aa r X i v : . [ m a t h . F A ] D ec Dunkl-spherical maximal function
Abdessattar JemaiJanuary 23, 2018
Abstract
In this paper, we study the L p -boundedness of the spherical max-imal function associated to the Dunkl operators. Keywords:
Dunkl operators; Dunkl transform; Dunkl translations; Sphericalmaximal function.
In [13], E. Stein introduced the spherical maximal function by M ( f )( x ) = sup r> Z S d − f ( x − ry ) dσ ( y ) , where dσ is the surface measure on S d − and showed that M is boundedfrom L p ( R d ) to L p ( R d ) for the optimal range p > dd − with d ≥
3. The case d = 2 was proved by Bourgain in [1]. The aim of this work is to extend theseresults to the Dunkl setting.To begin, we recall some results in Dunkl theory (see [3, 4, 6, 7, 9, 15])and we refer for more details to the survey [8].Let G ⊂ O ( R d ) be a finite reflection group associated to a reduced rootsystem R . For α ∈ R , we denote by H α the hyperplane orthogonal to α .We denote by k a nonnegative multiplicity function defined on R with theproperty that k is G -invariant. For a given β ∈ R d \ S α ∈ R H α , we fix a1 A. Jemai positive subsystem R + = { α ∈ R : h α, β i > } . We associate with k theindex γ = X ξ ∈ R + k ( ξ ) and a weighted measure ν k given by dν k ( x ) := w k ( x ) dx where w k ( x ) = Y ξ ∈ R + |h ξ, x i| k ( ξ ) , x ∈ R d . Further, we introduce the Mehta-type constant c k by c k = (cid:18)Z R d e − k x k w k ( x ) dx (cid:19) − . For every 1 ≤ p ≤ + ∞ , we denote by L pk ( R d ), the spaces L p ( R d , dν k ( x )) , andwe use k k p,k as a shorthand for k k L pk ( R d ) .By using the homogeneity of degree 2 γ of w k , for a radial function f in L k ( R d ), there exists a function F on [0 , + ∞ ) such that f ( x ) = F ( k x k ),for all x ∈ R d . The function F is integrable with respect to the measure r γ + d − dr on [0 , + ∞ ) and we have Z R d f ( x ) dν k ( x ) = Z + ∞ (cid:16) Z S d − f ( ry ) w k ( ry ) dσ ( y ) (cid:17) r d − dr = Z + ∞ (cid:16) Z S d − w k ( ry ) dσ ( y ) (cid:17) F ( r ) r d − dr = d k Z + ∞ F ( r ) r γ + d − dr, where S d − is the unit sphere on R d with the normalized surface measure dσ and d k = Z S d − w k ( x ) dσ ( x ) = c − k γ + d − Γ( γ + d ) . The Dunkl operators T j , ≤ j ≤ d are the following k -deformations ofdirectional derivatives ∂x j given by: T j f ( x ) = ∂f∂x j ( x ) + X α ∈ R + k ( α ) α j f ( x ) − f ( ρ α ( x )) h α, x i , f ∈ E ( R d ) , x ∈ R d , where ρ α is the reflection on the hyperplane H α and α j = h α, e j i , ( e , . . . , e d )being the canonical basis of R d .Notice that in the case k ≡
0, the weighted function w k ≡
1, the measure ν k coincide with the Lebesgue measure and the operator T ξ reduced to thecorresponding partial derivatives ∂x j . Therefore Dunkl analysis can be viewedas a generalization of classical Fourier analysis.For y ∈ C d , the system T j u ( x, y ) = y j u ( x, y ) , ≤ j ≤ d ,u (0 , y ) = 1 . admits a unique analytic solution on R d , denoted by E k ( x, y ) and called theDunkl kernel. This kernel has a unique holomorphic extension to C d × C d .We have for all λ ∈ C and z, z ′ ∈ C d , E k ( z, z ′ ) = E k ( z ′ , z ), E k ( λz, z ′ ) = E k ( z, λz ′ ) and for x, y ∈ R d , | E k ( x, iy ) | ≤ F k is defined for f ∈ D ( R d ) by F k ( f )( x ) = c k Z R d f ( y ) E k ( − ix, y ) dν k ( y ) , x ∈ R d . We list some known properties of this transform:i) The Dunkl transform of a function f ∈ L k ( R d ) has the following basicproperty kF k ( f ) k ∞ ,k ≤ k f k ,k . ii) The Dunkl transform is an automorphism on the Schwartz space S ( R d ).iii) When both f and F k ( f ) are in L k ( R d ), we have the inversion formula f ( x ) = Z R d F k ( f )( y ) E k ( ix, y ) dν k ( y ) , x ∈ R d . iv) (Plancherel’s theorem) The Dunkl transform on S ( R d ) extends uniquelyto an isometric automorphism on L k ( R d ).The Dunkl translation operators τ x , x ∈ R d is defined on L k ( R d ) by F k ( τ x ( f ))( y ) = E k ( ix, y ) F k ( f )( y ) y ∈ R d . A. Jemai
As an operator on L k ( R d ), τ x is bounded. According to ([15], Theo-rem 3.7), the operator τ x can be extended to the space of radial functions L pk ( R d ) rad , ≤ p ≤ f in L pk ( R d ) rad , k τ x ( f ) k p,k ≤ k f k p,k . It was shown in [10] that if f is a radial function in S ( R d ) with f ( y ) = e f ( k y k ),then τ x ( f )( y ) = Z R d e f ( A ( x, y, η )) dµ x ( η ) (1.1)where A ( x, y, η ) = p k x k + k y k − < y, η > and µ x is a probability mea-sure supported in the convex hull co ( G.x ) of the G -orbit of x in R d . Weobserve that, η ∈ co ( G.x ) = ⇒ min g ∈ G k g.x − y k ≤ A ( x, y, η ) ≤ max g ∈ G k g.x − y k . (1.2)We collect below some useful facts:i) For all x, y ∈ R d , τ x ( f )( y ) = τ y ( f )( x ).ii) For f ∈ L k ( R d ) ∩ L k ( R d ) Z R d τ x ( f )( y ) dν k ( y ) = Z R d f ( y ) dν k ( y ) . (1.3)iii) For all x ∈ R d and f, g ∈ L k ( R d ) Z R d τ x ( f )( y ) g ( y ) dν k ( y ) = Z R d f ( y ) τ x ( g )( y ) dν k ( y ) . (1.4)The Dunkl convolution product ∗ k of two functions f and g in L k ( R d ) isgiven by ( f ∗ k g )( x ) = Z R d τ x ( f )( − y ) g ( y ) dν k ( y ) , x ∈ R d . The Dunkl convolution product is commutative and for f, g ∈ D ( R d ), wehave F k ( f ∗ k g ) = F k ( f ) F k ( g ) . (1.5)It was proved in ([15], Theorem 4.1) that when g is a bounded radial functionin L k ( R d ), then the application f −→ f ∗ k g initially defined on the inter-section of L k ( R d ) and L k ( R d ) extends to L pk ( R d ), 1 ≤ p ≤ + ∞ as a boundedoperator. In particular, k f ∗ k g k p,k ≤ k f k p,k k g k ,k . (see [10, 11] ) The Dunkl transform of σ is given by F k ( σ )( x ) = 1 c k Z S d − E k ( − ix, y ) ω k ( y ) dσ ( y ) = c γ j γ + d − ( k x k ) , (1.6)where j γ + d − is the Bessel function of the first type and c γ = γ + d − Γ( γ + d ) .In particular (see [14]), the function F k ( σ )( rξ ) = O ( r − γ + d − ) , r → + ∞ . (1.7)1 r ∂∂r F k ( σ )( rξ ) = O ( r − γ + d +12 ) , r → + ∞ . (1.8)Along this paper we use C to denote a suitable positive constant which isnot necessarily the same in each occurrence and we write for x ∈ R d , k x k = p h x, x i . Furthermore, we denote by • E ( R d ) the space of infinitely differentiable functions on R d . • S ( R d ) the Schwartz space of functions in E ( R d ) which are rapidlydecreasing as well as their derivatives. • D ( R d ) the subspace of E ( R d ) of compactly supported functions. For f ∈ L k ( R d ), we define the maximal function M k f by M k ( f )( x ) = sup r> ν k ( B (0 , r )) (cid:12)(cid:12)(cid:12)(cid:12)Z B (0 ,r ) τ x f ( y ) dν k ( y ) (cid:12)(cid:12)(cid:12)(cid:12) , (2.1)where B (0 , r ) is the ball of radius r centered at 0 and x ∈ R d .It was proved in [15] that the maximal function is bounded on f ∈ L pk ( R d )for 1 < p ≤ ∞ , and of weak type (1 , f ∈ L k ( R d ) that is a > Z E ( a ) dν k ( x ) ≤ Ca k f k ,k (2.2) A. Jemai where E ( a ) = { x : M k f ( x ) > a } and C is a constant independent of a and f . As in [5], we define the spherical mean operator on A k ( R d ) = { f ∈ L k ( R d ) : F k ( f ) ∈ L k ( R d ) } , for x ∈ R d and r > S r ( f )( x ) = 1 d k Z S d − τ x ( f ( − ry )) ω k ( y ) dσ ( y )= 1 d k Z r S d − τ x ( f ( − y )) ω k ( yr ) dσ r ( y ) . Now, the Dunkl-spherical maximal function M ( f ) is given by M ( f )( x ) = sup r> | S r ( f )( x ) | , x ∈ R d . Theorem 2.1
Let γ + d ≥ and γ + d γ + d − < p < γ + d . Then there existsa constant C > such that for all f ∈ L pk ( R d ) , k M ( f ) k p,k ≤ C k f k p,k . (2.3)Before proving the theorem, we need to establish some useful results. Infact, fix a function ψ ∈ S ( R d ) which is the Dunkl transform of a C ∞ -radialfunction with compact support such that: ψ (0) = 1 , (cid:0) ∂ i ∂r i ψ (cid:1) (0) = 0 , (2.4)for 1 ≤ i < γ + d and where ∂∂r denotes the derivation in the radial direction.To obtain a such function, taking ψ ∈ S ( R d ) with F k ( ψ ) is a C ∞ - radialfunction with compact support and ψ (0) = 0 and ψ ( rξ ) = (cid:0) [ γ + d ] X j =0 a j r j (cid:1) ψ ( rξ ) , for ξ ∈ S d − . The coefficients a j are solutions of triangular system given bythe conditions (2.4) and [ γ + d ] is the least integer not less than γ + d .Now set the functions ψ j with ψ ( y ) = ψ ( y − ψ ( y ) and ψ j ( y ) = ψ (2 − ( j − y ) , j ≥ . (2.5)Thus, ψ j is radial and F k ( ψ j ) is a compact supported functions. It followsthat for some constants t, C > | ψ ( y ) | ≤ C k y k γ + d , if k y k ≤ t, (2.6)and + ∞ X j =0 ψ j ( y ) = 1 , y ∈ R d . (2.7)Let m j = F k ( σ ) ψ j and let ϕ j the function in S ( R d ) such that F k ( ϕ j ) = m j .It follows that for f ∈ S ( R d ), S r ( f )( x ) = + ∞ X j =0 f ∗ k ϕ j . In fact, this can be done because from (2.7), we have F k ( σ )( y ) = + ∞ X j =0 F k ( σ )( y ) ψ j ( y ) = + ∞ X j =0 F k ( ϕ j )( y )and we can write F k ( S r ( f ))( y ) = + ∞ X j =0 F k ( f ∗ k ϕ j )( y ) , which implies with the inversion formula that S r ( f )( y ) = F − k (cid:16) + ∞ X j =0 F k ( f ∗ k ϕ j ) (cid:17) ( y ) = Z R d + ∞ X j =0 F k ( f ∗ k ϕ j )( z ) E k ( iy, z ) dν k ( z ) . To interchange the sum and the integral, we proceed as follows:From (1.5), we have
A. Jemai + ∞ X j =1 (cid:16) Z R d (cid:12)(cid:12)(cid:12) F k ( f ∗ k ϕ j )( z ) E k ( iy, z ) (cid:12)(cid:12)(cid:12) dν k ( z ) (cid:17) ≤ + ∞ X j =1 (cid:16) Z R d |F k ( ϕ j )( z ) ||F k ( f )( z ) | dν k ( z ) (cid:17) ≤ + ∞ X j =1 (cid:16) Z R d | ψ j ( z ) ||F k ( f )( z ) | dν k ( z ) (cid:17) ≤ + ∞ X j =1 (cid:16) Z k z k≤ j − t | ψ j ( z ) ||F k ( f )( z ) | dν k ( z ) + Z k z k≥ j − t | ψ j ( z ) ||F k ( f )( z ) | dν k ( z ) (cid:17) . By using (2.5) and (2.6), we obtain Z k y k≤ j − t | ψ j ( z ) ||F k ( f )( z ) | dν k ( z ) ≤ c − ( j − γ + d Z k y k≤ j − t k z k γ + d |F k ( f )( z ) | dν k ( z ) ≤ c − ( j − γ + d Z R d k z k γ + d |F k ( f )( z ) | dν k ( z ) , and Z k z k≥ j − t | ψ j ( z ) ||F k ( f )( z ) | dν k ( z ) = Z k z k j − t ≥ | ψ j ( z ) ||F k ( f )( z ) | dν k ( z ) ≤ − ( j − t Z R d k z k| ψ j ( z ) ||F k ( f )( z ) | dν k ( z ) ≤ − ( j − t Z R d k z k|F k ( f )( z ) | dν k ( z ) , therefore + ∞ X j =1 (cid:16) Z R d (cid:12)(cid:12)(cid:12) F k ( f ∗ k ϕ j )( z ) E k ( iy, z ) (cid:12)(cid:12)(cid:12) dν k ( z ) (cid:17) converge.For all j ≥ r >
0, we define the function ϕ j,r ( x ) = r − γ − d ϕ j ( xr ). Thenwe can write, S r ( f )( x ) = + ∞ X j =0 f ∗ k ϕ j,r . Hence for f ∈ S ( R d ), we obtain M ( f ) ≤ + ∞ X j =0 M ϕ j ( f ) , (2.8)where M ϕ j ( f )( x ) = sup r> | f ∗ k ϕ j,r ( x ) | = sup r> (cid:12)(cid:12)(cid:12)(cid:12)Z R d τ x ( f )( y ) ϕ j,r ( y ) dν k ( y ) (cid:12)(cid:12)(cid:12)(cid:12) . To prove the theorem, it suffices to establish an inequality of the form k M ϕ j ( f ) k p,k ≤ C j,p k f k p,k , with + ∞ X j =0 C j,p < + ∞ . Lemma 2.1
There exists a constant
C > such that, for any x ∈ R d and j ≥ , | ϕ j ( x ) | ≤ C j (1 + k x k ) γ + d +1 . (2.9) Proof. 2.1
Remember that ϕ j = S r ( F − k ( ψ j )) for j ≥ . We have, F − k ( ψ j )( x ) = 2 ( j − γ + d ) F − k ( ψ )(2 j − x ) . Since F − k ( ψ ) and F − k ( ψ ) are in S ( R d ) , then |F − k ( ψ )( x ) | and |F − k ( ψ )( x ) | are bounded by C (1 + k x k ) γ + d +1 . (2.10) Taking φ j ( x, y, ξ ) = F − k ( ψ )(2 j − p k x k + k y k − < y, ξ > ) for j ≥ andusing (1.1), we get | ϕ j ( x ) | = (cid:12)(cid:12)(cid:12) Z S d − τ x ( F − k ( ψ j ))( − y ) ω k ( y ) dσ ( y ) (cid:12)(cid:12)(cid:12) ≤ ( j − γ + d ) Z S d − (cid:16) Z R d (cid:12)(cid:12)(cid:12) φ j ( x, y, ξ ) (cid:12)(cid:12)(cid:12) dµ x ( ξ ) (cid:17) ω k ( y ) dσ ( y ) . (2.11) For j = 0 , we have by (1.1) | ϕ ( x ) | = (cid:12)(cid:12)(cid:12) Z S d − τ x ( F − k ( ψ ))( − y ) ω k ( y ) dσ ( y ) (cid:12)(cid:12)(cid:12) ≤ Z S d − (cid:16) Z R d (cid:12)(cid:12)(cid:12) F − k ( ψ )( p k x k + k y k − < y, ξ > ) (cid:12)(cid:12)(cid:12) dµ x ( ξ ) (cid:17) ω k ( y ) dσ ( y ) . (2.12)0 A. Jemai
From (1.2), (2.10), (2.11) and (2.12), we get for j ≥ | ϕ j ( x ) |≤ C j (2 γ + d ) Z S d − (cid:16) Z R d j min g ∈ G k gx − y k ) γ + d +1 dµ x ( ξ ) (cid:17) ω k ( y ) dσ ( y ) ≤ C j (2 γ + d ) Z S d − j min g ∈ G k gx − y k ) γ + d +1 ω k ( y ) dσ ( y ) . If k x k > , then k gx − y k ≥ k x k − ≥ k x k and we have | ϕ j ( x ) | ≤ C j (2 γ + d ) (1 + 2 j − k x k ) γ + d +1 ≤ C − j k x k γ + d +1 ≤ C j (1 + k x k ) γ + d +1 . (2.13) If k x k ≤ then, | ϕ j ( x ) | ≤ Z S d − j (2 γ + d ) (1 + 2 j min g ∈ G k gx − y k ) γ + d +1 ω k ( y ) dσ ( y ) ≤ j (2 γ + d ) Z { y ∈ S d − ; min g ∈ G k gx − y k ≤ − j } ω k ( y ) dσ ( y )+ 2 j (2 γ + d ) + ∞ X i =0 − (2 γ + d +1) i Z { y ∈ S d − ; min g ∈ G k gx − y k ≤ i +1 − j } ω k ( y ) dσ ( y ) ≤ C (cid:16) j + 2 j + ∞ X i =0 − i (cid:17) ≤ C j (1 + k x k ) γ + d +1 . (2.14) From (2.13) and (2.14), we obtain | ϕ j ( x ) | ≤ C j (1 + k x k ) γ + d +1 . Lemma 2.2
There exists a constant
C > such that, for any f ∈ S ( R d ) , j ≥ and α > , Z e E ( α ) dν k ( x ) ≤ C j α k f k ,k . where e E ( α ) = { x ∈ R d ; M ϕ j ( f )( x ) > α } and C is a constant depending onlyon d . Proof. 2.2
Let first prove that, for f ∈ S ( R d ) and x ∈ R d , Z R d τ x ( | ϕ j,r | )( y ) | f ( y ) | dν k ( y ) ≤ C j M k ( | f | )( x ) j ≥ , (2.15) where M k ( f ) is given by (2.1).We denote by A i = { y ∈ R d , r i ≤ k y k < r i +1 } , then τ x ( | ϕ j,r | )( y ) = τ x ( + ∞ X i = −∞ | ϕ j,r | .χ A i )( y ) = + ∞ X i = −∞ τ x ( | ϕ j,r | .χ A i )( y ) . From (2.9), one has ( | ϕ j,r | .χ A i )( y ) ≤ C j r − (2 γ + d ) (1 + k y k r ) γ + d +1 χ A i ( y ) ≤ C j r − (2 γ + d ) (1 + 2 i ) γ + d +1 χ A i ( y ) , and since | ϕ j,r | .χ A i is a radial function, this implies that τ x ( | ϕ j,r | .χ A i )( y ) ≤ C j r − (2 γ + d ) (1 + 2 i ) γ + d +1 τ x ( χ A i )( y ) . Using (1.4), we obtain Z R d τ x ( | ϕ j,r | )( y ) | f ( y ) | dν k ( y ) ≤ C j Z R d + ∞ X i = −∞ r − (2 γ + d ) (1 + 2 i ) γ + d +1 τ x ( χ A i )( y ) | f ( y ) | dν k ( y ) ≤ C j + ∞ X i = −∞ r − (2 γ + d ) (1 + 2 i ) γ + d +1 Z R d χ A i ( y ) τ x ( | f | )( y ) dν k ( y ) ≤ C j + ∞ X i = −∞ r − (2 γ + d ) (1 + 2 i ) γ + d +1 Z B (0 ,r i +1 ) τ x ( | f | )( y ) dν k ( y ) ≤ C j + ∞ X i = −∞ r − (2 γ + d ) (1 + 2 i ) γ + d +1 ( r i +1 ) γ + d M k ( | f | )( x ) ≤ C j M k ( | f | )( x ) . A. Jemai
By the fact that, (cid:12)(cid:12)(cid:12)(cid:12)Z R d τ x ( ϕ j,r )( y ) f ( y ) dν k ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z R d | τ x ( ϕ j,r )( y ) || f ( y ) | dν k ( y ) ≤ Z R d τ x ( | ϕ j,r | )( y ) | f ( y ) | dν k ( y ) , we deduce M ϕ j ( f )( x ) ≤ C j M k ( | f | )( x ) . From (2.2), we conclude the proof of the lemma.
Lemma 2.3
There exists a constant
C > such that, for any f ∈ S ( R d ) , j ≥ , k M ϕ j ( f ) k ,k ≤ C − j γ + d − k f k ,k . Proof. 2.3 (See [13], [16]) For f ∈ S ( R d ) , we have from (1.5) F k ( f ∗ k ϕ j,r )( x ) = F k ( f )( x ) F k ( ϕ j,r )( x ) = F k ( f )( x ) m j ( rx ) . Put g j ( f )( x ) = (cid:16) Z + ∞ | f ∗ k ϕ j,r ( x ) | drr (cid:17) , the Littlewood-Paley function associated to the function ϕ j,r .Using the Plancherel theorem and by (1.5), we obtain k g j ( f ) k ,k = Z R d | g j ( f )( x ) | dν k ( x )= Z R d (cid:16) Z + ∞ | f ∗ k ϕ j,r ( x ) | drr (cid:17) dν k ( x )= Z + ∞ (cid:16) Z R d |F k ( f ∗ k ϕ j,r )( x ) | dν k ( x ) (cid:17) drr = Z + ∞ (cid:16) Z R d |F k ( f )( x ) | | m j ( rx ) | dν k ( x ) (cid:17) drr ≤ k f k ,k sup x =0 Z + ∞ | m j ( rx ) | drr . Since the function m j is radial, the integral is independent of x .By using the definition of m j , we have that Z + ∞ | m j ( rx ) | drr = Z + ∞ |F k ( σ )( rx ) | | ψ j ( rx ) | drr = Z + ∞ |F k ( σ )( rx ) | | ψ (2 − j rx ) | drr = Z + ∞ |F k ( σ )(2 j rx ) | | ψ ( rx ) | drr . From (1.6), (1.7) and (2.6), we obtain Z + ∞ | m j ( rx ) | drr ≤ C − j (2 γ + d − Z + ∞ | ψ ( rx ) | r γ + d − drr ≤ C − j (2 γ + d − , this gives k g j ( f ) k ,k ≤ C − j (2 γ + d − k f k . (2.16) Put now ˜ ϕ j,r ( x ) = r ddr ϕ j,r ( x ) and ˜ g j ( f ) the Littlewood-Paley function associ-ated to ˜ ϕ j,r , then ˜ g j ( f )( x ) = (cid:16) Z + ∞ | f ∗ k ˜ ϕ j,r ( x ) | drr (cid:17) = (cid:16) Z + ∞ r | ddr ϕ j,r ∗ k f ( x ) | dr (cid:17) . Similarly, with the use of (1.6) and (1.8), we have k ˜ g j ( f ) k ,k = Z R d (cid:16) Z + ∞ r | ddr ϕ j,r ∗ k f ( x ) | dr (cid:17) dν k ( x )= C − j (2 γ + d − k f k ,k . (2.17) Since lim r → + ∞ f ∗ k ϕ j,r ( x ) = 0 , we have | f ∗ k ϕ j,r ( x ) | = − Re (cid:16) Z + ∞ r ϕ j,s ∗ k f ( x ) dds ϕ j,s ∗ k f ( x ) ds (cid:17) = − Re (cid:16) Z + ∞ r ϕ j,s ∗ k f ( x ) ˜ ϕ j,s ∗ k f ( x ) dss (cid:17) ≤ Z + ∞ r | ϕ j,s ∗ k f ( x ) || ˜ ϕ j,s ∗ k f ( x ) | dss . A. Jemai
Using Cauchy-Schwartz’s inequality, we deduce that sup r> | f ∗ k ϕ j,r ( x ) | ≤ g j ( f )( x ) ˜ g j ( f )( x ) . Integrating over R d and using again the Cauchy-Schwartz inequality, we ob-tain from (2.16) and (2.17) k M ϕ j ( f ) k ,k ≤ C − j γ + d − k f k ,k . Lemma 2.4
There exists a constant
C > such that, for f ∈ S ( R d ) k M ϕ j ( f ) k ∞ ,k ≤ C j k f k ∞ ,k j ≥ . Proof. 2.4
From (1.3) and (2.9), one has for f ∈ S ( R d ) and x ∈ R d , | f ∗ k ϕ j,r ( x ) | ≤ Z R d | f ( y ) || τ x ϕ j,r ( y ) | dν k ( y ) ≤ k f k ∞ ,k Z R d | ϕ j,r ( y ) | dν k ( y ) ≤ C j k f k ∞ ,k Z R d k x k ) γ + d +1 dν k ( y ) ≤ C j k f k ∞ ,k , which gives the result. Remark 2.1
We observe that for j = 0 and from Lemmas 2.2, 2.4, weobtain by interpolation (see [14]) that M ϕ j ( f ) is of strong-type (p,p) with < p ≤ + ∞ . Proof of theorem 2.1.
According to Remark 2.1, we deduce by interpola-tion from Lemmas 2.2, 2.3 that k M ϕ j ( f ) k p,k ≤ C − j (2 γ + d − γ + dp − k f k p,k , for 1 < p ≤ j ≥ k M ϕ j ( f ) k p,k ≤ C − j ( γ + dp − k f k p,k , ≤ p ≤ + ∞ and j ≥ γ + d γ + d − < p < γ + d , we have + ∞ X j ≥ − j (2 γ + d − γ + dp − < + ∞ and + ∞ X j ≥ − j ( γ + dp − < + ∞ . This yields using (2.8) k M ( f ) k p,k ≤ C k f k p,k , which completes the proof. Remark 2.2
The case γ + d = 2 implies that k ≡ and was proved byBourgain in [1]. References [1]
J. Bourgain,
Averages in the plane over convex curves and maximaloperators, J. Analyse Math. 47 (1986), 69-85. [2]
F. Dai and H. Wang,
A transference theorem for the Dunkl transformand its applications, Journal of Functional Analysis 258 (2010), no. 12,4052–4074. [3]
C. F. Dunkl,
Differential–Difference operators associated to reflextiongroups, Trans. Amer. Math. 311 (1989), no. 1, 167–183. [4]
M.F.E. de Jeu,
The Dunkl transform, Invent. Math. 113 (1993), no. 1,147-162. [5]
H. Mejjaoli and K. Trimche,
On a mean value property associatedwith the Dunkl Laplacian operator and applications, Integral Transform.Spec. Funct. 12 (2001), 279-302. [6]
M. R¨osler,
Generalized Hermite polynomials and the heat equation forDunkl operators, Comm. Math. Phys. 192 (1998), no. 3, 519–542. [7]
M. R¨osler,
Positivity of Dunkl’s intertwining operator, Duke Math. J.98 (1999), 445–463. A. Jemai [8]
M. R¨osler,
Dunkl operators : theory and applications, Orthogonal poly-nomials and special functions (Leuven, 2002), Lect. Notes Math. 1817,Springer–Verlag (2003), 93–135. [9]
M. R¨osler,
Bessel–type signed hypergroup on R , Probability measures ongroups and related structures, XI (Oberwolfach, 1994), World Sci. Publ.,River Edge, NJ, 1995, pp. 292304. [10] M. R¨osler,
A positive radial product formula for the Dunkl kernel,Trans. Amer. Math. Soc. 355 (2003), no. 6, 2413–2438. [11]
M. R¨osler,
Markov Processes Related With Dunkl Operators, Advancesin Applied Mathematics 21, 575–643 (1998). [12]
E.M. Stein,
Maximal functions: spherical mean, Proc. Nat. Acad. Sci.U.S.A. 73(1976), 2174-2175. [13]
E.M. Stein,
Singular Integrals and Differentiability Properties of Func-tions, Princeton Mathematical Series 30. Princeton University Press,Princeton, NJ, 1970. [14]
E.M. Stein and G. Weiss,
Introduction to Fourier Analysis on Eu-clidean Spaces, Princeton Mathematical Series 32. Princeton UniversityPress, Princeton, NJ, 1971. [15]
S. Thangavelu and Y. Xu,
Convolution operator and maximal func-tion for the Dunkl transform, J. Anal. Math. 97 (2005), 25–56. [16]