 x )

So by derivation of formula (11) and using equation (12) , we obtain

d 2 d a + 112 - ( x ) = ( i A) [z *; ( x ) - 2 ( a + l ) ( a + 2 ) ( i ~ ) ~ x j ~ + ~ ( A x ) ] . d x 2

From formulas ( 3 ) and ( l l ) , we deduce that

d 2 - dx2 c : ( ~ ) = (a). [ Q ; ( ~ ) -

So equation (8) is true for n = 1 with

a + 112 C2,o = 1 , C2,l = --.

a + l Suppose now, that

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A. DACHRAOUI

where CZn.k are given by the relation (10). Then after two derivations, and using formula (13), we deduce that

If we remark that

a + + 112 - - (-l)k+l (n) r ( a + 1) r ( a + k + 312) (-1) C2n.k a + + k r ( a + 112) r ( a + k + 2)

then we have

Using

we obtain

n+l + c ( a ) 2 n + 2 ( - ~ ) k ( ) r ( a + l ) r(a+k+1/2)

k-1 r (a+1/2) r ( a + k + l ) ~ ; l + ~ ( x )

k = l

= ( i ~ ) ~ " + ~ Q ~ ( x ) + 2 (-l)k(iX)2nt2 [(;) + (kll)] k = l

X r ( a + 1) r ( a + k + 1/2)

~ a + ~ ( x ) r ( a + 112) r ( a + k + 1)

+ (-l)"+'(i~)~"+ r ( a + l ) r(a+(n+1)+1/2) *a+n+l r (a+l /2) r ( a + ( n + l ) + l ) ( 4 .

(3 + (A) = (n:l)?

d2(n+ 1) n + l *z(x) = (iX) 2(n+l)

dx2(n+l) [z c2(n+l),k*:+k(x~]-

So we have proved by induction that formulas (8) and (10) are true for all n E N. By deriving formula (8) and using equation ( l l ) , we deduce formula (9) and the proof of Proposition 2.1 is completed.

In the following we give the product formula for 3; established by M. Rijsler in [7].

We begin with some notations.

Notations. We denote by the following.

pa the measure defined on W by

dpa (x) = 1x120+1

dx, where a > -112; 2at11'(a + 1)

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LT(dpa), 1 I r 5 +m, the space of measurable functions f on W such that l/r

IIfIIr.a = [ l f ( ~ ) l ' d ~ ( ~ ) ] < + m if 1 5 r < +m, (16) R

llf llm = SUP If(x)l < +oo; .VCR

(17)

Vx,y,zEW,

L o otherwise;

Here Ka is the kernel of the product formula for the functions ja(Xx) given by

where 1~ is the indicator function of A. From formula (la), we have for all z E C such that lzl E [11x1 - Ivl(, 1x1 + Iyl]

and from (19) and (20) we deduce that

p;,, the measure given by

Wa(x, y, 2) d/.~a(z) if 2, Y # 0,

~PZ,&) = { :; if y = 0,

if x = 0, where 6, is the Dirac measure.

The measure p;,, satisfies

i) v X , Y # 0, SUPP p:,, = [-I4 - I Y L - ll4 - ly111 u [I14 - Ivl(,14 + lY117

ii) V x, y E R, pz,, (W) = 1, and

Proposition 2.2. For a > -112, X E C, and x,y E W, the eigenfunctions '3: satisfy the following product formula

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A. DACHRAOUI

Remark 2.1. This formula permits M. Risler [7] to define a generalized convolution product *D given for Dirac measures 6, and 6,, x, y E W, by

On the other hand using the kernel Ka given by formula (21), the generalized convolution product *a associated with the Bessel operator is given for 6, and 6,, where x, y E W+ = [0, +m[, by

We remark that (W, *D) is not a hypergroup but it is a signed hypergroup, while (W+,*B) is a hypergroup called the Bessel-Kingman hypergroup (see [3], (71).

A harmonic analysis associated with this hypergroup is studied in [a], [9]. In particular, we define the generalized convolution product f *B g of f even in Lr (dpa), r 2 1, and g even in L' (dp,) by

+m +03

2 0 + 1 2a+ld f *B g(x) = / / K a ( x y y , ~ ) f(z)g(y) Y Z ~ Y , for a.e. xEW+. (26)

0 0

We remark that this function satisfies the following inequality.

3. T H E DUNKL T R A N S F O R M

Notations. We denote by the following. - E(W) the space of C"-functions on W, equipped with the topology of uniform

convergence on all compacts for the functions and their derivatives. - V(W) the space of Cm-functions on W with compact support

V(W) = u 'Da (W), a20

where Da(W) is the space of Cm-functions on W with support in the interval [-a,a]. The topology on Va(W) is defined by the semi-norms

The space V(W) equipped with the limit inductive topology is a Fr6chet space.

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PSEUDODIFFERENTIAL -DIFFERENCE OPERATORS 167

- D. ( W ) the subspace of D ( W ) , consisting of even functions on W . - S ( W ) the space of Cm-functions on W , rapidly decreasing together with their

derivatives

v ( k , e ) E N', P k , t ( f ) = sup(1 + x ' ) ~ 1% (X) l < f m. ( 2 8 ) ZER

The seminorms P k , t , ( k , ! ) E N 2 , define the topology of S ( W ) . Equipped with this topology S ( W ) is a RBchet space.

- S , ( W ) the subspace of S ( W ) consisting of even functions on W . - H ( C ) the space of entire functions on C , which are of exponential type and

rapidly decreasing. We have

!HI(@) = U H a ( @ ) , 0 2 0

where H a ( @ ) is the space of entire functions on @ satisfying

The topology on I&,(@) is defined by the seminorms P,, m E N. The space !HI(@) equipped with the limit inductive topology is a RBchet space.

- H.(C) the subspace of !HI(@) consisting of even functions on C .

Definition 3.1. The Dunk1 transform is defined on L1(dp , ) by

V X E C , F ~ ( f ) ( . \ ) = / f ( x ) ~ ? ( x ) d p ~ ( ~ ) . ( 3 0 ) R

Remark 3.1.

i) Let f be an even function in ~ ' ( d ~ , ) , then F D ( ~ ) is even and for all X in W

F o ( f ) ( N = G ( f ) ( N , (31) where F g ( f ) is the Fourier-Bessel transform of f defined by

ii) Let f be an odd function in S ( W ) , then F o ( f ) is odd, and for all X in W * , we have

Theorem 3.1. (Paley-Wiener theorem for F B . ) T h e Fourier-Bessel t r a n s f o r m fi i s a topological i s o m o r p h i s m f r o m - S, (W) o n t o i t se l f ; - D,(W) o n t o H.(C). T h e inverse t r a n s f o r m ( s ) - ' satisfies (= ) - I = 7;.

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168 A . DACHRAOUI

Theorem 3.2. The Dunkl transform F D is a topological isomorphism from

- S ( R ) onto itself; - V(R) onto W(C) .

The inverse Dunkl transform is given b y

V x E R , . F i l ( f ) ( x ) = F D ( ~ ) ( - x ) . (34)

Proof. Let g be in S ( R ) (resp W ( C ) ) then g = g, +,go, where g, is even, and go is odd, so the functions g, and X -+ go(X)/X belong to S.(R) (resp. IHI.(C)). We conclude the proof by using the Paley--Weiner theorem for F;, and Fgfl and formulas (31), (33). 0

Proposition 3.1. Suppose that f E E(R) , and a > -112. Then we have the following.

i) For all n E W, A: ( f ) defined by A: ( f ) = f and A:( f ) = A, (A : - I f ) is a Cm-function on R.

ii) The operator A, maps continuously S ( R ) into itself.

Proof.

i) Let a > -112, x E R, and f E E(R) . We have 1

f(.) - f ( - x ) = 1 f l ( z t ) x dt. - 1

So we deduce that

Thus iim + ( x ) = 2 f i ( 0 ) z+O

and from formula (1 )

Iim A, f ( x ) = (2a + 2) f ' ( 0 ) . x+o

So A,( f ) can be extended to R. Using derivation under integral sign in formula (35), we have for all I; E N

1

$ ( ' ) (x ) = / t k f (L i l ' (x t ) dt - 1

and from (36) , we deduce that

So the function 4 extends to a Cm-function on R and we have the same result for A, f .

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PSEUDODIFFERENTIAL -DIFFERENCE OPERATORS

ii) Let f E S(W), k , n E N. Then from (1) and (35), we have

Using formula (36), we have

For 1x1 2 1, using Leibnitz formula and (35), there exists a positive constant C such that

sup l(1 + x2In 3 ( x ) 1 b l l l

k

5 C C sup (1 + x2)" {I f(k- ')(x)l + 1 f ( k - f ) ( - x ) l } (39) L=O 1421

where Pn,k-( is the semi-norm given by (28). Using formulas (37), (38), and (39) we deduce that A,, is continuous from S(W) into itself. 0

Proposi t ion 3.2.

i) For all f in S(W), g in &(W) , we have

iii) For all f in ~ ' ( d ~ , ) , the Dunkl transform F D ( ~ ) is a continuous function on W and satisfy

11F~(f)lloo,a 5 I l f 111,~. (42) T h e o r e m 3.3.

i) Plancherel formula for F D : for all f in D(W) we have

/ I ~ ( X ) I ~ I X I ~ " + ~ ~ X = / I F D D ( ~ ) ( u ~ ~ P ~ ( A ) . W x

ii) Plancherel theorem for F D : the Dunkl transform FD eztends uniquely to an isomorphism on L2(d/.ja).

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170 A. DACHRAOUI

4. PSEUDODIFFERENTIAL-DIFFERENCE OPERATORS ASSOCIATED WITH DUNKL OPERATORS

Definition 4.1. Let 7n be a real. The function p : R x C -i @ is called a symbol in the class SF, if it satisfies the following.

i) For fixed t in W, the function X -i p(x, A) is Cm on W.

ii) For fixed X in W, the function x + p(x, A) is Cm on W. iii) For all k, n in N, there exists Ck,n,m > 0, such that for all x E W and X E C

Definition 4.2. Let m be a real. The function p : W x C + C is called a symbol in the class Sm, if it satisfies i), ii) of Definition 4.1 and iii)'.

iii)' For all k, e, n in N, there exists C ~ J , ~ , , , > 0 such that

Definition 4.3. Let p E Sr. The pseudodifferential-difference operator associated with the symbol p is defined on S(R) by

(45) W

In certain cases the operator Hp takes simple forms. In the following remark, we give some of these forms.

Remark 4.1. If p E Sr, having the form p(x, A) = q ( X ) then using formula (34), we have clearly

Hp(f)(x) = 3 i 1 ( q 3 ~ ( f ))(x).

If p(X) = -iX, from formula (41), we recover the Dunk1 operator

Hp(f = A a ( f 1.

The aim of this section is to establish that for p E Sr , the pseudodifferential- difference operator Hp associated with p is a linear continuous mapping from S(W) into itself. For the proof of this result, we need the following propositions.

Proposition 4.1. Let f E S(W) and p E SE;(W). Then for all n , j , k , e E N, there exists a positive constant C and s E N, such that:

where Pr,t is the semi-norm given by (28).

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Proof. Let r , t,!, k in W, f E S(W) and p E S F . Using Leibnitz formula, Theorem 3.2, and formula (43), there exist positive constants k l , k2 and sl E N such that

ak l(i + ~ 2 ) ~ & b l ~ D ( f ) ~ ) p ( x , ~ ) ] 1

We conclude the proof of Proposition 4.1, using the continuity of on S(W) given by Proposition 3.1 ii). 0

Remark 4.2. Rom (47), we deduce that for fixed x in W, the function

belongs to S(W) uniformly with x.

Proposition 4.2. Let a > -112, k , t , n E PI and r E N, with 0 5 r 5 t , 0 5 2.t 5 k. Then for f E S(W) and p E S F , there exist C > 0, and s E N, such that the integral

Proof. Rom formulas (4 ) , (5 ) and (2 ) , we remark that

2n a+r 2n a+r ( i x ) \k-x ( x ) = ( i x ) \k-, (A) = A?+~,A [Q$] (A). (50)

Let f E S(W) and p E SF . Using formulas ( I S ) , (40) and (50) we deduce that

Let q E W, q > a + r + 1, then using formula (7 ) and Proposition 4.1, we deduce that there exist C > 0 , and s E N, such that

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172 A . DACHRAOUI

Proposition 4.3. Let a 1 -112, k,n,P,r E N with 0 5 r 5 1, 0 < 21+ 1 I: k, then for p E SF and f E S(IR), there exist C > 0 and .9 E R, such that the integral

satisfies

Proof. First, we recall that for a > -112, r E N, the function X -+ j,+,+l(Xx) is the unique solution of the differential equation on [0, +m[, given by

where La is the Bessel operator defined by

So from (53), we have

i)

ii)

( i ~ ) ~ " j a + r + i ( X x ) = L l + r + ~ , ~ [ ja+r+i(X~)] .

We suppose first that 0 < r < C, r E N. Wc denote by

then from formula (47), the function X -t Ft(X) + FIT(-A) belongs to S.(R), and we have from (54)

+w

R.,t(x) = A, . / L;+.+~,A [Fz(h) + &(-A)] ja+,+l(Ax) d~n+i+l( . \ ) . 0

Using the continuity of La+,+1 on S,(R), and inequality (47), we concliide the proof of Proposition 4.3 for r < !. If r = e, the integral R,.,,(x) is given by

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Using the same proof as for inequalities (35), (38), (39) and (47), the function belongs to S*(W). From the continuity of FD on S(W) and formula (43),

we deduce that for 81,s~ E N there exist s3 E N and C > 0, independant of x, such that the semi-norm

From formulas (54), (55) and (56), we have

%r (XI = A w )mL~+r+8 ,A(h) (~) ja+r+l(Xx) dpa+T+i (A) 0

and for q E N, q > a + r + 2 , we deduce that

We conclude the proof using the continuity of L,+r+l,x on &(R) and formula (58).

Theorem 4.1. Let p E Sp, t hen the pseudodiflerential-dzfference operator H p associated wi th p is a linear continuous mapping f rom S(R) in to itself.

Proof. Let f E S ( W ) and p E SF. Using formulas ( 5 ) , (43), (45), derivation under integral sign and Leibnitz formula, we have for all n, k E N

Using formulas (8), (9), (48), (51) we deduce that

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174 A. DACHRAOUI

dk ( i ~ ) ~ " - Hp (f) (x) =

dxk x x (:!) c2 tqr~7 ,2 t (x) 0<215k O<r<C

f f+r+1/2 (59)

+ c x ( 2 ~ 1 ) c 2 ~ , ~ {~r,Zl+l(x)- f f+r+ l 0<2C+l<k o < r < e

Using inequalities (52), (49) and formula (59), we deduce that Ifp is continuous from S ( R ) into itself. 0

5. SOBOLEV TYPE SPACES

Definition 5.1. The space G;", s E R , r E [I , +m], is defined as the closure of V(R) with respect to the norm

The spaces G;' satisfy the following properties.

i) The space S ( R ) is included in G;'(IIS), r E [I , +m].

ii) Let s E R and f E G;'. Then for all TL E N, n 5 S, the function A: f belongs t o L2(dP,). Indeed we have

So from formula (41), 3~ (AEf) belongs to L2(dp,). We deduce the result by using Plancherel Theorem (Theorem 3.3).

iii) We have G$2 = ~ ~ ( d , u , ) . In the following, we study the Dunkl transform of a pseudodifferential-difference operator H,,, associated with p E Sm, and we prove that H,(f) belongs to some Sobolev type space.

Proposition 5.1. Let p be a symbol in S"'. Then for all k E N, there exists Cm,k > 0, such tha t for all XI, A 2 E R, we have

Proof. Let p E Sm. From Definition 4.2, for all k, j E N there exists C m , k , j > 0, S U C ~ that D

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Thus for all X E W, the function x + p(x , X I ) belongs to S(W). For all n E N, from formulas (41) , (7), we have for j E W, j > a + 1,

) ( i ~ z ) ' " FD(P( . , ~ 1 ) ) ( ~ 2 ) ) = ) F D (A&P(., X i ) ) ( X 2 ) 1

Using the continuity of A,,, on S(W) and formula (62), we deduce the inequality (61) . 0

Proposi t ion 5.2. Let p(x, A) be a symbol in S m , and f E S(W). Then for all X in W, we have

where W, is the kernel given by formula (20) .

Proof. From formulas (30) and (45) , we have for p E Sm and f in S(W)

T h e o r e m 6.1. Let p be a symbol in S m , and H p the pseudodiflerential-difference operator associated with p. Then, there exists C > 0 , such that:

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176 A. DACHRAOUI

i) for all f in GT'~" we have

Proof. Using formulas (61), (63) and (24), we deduce that there exists C,," > 0 such that

So we deduce formula (64). Let k E N, k > a + I , then from formulas (63), (61) and (22) we have

We denote by

Then for all X E R, from (66), we have

where * B is the convolution product related to the Bessel operator and defined by (26). Using formulas (68), (27) we have

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But from formula (67), we have

So there exists a positive constant C such that for r 2 1

ACKNOWLEDGEMENTS

The author is thankful to Professor K. Trimbche for the interest that he yields for this work.

REFERENCES

1. N. Ben Salem and A. Dachraoui, Pseudodifferential operators associated with the Jacobi differential operator, J . Math. Anal. Appl., 220 (1998), 365-381.

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psedidifferential-difference operators associated with dunkl operators

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