J. Pseudo-Differ. Oper. Appl. DOI 10.1007/s11868-016-0149-9

Generalized Dunkl-Lipschitz spaces Samir Kallel1

Received: 4 October 2015 / Revised: 5 February 2016 / Accepted: 8 February 2016 © Springer International Publishing 2016

Abstract This paper deals with generalized Lipschitz spaces ∧kα, p,q (R) in the context of Dunkl harmonic analysis on R , for all real α. It also introduces a generalized Dunkl-Lipschitz spaces T ∧kα, p,q (R2+ ) of k-temperature on R2+ . Some properties and continuous embedding of these spaces and the isomorphism of T ∧kα, p,q (R2+ ) and ∧kα, p,q (R) are established. Keywords Dunkl operator · Poisson transform · Heat transform · Dunkl transform · Generalized Dunkl-Lipschitz space Mathematics Subject Classification

42A38 · 46E30 · 46E35 · 46F12

1 Introduction In [16], we have introduced and characterized for α > 0 and 1 ≤ p, q ≤ ∞ the generalized Dunkl-Lipschitz spaces ∧kα, p,q (R) associated with the Dunkl operator with parameter k ≥ 0 Dk f (x) = f  (x) + k

f (x) − f (−x) , x

f ∈ C 1 (R).

We were interested in characterizing the functions f ∈ ∧kα, p,q (R) for α > 0 in p terms of their k-Poisson transform and the second order L k -modulus of continuity. k It is natural to extend the theory of the spaces ∧α, p,q (R) for all real α. To get this

B 1

Samir Kallel [email protected] Department of Mathematics, Faculty of Sciences of Tunis, University Campus, 2092 Tunis, Tunisia

S. Kallel

extension we use the k-heat transforms, since it is better suited in the treatment of tempered distributions than the k-Poisson transforms. More precisely, we define the spaces ∧kα, p,q (R) for α ≤ 0 as spaces of tempered distributions T that belongs to an appropriate Lebesgue space for which the k-heat transform G kt (T ) of T satisfies the condition 

1

t

q(n− 21 α)

q dt ∂tn G kt (T )k, p

t

0

 q1

< ∞, if 1 ≤ q < ∞

and 1

sup t n− 2 α ∂tn G kt (T )k, p < ∞, if q = ∞,

0
where n = ( α2 ) and α is the smallest non-negative integer larger than α. The first goal of this paper is to study these spaces. As it is well known, the fractional integral operators play an important role in this theory. Here we use the Dunkl-Bessel potential Jαk which we show that Jβk is a topological isomorphism from ∧kα, p,q (R) onto ∧kα+β, p,q (R), with 1 ≤ p, q ≤ ∞ and α, β ∈ R. Next, certain properties and continuous embedding for ∧kα, p,q (R) are given. Our second objective will study the generalized Dunkl-Lipschitz spaces of ktemperatures (i.e., solutions of the Dunkl-type heat equation (Dk2 − ∂t )U = 0, for more details we can see for instance Mejjaoli [21] and Rösler [24]) on the whole halfplane R2+ = {(x, t) : x ∈ R, t > 0} which denote by T ∧kα, p,q (R2+ ), 1 ≤ p, q ≤ ∞. In Theorem 7.9, we prove some basic properties of the space T ∧kα, p,q (R2+ ) in which the most important one is the fact that the topological property of the space T ∧kα, p,q (R2+ ) does not depend on the (Lipschitz) index α. Thus, we should ask what relations there are between the generalized Dunkl-Lipschitz spaces ∧kα, p,q (R) and the generalized Dunkl-Lipschitz spaces of k-temperatures T ∧kα, p,q (R2+ ). To reply to this question we must use the k-heat transforms. In Theorem 7.10, we establish that a k-temperature U belongs to T ∧kα, p,q (R2+ ) if and only if it is the k-heat transform of an element of ∧kα, p,q (R). So that the spaces ∧kα, p,q (R) for α ≤ 0, which consist of tempered distributions, can be realized as spaces of functions. Similar results have been obtained by Fleet and Taibleson [13,25] in the framework of classical case k = 0. Later, Johnson [15], adopting Flett’s idea, defined a space of temperatures which is isomorphic to the Lipschitz space of Herz. His method leaned on a theory of Riesz potentials for temperatures. Additionally, for α > 0, the generalized Dunkl-Lipschitz spaces or Besov-Dunkl spaces have been studied extensively by several mathematicians and characterized in different ways by many authors (see [1–3,16–18]). Also, for α ∈ R, Bouguila, Lazhari and Assal [8], Kawazoe and Mejjaoli [19] have studied the Besov spaces in the Dunkl setting. In this work, it is important to mention that the 1D restriction is due to the fact that Dunkl translations operations in higher dimension are not yet known to be bounded p on L k apart from p = 2.

Generalized Dunkl-Lipschitz spaces

The organization of this paper is as follows. In Sect. 2, we recall some basic harmonic analysis results related to Dunkl operator. In Sect. 3, we recall some properties of the k-heat transform of a measurable function. In Sect. 4, a semi-group formula for k-temperatures is proved which will be used frequently. In Sect. 5, the Dunkl-Bessel potential is defined and related properties are investigated. In Sect. 6, ∧kα, p,q (R) for real α is defined and its properties have been obtained. In this section we also proved that Jβk is a topological isomorphism from ∧kα, p,q (R) onto ∧kα+β, p,q (R), α, β ∈ R, and a variety of equivalent norms for ∧kα, p,q (R) are given. The remainder of this section is devoted to some properties and continuous embedding for ∧kα, p,q (R). In Sect. 7, we defined the space T ∧kα, p,q (R2+ ), the equivalence of several norms on T ∧kα, p,q (R2+ ) is proved and some properties of this space are studied. At the end, the isomorphism of T ∧kα, p,q (R2+ ) and ∧kα, p,q (R) is established. In what follows, B represents a suitable positive constant which is not necessarily the same in each occurrence.

2 Preliminaries in the Dunkl setting on R In this section we state some definitions and results which are useful in the sequel and we refer for more details to the articles [7,9–12,26] and [28]. We first begin by some notations. Notations • C(R) is the space of continuous functions on R. • Cb (R) is the space of bounded continuous functions on R. • C0 (R) is the space of continuous functions vanishing at infinity, equipped with the usual topology of uniform convergence on R. • C p (R) is the space of functions of class C p on R, p ∈ N. • E(R) is the space of C ∞ -functions on R, endowed with the usual topology of uniform convergence of any derivative on compact subsets of R. • S(R) is the space of C ∞ -functions on R which are rapidly decreasing as well as their derivatives, endowed with the topology defined by the semi-norms ρs,l (ϕ) :=

j

sup (1 + x 2 )l |Dk ϕ(x)|, s, l ∈ N.

x∈R, j≤s

• S  (R) is the space of tempered distributions on R which is the topological dual of S(R). The Dunkl operator Dk with parameter k ≥ 0 is given by Dk f (x) := f  (x) + k

f (x) − f (−x) , x

f ∈ C 1 (R).

For k = 0, D0 reduces to the usual derivative which will be denoted by D. The Dunkl intertwining operator Vk is defined in [11] on polynomials f by Dk Vk f = Vk D f, and Vk 1 = 1.

S. Kallel

For k > 0, Vk has the following representation (see [11], Theorem 5.1) Vk f (x) :=



2−2k (2k + 1) (k)(k + 1)

1

f (xt)(1 − t 2 )k−1 (1 + t)dt.

−1

(1)

The operator Vk is a topological automorphism of E(R) satisfying ∀ f ∈ E(R), Dk (Vk f ) = Vk

d f. dx

The operator t Vk is defined on D(R) by  ∀ y ∈ R,

t

Vk ( f )(y) :=

|x|≥|y|

K (x, y) f (x)|x|2k d x,

where K is the kernel of the integral representation of Vk given in the formula (1). The operator t Vk is a topological automorphism of D(R). The operators Vk and t V possess the following property : For all f ∈ D(R) and g ∈ E(R), we have k 

 t

R

Vk ( f )(y)g(y)dy =

R

f (x)Vk (g)(x)|x|2k d x,

(see [7] and [28]). For k ≥ 0, and λ ∈ C, I the initial problem 

Dk u(x) = λu(x), x ∈ R, u(0) = 1,

has a unique analytic solution u(x) = E k (λ, x), called Dunkl kernel [11] and given by E k (λ, x) := jk− 1 (iλx) + 2

λx j 1 (iλx), 2k + 1 k+ 2

1 where jα is the normalized Bessel function, defined for α ≥ − by 2 jα (z) := (α + 1)

+∞  (−1)n n=0

n!

( 2z )2n , z ∈ C. I (n + α + 1)

We remark that E k (λ, x) = Vk (eλ. )(x). Formula (1) and the last result imply that | E k (λ, x) |≤ e|x||Reλ| , | E k (−i y, x) |≤ 1, for all x, y ∈ R and λ ∈ C. I

(2)

Generalized Dunkl-Lipschitz spaces

For all f and g in C 1 (R) with at least one of them is even, we have Dk ( f g) = (Dk f )g + f (Dk g). For f ∈ Cb1 (R) and g in S(R), we have 

 R

Dk f (x)g(x)|x|2k d x = −

R

f (x)Dk g(x)|x|2k d x.

Hereafter, we denote by L p (R, |x|2k d x), p ∈ [1, ∞], the space of measurable functions on R such that   f k, p :=

1 | f (x)| |x| d x p

R

2k

p

< +∞, 1 ≤ p < ∞,

and  f k,∞ := ess sup | f (x)| < +∞. x∈R

The Dunkl kernel gives rise to an integral transform, called Dunkl transform on R, which was introduced by Dunkl in [12], where already many basic properties were established. Dunkl’s results were completed and extended later on by de Jeu in [9]. The Dunkl transform of a function f ∈ L 1 (R, |x|2k d x) is given by  ∀y ∈ R, Fk ( f )(y) := ck where ck :=

1 1

2k+ 2 (k+ 21 )

R

f (x)E k (x, −i y)|x|2k d x,

(3)

.

We summarize the properties of Fk ( f ) in the following proposition: Proposition 2.1 [9] (i) For all f ∈ S(R), we have Fk (Dk f )(x) = i xFk ( f )(x), x ∈ R. (ii) Inversion formula: For all f ∈ L 1 (R, |x|2k d x) such that Fk ( f ) belongs to L 1 (R, |x|2k d x), we have  E k (x, i y)Fk ( f )(y)|y|2k dy, a.e. f (x) = R

(iii) Plancherel’s Theorem: The Dunkl transform extends to an isometry of L 2 (R, |x|2k d x). In particular, we have the following Plancherel’s formula  f k,2 = Fk ( f )k,2 ,

f ∈ L 2 (R, |x|2k d x).

S. Kallel

Definition 2.2 Let f ∈ C(R) and y ∈ R. Then T yk f (x) = u(x, y) is the unique solution of the following Cauchy problem 

Dk,x u(x, y) = Dk,y u(x, y), u(x, 0) = f (x).

T yk is called the Dunkl translation operator. Remark 2.3 In what follows we point out some remarks. • The operator Txk admits the following integral representation  T yk

π

f (x) := dk 0

 +

π

f e (G(x, y, θ ))h e (x, y, θ ) sin2k−1 θ dθ  f o (G(x, y, θ ))h (x, y, θ ) sin o

2k−1

θ dθ ,

(4)

0

where dk :=

(k + 21 ) (k)( 21 )

, G(x, y, θ ) =



x 2 + y 2 − 2|x y| cos θ ,

h e (x, y, θ ) = 1 − sgn(x y) cos θ, ⎧ e ⎨ (x + y)h (x, y, θ ) , o h (x, y, θ ) = G(x, y, θ ) ⎩ 0, f e (x) =

if (x, y) = (0, 0), otherwise,

1 1 ( f (x) + f (−x)), and f o (x) = ( f (x) − f (−x)). 2 2

• There is an abstract formula for T yk , y ∈ R, given in terms of the intertwining operator Vk and its inverse, (see [7,28]). It takes the form of

T yk f (x) := (Vk )x ⊗ (Vk ) y (Vk )−1 ( f )(x + y) , x ∈ R,

f ∈ E(R).

• The Dunkl translation operators satisfy for x, y ∈ R the following relations Txk f (y) = T yk f (x), T0k f (y) = f (y), Txk T yk = T yk Txk , Txk Dk = Dk Txk . • For each y ∈ R, the Dunkl translation operator T yk extends to a bounded operator on L p (R, |x|2k d x). More precisely T yk f k, p ≤

√

2 f k, p , 1 ≤ p ≤ ∞.

(5)

Generalized Dunkl-Lipschitz spaces

• Unusually, T yk is not a positive operator in general (see [23]) but if f is even, π then T yk f (x) = dk 0 f (G(x, y, θ ))h e (x, y, θ ) sin2k−1 θ dθ , which shows that T yk f (x) ≥ 0 whenever f is non-negative. • From the generalized Taylor formula with integral remainder (see [22], Theorem 2, p. 349), we have for f ∈ E(R) and x, y ∈ R 

Txk





f − f (y) =

|x| −|x|



 sgn(x) sgn(z) Tzk (Dk f )(y)|z|2k dz. − 2|x|2k 2|z|2k

(6)

Associated to the Dunkl translation operator T yk , the Dunkl convolution f ∗k g of two appropriate functions f and g on R defined by  f ∗k g(x) :=

R

Txk f (−y)g(y)|y|2k dy, x ∈ R.

The Dunkl convolution preserves the main properties of the classical convolution which corresponds to k = 0. For S ∈ S  (R) and f ∈ S(R), we define the Dunkl convolution product S ∗k f by S ∗k f (x) := S y , Txk f (−y).

3 The k-heat transforms of a function We recall some properties of the k-heat transforms of a measurable function f and we refer for more details to the survey [6] and the references therein. – For t > 0, let Ftk be the function defined by 1

x2

Ftk (x) := (2t)−(k+ 2 ) e− 4t

which is a solution of the Dunkl-type heat equation (Dk2 − ∂t )U = 0 on the halfplane R2+ .1 The function Ftk may be called the heat kernel associated with Dunkl operator or the k-heat kernel and it has the following basic properties : Proposition 3.1 For all t > 0 and n, m ∈ N, we have  2 (i) Fk (Ftk )(x) = e−t x and R Ftk (x)|x|2k d x = ck−1 .  n 1  2 (ii) R |Dkn Ftk (x)||x|2k d x ≤ B(k, n)t − 2 , where B(k, n) = 2k+ 2 R e−x |Pn (x)||x|2k d x with Pn is a polynomial of degree n. 2 (iii) ∂tm Ftk (x) = t −m Rm ( x4t )Ftk (x), where Rm is a polynomial of degree m with coefficients depending only on m and k.   (iv) R |∂tm Ftk (x)||x|2k d x ≤ B(k, m)t −m and R ∂tm Ftk (x)|x|2k d x = 0. 1 R2 = {(x, t) : x ∈ R, t > 0}. +

S. Kallel

Definition 3.2 The k-heat transform of a smooth measurable function f on R is given by G kt ( f )(x) := Ftk ∗k f (x), t > 0. Theorem 3.3 Let f be a measurable bounded function on R. Then, (i) (x, t) −→ G kt ( f )(x) is infinitely differentiable on R2+ and it is a solution of the Dunkl-type heat equation. Further, if n, m ∈ N, then for all t > 0 Dkn G kt ( f ) = Dkn Ftk ∗k f and ∂tn G kt ( f ) = ∂tn Ftk ∗k f.  k F k (x)G k ( f )(y) (ii) For all s, t > 0 and x ∈ R, we have G kt+s ( f )(x) = R T−y t s |y|2k dy. (iii) If f ∈ Cb (R), then G kt ( f )(x) −→ f (ξ ) as (x, t) −→ (ξ, 0). Theorem 3.4 Let p ∈ [1, ∞] and let f ∈ L p (R, |x|2k d x). Then the k-heat transform G kt ( f ) of f has the following properties: (i) For all t > 0 and m ∈ N, we have G kt ( f )k, p ≤ ck−1  f k, p and ∂tm G kt ( f )k, p ≤ B(k, m)t −m  f k, p . (ii) If 1 ≤ p < r < ∞ and δ =

1 p

− r1 , then for all t > 0

G kt ( f )k,r ≤ t −(k+ 2 )δ ckδ−2  f k, p 1

1

and G kt ( f )k,r = ◦(t −(k+ 2 )δ ),2 as t −→ 0+ . Definition 3.5 For any T ∈ S  (R), the k-heat transform of T is given by G kt (T )(x) := T ∗k Ftk (x), x ∈ R.

4 A semi-group formula for k-temperatures Hereafter we shall be concerned mostly with temperatures associated with the Dunkl setting on R which we recall the k-temperatures, satisfying a property which we call “semi-group formula”. Definition 4.1 A function U on R2+ is said to be a k-temperature if it is indefinitely differentiable on R2+ and satisfies at each point of R2+ the Dunkl-type heat equation i.e., Dk2 U(x, t) = ∂t U(x, t). 2 f (x) = ◦(g(x)), x −→ a, means f (x)/g(x) −→ 0 as x −→ a.

Generalized Dunkl-Lipschitz spaces

– We consider the following initial-value problem for the k-heat equation :  (I V P)

(Dk2 − ∂t )U = 0 U(., 0) = f

on R2+

with initial data f ∈ Cb (R). For f ∈ C0 (R), the function  Ht f (x) :=

R

k T−y Ftk (x) f (y)|y|2k dy, t > 0,

solves initial value problem (IVP) (see [24]). Lemma 4.2 Let f be in E(R), let c > 0, a > 0 and let S = R×]0, c[. Then there exists at most one k-temperature U on S which is continuous on S and satisfies the conditions that U(x, 0) = f (x), x ∈ R and  0

c

 |U(x, t)|e

R

−ax 2

 |x| d x dt < ∞. 2k

Proof Since Vk is a topological automorphism to the space E(R), then from Theorem 16 of Friedman [14] (see also Lemma 5 of Flett [13]), there exists at most one classical temperature U˜ on S which is continuous on S and satisfies the conditions that ˜ U(x, 0) = Vk−1 ( f )(x), x ∈ R and



c

 R

0

 −ax 2 ˜ |U(x, t)|e d x dt < ∞.

˜ t))(x) is a k-temperature on S which is continuous Thus, (x, t) −→ U(x, t) = Vk (U(., on S and U(x, 0) = f (x), x ∈ R. From the formula (1) we deduce that for x = 0 ˜ t))(x) = B(k)|x|−2k sgn(x) Vk (U(.,



|x| −|x|

˜ U(y, t)(x 2 − y 2 )k−1 (x + y)dy.

(7)

Then according to formula (7) and Fubini-Tonelli’s theorem, we obtain  0

c



   c  2 ˜ t)|)(x)e−ax 2 |x|2k d x dt |U(x, t)|e−ax |x|2k d x dt ≤ Vk (|U(., 0 R R   c  2 ˜ |U(y, t)| t Vk (e−a|.| )(y)dy dt. = 0

R

Hence by change of variables ξ = x 2 and formula (11) given in [5] p. 202, we deduce that t

Vk (e−a|.| )(y) = 2

which establishes the lemma.

2−2k (2k + 1) −k −ay 2 , a e (k + 1)  

S. Kallel

Remark 4.3 For f ∈ Cb (R), Rösler [24] proved that U f (x, t) = Ht ( f )(x) is bounded continuous on R × [0, +∞[ and U f is the unique solution of (IVP) on S within the class of functions U ∈ C 2 (S) ∩ C(S) which satisfy the following exponential growth condition : There exist positive constant C, λ, r such that |U(x, t)| ≤ Ceλx

2

for all (x, t) ∈ S with |x| > r.

Theorem 4.4 Let p ∈ [1, ∞] and let U be a k-temperature on R2+ such that the function t −→ U(., t)k, p is locally integrable on ]0, ∞[. Hence (i) for all s > 0 and (x, t) ∈ R2+ ,  U(x, s + t) =

k T−y Ftk (x)U(y, s)|y|2k dy.

R

(8)

(ii) t −→ U(., t)k, p is decreasing and continuous on ]0, ∞[. Further, for each (n, m) ∈ N × N the function t −→ Dkn ∂tm U(., t)k, p is decreasing and continuous on ]0, ∞[. Proof It is obtained in the same way as for Theorem 4 of Flett [13] by using Lemma 4.2.   Remark 4.5 The Eq. (8) is called the “semi-group formula” hereafter.

5 Dunkl-Bessel potentials The aim of this section is to define the Bessel potential of some classes of k-temperature associated with the Dunkl setting on R and to prove related properties needed later. We adopt the method used by Flett [13] and Johnson [15] in treating classical temperatures. Definition 5.1 For any f ∈ L p (R, |x|2k d x), where 1 ≤ p ≤ ∞ and for any α > 0, the Dunkl-Bessel potential Jαk f of order α of f is given by Jαk f := Bαk ∗k f, with the kernel function Bαk (x)

:= =



1 ( α2 )

2

k+ 21

2

α 2 −1

1

+∞

x2

e−t e− 4t t −k+

|x| 2 (α−1)−k K α − 1 −k (|x|). 2

2

Here K β (z) :=

π 2

dt

0 1

( α2 )

(α−1) 2 −1



 J−β (z) − Jβ (z) , sin βπ

(9)

Generalized Dunkl-Lipschitz spaces

where Jβ is the modified Bessel function of the first kind with series expansion Jβ (z) :=

+∞  n=0

( 21 z)β+2n . n!(β + n + 1)

The Bessel potentials associated with the Dunkl setting on R which we recall the kBessel potentials are bounded operators from L p (R, |x|2k d x) to itself for 1 ≤ p ≤ ∞ (see [27]), i.e., if f ∈ L p (R, |x|2k d x) and α > 0, then Jαk f ∈ L p (R, |x|2k d x) and Jαk f k, p ≤  f k, p . Further, for α, β > 0 k f. Jαk (Jβk f ) = Jα+β

By using the well-known asymptotic behavior of the function K ν , ν ∈ R (see [4], page 415), we deduce that (a) Bαk (x) ∼ (b)

k (x) B1+2k

(c) Bαk (x) ∼ (d) Bαk (x) ∼

( 1−α 2 + k) 2

α− 21 −k

( α2 )

|x|α−1−2k ,3 as |x| −→ 0, for 0 < α < 2k + 1. 

1

1 ∼ log 1 k− 21 |x| 2 (k + 2 )

( α−1 2 −k) 1

2 2 +k ( α2 )

√ π

α−1 2

 as |x| −→ 0.

as |x| −→ 0, for α > 2k + 1. α

( α2 )

|x| 2 −1−k e−|x| as |x| −→ ∞, for α > 0.

2 Using the formula (9) and the following identity t

−a

1 = (a)



+∞

e−tδ δ a

0

dδ , with a > 0, δ

(10)

we obtain 1

Bαk (x)

2k+ 2 −α (k + ≤ ( α2 )

1−α 2 )

|x|α−1−2k , if 0 < α < 1 + 2k.

(11)

By differentiation under the integration sign of formula (9) and relation (10) again, we show that 3

|Dk Bαk (x)| <

2k+ 2 −α (k + ( α2 )

3−α 2 )

|x|α−2−2k , if 0 < α < 2k + 3.

f (x) 3 As usual, we write f (x) ∼ g(x) as x −→ a if lim x−→a g(x) = 1.

(12)

S. Kallel

Added to this, we can see that the kernel Bαk , α > 0, satisfies (i) (ii) (iii) (iv)

Bαk (x) ≥ 0, for all x ∈ R. Bαk k,1 = 1. α Fk (Bαk )(x) = (1 + x 2 )− 2 , x ∈ R. Bαk 1 +α2 = Bαk 1 ∗k Bαk 2 , if α1 , α2 > 0.

The next theorem is the basis of our definition of the Dunkl-Bessel potential for k-temperatures. Theorem 5.2 [6] Let α > 0, 1 ≤ p ≤ ∞ and let f ∈ L p (R, |x|2k d x), then (i) The k-Bessel potential Jαk f of order α of f is given for almost all x by Jαk f (x) =

1 ( α2 )



+∞ 0

α

t 2 −1 e−t G kt ( f )(x)dt,

(13)

where G kt ( f ), t > 0, is the k-heat transform of f on R. (ii) The k-heat transform of Jαk f , α > 0, on R is the function G ks (Jαk f ) given by G ks (Jαk

1 f )(x) = ( α2 )



+∞ 0

α

t 2 −1 e−t G ks+t ( f )(x)dt.

(14)

Moreover, for each s > 0, the function x → G ks (Jαk f )(x) is the k-Bessel potential of x → G ks ( f )(x). Definition 5.3 Let k (R2+ ) denote the linear space of k-temperatures U on R2+ with the properties that if (n, m) ∈ N × N, b > 0, c > 0, and S is a compact subset of R, then there is a positive constant C such that |Dkn ∂tm U(x, t)| ≤ Ct −b et , for all (x, t) ∈ S × [c, ∞[. Definition 5.4 For any U in k (R2+ ) and any real number α, Jαk U is the function defined on R2+ by (i) J0k (U) = U; (ii) if α > 0, Jαk (U)(x, s) =

1 ( α2 )



+∞

α

t 2 −1 e−t U(x, s + t)dt;

0

(iii) if α is a negative even integer, say α = −2m, then k Jαk (U)(x, s) = J−2m (U)(x, s) = (−1)m es ∂sm {e−s U(x, s)};

Generalized Dunkl-Lipschitz spaces

(iv) if α = −β < 0 and β is not an even integer, then   k k k J−2m (U) = J2m−β (U) ; Jαk (U) = J−β k k and J−2m are defined as in (ii) and where m = [ 21 β] + 1, 4 and where J2m−β (iii).

Theorem 5.5 [6] Let U ∈ k (R2+ ) and α, β be real numbers. and Jαk (U) ∈ k (R2+ ), (i) Jαk (U)  is well-defined    k (U) = J k J k (U) . (ii) Jαk Jβk (U) = Jα+β α β Corollary 5.6 For each real number α, Jαk is a linear isomorphism of k (R2+ ) onto k . itself, with inverse J−α Theorem 5.7 Let f be in L p (R, |x|2k d x), 1 ≤ p ≤ ∞, α > 0, and let G kt ( f ) be the k-heat transform of f on R2+ . Then for t > 0 (i) Jαk G kt ( f )k, p ≤ ck−1  f k, p ;

1

− α k G k ( f ) (ii) J−α k, p ≤ B(k, α)(t 2 + 1) f k, p ; t (iii) furthermore, if 1 ≤ p < ∞ then 1

k J−α G kt ( f )k, p = ◦(t − 2 α ), as t −→ 0+ .

Proof Part (i) follows from relation (14), Minkowski’s integral inequality and Theorem 3.4(i). According to the fact that k G kt ( f ) = J−2m

m  m (−1)i ( )∂ti G kt ( f ), m ∈ N, i i=0

Minkowski’s inequality, Theorem 3.4(i) and the following inequality (a + b)s ≤ 2s−1 (a s + bs ), s ∈ [1, +∞[, a, b ≥ 0,

(15)

yield the part (ii) when α = 2m. Supposing that α is not an even integer and let m = [ 21 α] + 1. Then for (x, s) ∈ R2+ k J−α G ks ( f )(x)

=



1 (m −

1 2 α)

+∞ 0

1

k t m− 2 α−1 e−t J−2m G ks+t ( f )(x)dt.

Hence, Minkowski’s integral inequality and the previous case when α = 2m yield −1α k G k ( f ) that J−α k, p ≤ B(k, α)(s 2 + 1) f k, p . We shall prove (iii) only when s 4 Here [x] stands for the greatest integer not exceeding x, x ∈ R.

S. Kallel

α = 2m, because the general case can be treated in the same manner. Let (x, t) be in R2+ . Thus by Proposition 3.1(iv) k J−2m G kt ( f )(x)

 m k = (−1) ( ) ∂ti Ftk (y)(T−y f (x) − f (x))|y|2k dy i R i=0 m 

i

which together with Minkowski’s integral inequality imply that k t m J−2m G kt ( f )k, p ≤ t m

 m  m k ( ) |∂ti Ftk (y)|T−y f − f k, p |y|2k dy i |y|<δ i=0

 m  m m k +t ( ) |∂ti Ftk (y)|T−y f − f k, p |y|2k dy = I1 (t) + I2 (t) (δ > 0). i |y|≥δ i=0

k f − f Since lim y−→0 T−y k, p = 0, for an arbitrary positive number , there exists a k δ > 0 such that T−y f − f k, p <  if |y| < δ. Therefore, from Proposition 3.1(iv) and inequality (15), we obtain I1 (t) ≤ B(k, m)(1 + t m ). By relation (5), Proposition 3.1(iii) and the change of variables, we have

I2 (t) ≤ B(k) f k, p

 +∞ m  m ( )t m−i 2 |Ri (σ )|e−σ σ k−1/2 dσ. i δ i=0

4t

Letting t → 0+ , the last integral approaches to 0. This proves the part (iii).

 

Corollary 5.8 Let α > 0, 1 ≤ p ≤ ∞, and U be in k (R2+ ). If U satisfies the semi-group formula, then for all s, t > 0 (i) Jαk U(., s + t)k, p ≤ U(., s)k, p . −1α k U(., s + t) (ii) J−α k, p ≤ B(k, α)(t 2 + 1)U(., s)k, p . Proof Let s be fixed. We may assume that U(., s)k, p is finite (otherwise the conclusion would be trivial). Then for all t > 0, by the semi-group formula for U yields  U(x, s + t) =

R

k T−y Ftk (x)U(y, s)|y|2k dy

which implies the corollary by analogous reasoning of Theorem 5.7.

 

Theorem 5.9 Let 1 ≤ p ≤ ∞, 1 ≤ q < ∞, β be a positive number and U be a k-temperature on R2+ such that 

+∞

C= 0

1

t 2 qβ−1 e−t U(., t)k, p dt q

 q1

< ∞.

Generalized Dunkl-Lipschitz spaces 1

1

1

−2β q Thus for t > 0, U(., t)k, p ≤ ( eqβ )C and U(., t)k, p = ◦(t − 2 β ) as 2 ) (1 + t + t −→ 0 . Moreover, if q < r < ∞, then



+∞

t

1 2 rβ−1

e

−t

0

U(., t)rk, p dt

 r1

≤ B(q, r, β)C.

Proof The proof is similar to the classical case (see Theorem 11, p. 405 in [13]).   Theorem 5.10 Let 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞, α be a real number, β > 0, β > α and U be a k-temperature on R2+ such that ⎧  ⎪ ⎪ ⎨

 q1 1 q t 2 qβ−1 e−t U(., t)k, p dt = C1 < ∞, (1 ≤ q < ∞), C := 0  1 ⎪ ⎪ ⎩ sup t 2 β e−t U(., t)k, p = C2 < ∞, (q = ∞). +∞

t>0

(i) U ∈ k (R2+ ) and ⎧  ⎪ ⎪ ⎨

 q1 1 q t 2 q(β−α)−1 e−t Jαk U(., t)k, p dt ≤ B(k, α, β, q)C1 , (1 ≤ q < ∞), 0  1 ⎪ ⎪ ⎩ sup t 2 (β−α) e−t Jαk U(., t)k, p ≤ B(k, α, β)C2 , (q = ∞). +∞

t>0

1

(ii) If 1 ≤ q < ∞, then Jαk U(., t)k, p = ◦(t − 2 (β−α) ) as t −→ 0+ . 1 (iii) If q = ∞ and U(., t)k, p = ◦(t − 2 β ) as t −→ 0+ , then Jαk U(., t)k, p = 1 ◦(t − 2 (β−α) ) as t −→ 0+ . Proof Clearly t −→ U(., t)k, p is locally integrable on ]0, ∞[, so that U ∈ k (R2+ ) and U(., t)k, p is decreasing. Therefore Jαk U is well defined. First, suppose that γ = −α > 0. Then by Corollary 5.8 we see that 1

Jαk U(., 2t)k, p ≤ B(k, α)(t 2 α + 1)U(., t)k, p

(16)

which implies that 

+∞

t 0

1 2 q(β−α)−1

e

−t

q Jαk U(., t)k, p dt

 q1

≤ B(k, α, β, q)C1 .

Next, we shall prove the result for the special case when α = 2 and β > 2. Since  J2k U(x, t)

+∞

= 0

e−ξ U(x, t + ξ )dξ,

(17)

S. Kallel

it follows from Minkowski’s integral inequality and Hardy’s inequality that 

+∞

t

1 2 q(β−2)−1

0

e



−qt

+∞

≤ B(k, β, q) 0

q J2k U(., t)k, p dt 1

 q1

t 2 qβ−1 e−qt U(., t)k, p dt q

1/q .

To prove the result for α = δ > 0, let γ be the least positive number such that γ + δ is an even positive integer. Then by applying part (i) in case α < 0, we have 

+∞

t

1 2 q(β+γ )−1

e

−t

0

q k J−γ U(., t)k, p dt

 q1

≤ B(k, γ , β, q)C1

and hence after repeated applications of part (i) in case α = 2, we obtain 

+∞

t

1 2 q(β−δ)−1

0

e

−t

q Jδk U(., t)k, p dt

 q1

≤ B(k, α, β, q)C1 .

It is easy to see  1  sup t 2 (β−α) e−t Jαk U(., t)k, p ≤ B(k, α, β)C2 t>0

from Corollary 5.8. The assertion (ii) then follows from part (i) and Theorem 5.9. Now, we shall prove the assertion (iii). First, assuming that α < 0, the result follows easily from the estimate (16). Next, we shall prove the result for the case when α = 2 and β > 2. It follows from relation (17) and Minkowski’s integral inequality that s

β 2 −1

J2k U(., s)k, p

≤s

β 2 −1

 e

s

+∞

e−t U(., t)k, p dt,

s

consequently the assertion is proved for the special case. In case α = δ > 0 and by choosing γ > 0, γ + δ is an even positive integer. Applying the above result for − 1 (β+γ ) k U(., t) ). Repeated use of the result for α < 0 we see that J−γ k, p = ◦(t 2 1

1

− (β+γ )+ 2 (γ +δ) k U(., t)) ) = α = 2 yields Jδk U(., t)k, p = Jγk+δ (J−γ k, p = ◦(t 2 1

◦(t − 2 (β−δ) ). Thus part (iii) is proved.

 

Definition 5.11 For any real number α and for any T ∈ S  (R), the k-Bessel potential of order α of T is the element Jαk (T ) of S  (R) given by the relation α

Fk (Jαk (T )) := (1 + (.)2 )− 2 Fk (T ), where the identity is to be understood in the sense of distributions.

Generalized Dunkl-Lipschitz spaces

Remark 5.12 We have • For all real α, β and all T ∈ S  (R) k Jαk (Jβk (T )) = Jα+β (T ).

• By definition Jαk (T ) = T ∗k Bαk , where Bαk is a tempered distribution whose Dunkl transform Fk (Bαk ) = α (1 + (.)2 )− 2 .5 • If f ∈ L p (R, |x|2k d x), where p ∈ [1, ∞] and α > 0, then Jαk ([ f ]) = Jαk ( f ) = f ∗k Bαk .

6 Generalized Dunkl-Lipschitz spaces, α real Our basic aim is to define Lipschitz spaces associated with the Dunkl operators for all real α. In the classical case, the heat (or Poisson) semi-group provides an alternative characterization of the Lipschitz spaces, we will follow this approach, using the kheat (or k-Poisson) semi-group, to define generalized Dunkl-Lipschitz spaces. One of the main result of this part is to show that Jβk is an isomorphism of ∧kα, p,q (R) onto ∧kα+β, p,q (R) for real α and β. The section closes by giving some properties and continuous embedding for the space ∧kα, p,q (R). We define for t > 0, the function Ptk on R by 1

Ptk (x) := c˜k

t 2k+ 2 (k + 1). , where c ˜ := k (t 2 + x 2 )k+1 ( 21 )

The function Ptk is called the k-Poisson kernel. We summarize the properties of Ptk in the following proportion : Proposition 6.1 For all t > 0, n ∈ N and x ∈ R, we have (i) (ii) (iii) (iv) (v)

k e−t|x| . Fk (Pkt )(x) = 2k dy = 1. P (y)|y| R t Ptk ∈ L p (R, |x|2k d x), 1 ≤ p ≤ ∞. Ptk1 +t2 = Ptk1 ∗k Ptk2 , if t1 , t2 > 0. ˜ ∂tn Ptk k,1 ≤ B(k, n)t −n , Dkn Ptk k,1 ≤ B(k, n)t −n and |∂tn Ptk (x)| ≤ B(k, n) t −2k−1−n .

5 [ f ] is the distribution on R associated with the function f . In addition [ f ] belongs to S  (R), when f ∈ L p (R, |x|2k d x) or f is slowly increasing.

S. Kallel

(vi) limt→0 Ptk f (x) = f (x), where the limit is interpreted in L kp -norm and pointwise a.e. For f ∈ C0 (R) the convergence is uniform on R. However, for t > 0 and for all f ∈ L p (R, |x|2k d x), p ∈ [1, ∞], we put Ptk f (x) := Ptk ∗k f (x), x ∈ R. The function Ptk f is called the Poisson transform of a function f associated with the Dunkl setting on R that’s why we may recall it the k-Poisson transform of f . A C 2 function U on R2+ satisfying (Dk2 + ∂t2 )U(x, t) = 0 is said to be k-harmonic. For p ∈ [1, ∞], we suppose that  A p := sup ck t>0

R

|U(x, t)| p |x|2k d x < ∞,

(18)

where the constant ck is defined in relation (3). Now, we need the following key results. Lemma 6.2 (Semi-group property) If U(x, t) is k-harmonic on R2+ and bounded in each proper sub-half space of R2+ , then for t0 > 0, U(x, t + t0 ) is identical with the k-Poisson transform of U(., t0 ), that is, U(x, t0 + t) = Ptk (U(., t0 ))(x), for t > 0. Furthermore, ∂t U(x, t0 + t) = ∂t Ptk (U(., t0 ))(x) = Ptk (∂t U(., t0 ))(x). Proof It is obtained in the same way as for Property 12, p. 417 in [25].

 

Theorem 6.3 (Characterization of k-Poisson transform) Let p ∈ [1, ∞] and let U(x, t) be k-harmonic on R2+ . Then (i) if 1 < p < ∞, U(x, t) is the k-Poisson transform of a function f ∈ L p (R, |x|2k d x) if and only if U(x, t) satisfies condition (18), moreover  f k, p = A. (ii) For p = 1, U(x, t) is the k-Poisson transform of f ∈ L 1 (R, |x|2k d x) if and only if U(x, t) satisfies condition (18) and U(., t1 ) − U(., t2 )k,1 , as t1 , t2 → 0. (iii) For p = ∞, U(x, t) is the k-Poisson transform of a function f ∈ L ∞ (R, |x|2k d x) if and only if there exists C > 0 such that U(., t)k,∞ ≤ C for all t > 0. Proof Parts (i) and (ii) are proved in [20] Theorem 4.16, p. 254. Part (iii) is proved in usual way (see [25], p. 416).   Remark 6.4 Analogously to the k-harmonic case, we can assert that Theorem 6.3 and Lemma 6.2 are true when we take U(x, t) k-temperature on R2+ and we replace k-Poisson transform by k-heat transform.

Generalized Dunkl-Lipschitz spaces

Before giving a central result of this section, we need to recall the definition of the spaces ∧kα, p,q (R) (see [16]) and the following auxiliary lemmas. Definition 6.5 The generalized Dunkl-Lipschitz spaces ∧kα, p,q (R), α ∈]0, 1[, 1 ≤ p, q ≤ ∞, is the set of functions f ∈ L p (R, |x|2k d x) for which the norm 6  f k, p +

 q  y,k f k, p R

|y|1+αq

1 q

dy

< ∞, if q < ∞

and  y,k f k, p < ∞, if q = ∞. |y|α |y|>0

 f k, p + sup Notations

• For any k-harmonic (or k-temperature) U on R2+ , we denote by ⎧  ⎪ ⎪ ⎨

∞

q dt U(., t)k, p k t A p,q (U) := 0 ⎪ ⎪ ⎩ sup U(., t)k, p

1 q

(1 ≤ q < ∞), (q = ∞),

t>0

and ⎧  ⎪ ⎪ ⎨

1

q dt U(., t)k, p k,∗ t A p,q (U) := 0 ⎪ sup ⎪ U(., t)k, p ⎩

 q1

(1 ≤ q < ∞), (q = ∞),

0
the value ∞ being allowed. • For α real, α will denote the smallest non-negative integer larger than α. Remark 6.6 ([16]) We have: • For α ∈]0, 1[ and q = ∞, f ∈ ∧kα, p,∞ (R) if and only if ∂t Ptk f k, p ≤ B(k, α)t −1+α . • For α > 0, p, q ∈ [1, ∞], we set ∧kα, p,q (R) :=



 f ∈ L p (R, |x|2k d x) : Akp,q (t α−α ∂tα Ptk ( f )) < ∞ .

The ∧kα, p,q -norms are defined by  f ∧kα, p,q :=  f k, p + Akp,q (t α−α ∂tα Ptk ( f )). 6  k y,k f = T y f − f .

S. Kallel

Lemma 6.7 We have Bαk ∈ ∧kα,1,∞ (R), if α > 0. Proof Let us first consider the case α ∈]0, 1[. Since Bαk ∈ L 1 (R, |x|2k d x), we can write   y,k Bαk k,1

=

|x|≤2|y|



+

|T yk Bαk (x) − Bαk (x)||x|2k d x

|x|>2|y|

|T yk Bαk (x) − Bαk (x)||x|2k d x

= I1 (y) + I2 (y). Bαk is an even function, then formula (4) yields  T yk Bαk (x) = dk

π

0

Bαk (G(x, y, θ ))h e (x, y, θ ) sin2k−1 θ dθ

which shows that T yk Bαk (x) ≥ 0 since Bαk is non-negative. Moreover, using the following inequalities G(x, y, θ ) ≥ ||x| − |y||, 0 ≤ h e (x, y, θ ) ≤ 2 and relation (9), we have (19) T yk Bαk (x) ≤ 2Bαk (|x| − |y|). Then, by inequalities (19) and (11), we have  I1 (y) ≤ B(k, α)

|x|≤2|y|

||x| − |y|||α−1−2k |x|2k d x +

≤ B(k, α)|y|α .

 |x|≤2|y|

|x|α−1 d x



By the generalized Taylor formula with integral remainder (6), we have  |T yk Bαk (x) − Bαk (x)|

≤

|y| −|y|

|Tzk (Dk Bαk )(x)|dz.

(20)

Since Dk Bαk is an odd function, formula (4) gives  Tzk (Dk Bαk )(x) = dk

0

π

Dk Bαk (G(x, z, θ ))h o (x, z, θ ) sin2k−1 θ dθ.

It is obvious to see that h o (x, z, θ ) ≤ 2 and 0 ≤ G(x, z, θ ) ≤ |x| + |z|. Thus, formula (12) yields |Tzk (Dk Bαk )(x)| ≤ B(k, α)(|x| + |z|)α−2−2k .

Generalized Dunkl-Lipschitz spaces

Hence, by relation (20) we obtain |T yk Bαk (x) − Bαk (x)| ≤ B(k, α)|y||x|α−2−2k and so I2 (y) ≤ B(k, α)|y|α . This completes the proof when α ∈]0, 1[. To pass to the general case for α > 0, we write t = t1 + t2 + · · · + tα and ti > 0. Then Ptk Bαk = Ptk1 Bβk ∗k Ptk2 Bβk ∗k · · · ∗k Ptkα Bβk , where β = αα ∈]0, 1[. Therefore ∂tα Ptk Bαk k,1 ≤ B(k, α)t α−α , whenever t1 = t2 =   · · · = tα = αt . This finishes the proof. Lemma 6.8 Let 1 ≤ p, q ≤ ∞, U(x, t) is k-harmonic on R2+ and bounded in each proper sub-half space of R2+ . Suppose we are given A > 0, α > 0, t0 > 0 and an integer n > α such that Akp,q (t n−α ∂tn U) ≤ A, U(., t)k, p ≤ A, t ≥ t0 . Then U(x, t) is the k-Poisson transform of a function f ∈ ∧kα, p,q (R) and : (a) ∂t U(., t)k, p = o(t −1 ), as t −→ 0, (b)  f ∧kα, p,q ≤ B(α, k, t0 , n)A. Proof Consider first the case α ∈]0, 1[. We are given U(., t) = O(1),7 as t −→ ∞, then from Lemma 6.2, Hölder’s inequality and Proposition 6.1(v), we get ∂tm−1 U(., t) = ◦(1), t −→ ∞, m ∈ N. Using the fact that  ∂tm−1 U(x, t) = −

∞ t

∂sm U(x, s)ds, m ∈ N,

and Minkowski’s integral inequality, we obtain  ∂tm−1 U(., t)k, p

+∞

≤ t

∂sm U(., s)k, p ds.

(21)

From Hardy inequality and relation (21), we deduce that Akp,q (t 1−α ∂t U) ≤ B(n, α)A. But t −→ ∂t U(., t)k, p is a non-increasing function, so that − q1 1−α

((1 − α)q)

s

 ∂s U(., s)k, p =

s

(t

1−α

∂s U(., s)k, p )

0

≤ B(n, α)A, if, α < 1, 7 f (x) = O(g(x)), x → a, means f (x) is bounded as x → a. g(x)

q dt

t

1 q

S. Kallel

which proves t∂t U(., t)k, p ≤ B(n, α, q)At α = ◦(1), as t −→ 0.

(22)

If α ≥ 1, it is easily to verify that 1

Akp,q (t n− 2 ∂tn U) ≤ B(n, k, q, t0 )A.

(23)

Then by relation (21), Hardy inequality and relation (23), we have 1

Akp,q (t 2 ∂t U) ≤ B(n, k, q, t0 )A. By the same reason for α ∈]0, 1[, we obtain t∂t U(., t)k, p = ◦(1) as t −→ 0+ which proves the part (a). To complete the proof it suffices to find a function p f ∈ L p (R, |x|2k d x) so that U(., t) = Ptk ( f ) converges in the L k -norm to f and U(., t)k, p ≤ B(α, k, t0 , n)A. Using inequality (22), we deduce that for t ≤ t0 

t0

U(., t)k, p ≤ U(., t0 )k, p +

∂s U(., s)k, p ds ≤ B(n, α, q, t0 )A.

t

On the other hand, by relation (22), we have  U(., t1 ) − U(., t2 )k,1 ≤

t2

∂s U(., s)k,1 ds

t1



≤ B(n, α, q, t0 )A

t2

s α−1 ds −→ 0, as t1 ≤ t2 −→ 0.

t1 p

According to Theorem 6.3, there exists f ∈ L k (it is uniformly continuous if p = ∞)   such that U(x, t) = Ptk f . This achieves the proof of the Lemma 6.8. Remark 6.9 We have • By proceeding in same manner as before, we can assert that the Lemma 6.8 is true when we take U(x, t) k-temperature on R2+ and we replace k-Poisson transform by k-heat transform. k ) as follows • If β > 0, we define Ptk (B−β k k k Ptk (B−β )(x) = Ptk (B2−β )(x) + ∂t2 Ptk (B2−β )(x), when 0 < β < 2,

and for arbitrary β > 0 by the rule k k k Ptk (B−β )(x) = P kt (B−γ ) ∗k P kt (B−δ )(x), whenever γ + δ = β. 2

2

(24)

Generalized Dunkl-Lipschitz spaces k f (x) for a function f ∈ • If β > 0, we define the k-Bessel potential J−β L p (R, |x|2k d x), 1 ≤ p ≤ ∞, by k k J−β f (x) = lim Ptk (B−β ) ∗k f (x), t−→0

p

where the limit is interpreted in L k -norm and pointwise a.e. Remark 6.10 For f ∈ L p (R, |x|2k d x), 1 ≤ p ≤ ∞ and β > 0, the k-Poisson k f , P k (J k ( f )), is k-harmonic on R2 and P k (J k ( f )) transform of J−β k, p ≤ t + t −β −β k J−β f k, p , for all t > t0 , with t0 > 0. We will study the action of the k-Bessel potential Jβk on the generalized DunklLipschitz spaces, ∧kα, p,q (R). Theorem 6.11 Let α > 0, β > 0 and 1 ≤ p, q ≤ ∞. Then Jβk is a topological isomorphism from ∧kα, p,q (R) onto ∧kα+β, p,q (R). Proof If f ∈ ∧kα, p,q (R), by Lemma 6.7, we have Jβk ( f )∧k

α+β, p,q

≤ B(k, β) f ∧kα, p,q

which implies the continuity of Jβk from ∧kα, p,q (R) into ∧kα+β, p,q (R). If f ∈ ∧kα+β, p,q (R), we may assume without loss of generality that β ∈]0, 2[. Applying the formula (24) and Lemma 6.7, we obtain k Ptk (B−β )k,1 ≤ 1 + B(k, β)t −β ≤ B(k, β), t ≥ 1.

(25)

Therefore, k J−β (Ptk ( f ))k, p ≤ B(k, β) f ∧k

α+β, p,q

, t ≥ 1.

From formula (25) and Proposition 6.1(v), a direct verification yields that α+β

Akp,q (t α+β−α ∂t

k Ptk (J−β ( f ))) ≤ B(k, α, β) f ∧k

α+β, p,q

.

On the other hand, by Remark 6.10 and Lemma 6.8, there exists a function g ∈ k ( f )) = P k (g). Consequently, we get ∧kα, p,q (R) satisfying Ptk (J−β t k k J−β ( f ) = g with g ∈ ∧kα, p,q (R) and J−β ( f )∧kα, p,q ≤ B(k, α, β) f ∧k

α+β, p,q

k from ∧k k which proves the continuity of J−β α+β, p,q (R) into ∧α, p,q (R). We now come k (J k ( f ))(x) = f (x) a.e., if f ∈ ∧k to show J−β α, p,q (R), α > 0, which follows from β the fact that

S. Kallel k (J k ( f )))(x) = P k ( f )(x) and similarly, J k (J k ( f ))(x) = f (x) a.e., if Ptk (J−β t β β −β f ∈ ∧kα+β, p,q (R), α > 0. This concludes the proof of the theorem.  

Before giving a formal definition of the generalized Dunkl-Lipschitz spaces, we p introduce the definition of the space Lα,k (R). Definition 6.12 The Lebesgue space   p Lα,k (R) := T ∈ S  (R) : T = Jαk (g), g ∈ L p (R, |x|2k d x) , for α real, 1 ≤ p ≤ ∞, is called the Dunkl-Sobolev space of fractional order α. Define T k, p,α := gk, p . p

Thus Lα,k (R) is a Banach space that is an isometric image of L p (R, |x|2k d x). Now, following the classical case, see for instance [13,25], we are going to define the generalized Dunkl-Lipschitz spaces ∧kα, p,q (R), for all real α. α Definition 6.13 Let p, q ∈ [1, ∞], α ∈ R and n = ( ). 2 (i) If α > 0, ∧kα, p,q (R) is the space of functions of f ∈ L p (R, |x|2k d x) for which the k-heat transform G kt ( f ) of f satisfies the condition that α

Akp,q (t n− 2 ∂tn G kt ( f )) < ∞. The space is given the norm α

 f ∧kα, p,q :=  f k, p + Akp,q (t n− 2 ∂tn G kt ( f )). p (R) α− 12 ,k

(ii) If α ≤ 0, ∧kα, p,q (R) is the space of tempered distributions T ∈ L which the k-heat transform G kt (T ) of T satisfies the condition that

for

α

n− 2 n k ∂t G t (T )) < ∞. Ak,∗ p,q (t

The space is given the norm α

n− 2 n k T ∧kα, p,q := T k, p,α− 1 + Ak,∗ ∂t G t (T )). p,q (t 2

p

Lemma 6.14 Let α < 0, 1 ≤ p ≤ ∞, T ∈ Lα,k (R) and let G kt (T ) be the k-heat transform of T on R2+ . Then G kt (T ) ∈ k (R2+ ) and 1

G kt (T )k, p ≤ B(k, α)(t 2 α + 1)T k, p,α .

Generalized Dunkl-Lipschitz spaces

 

Proof From Theorem 3.12 of [6] and Theorem 5.7, the result is proved.

Now, we want to extend the Theorem 6.11 for all real α and β. For this, we need the following auxiliary lemmas. Lemma 6.15 Let H (x, t) be absolutely continuous as a function t for (x, t) ∈ R2+ , t ≤ 1. Then for α > 0, p, q ∈ [1, ∞],

α k,∗ α+1 ∂t H ) + H (., 1)k, p . Ak,∗ p,q (t H ) ≤ B(α, q) A p,q (t Proof We shall prove the Lemma only when q ∈ [1, ∞[, the case q = ∞ can be similarly treated. We can write 

1

H (x, t) = H (x, 1) −

∂s H (x, s)ds.

t

From Minkowski’s integral inequality, we obtain α Ak,∗ p,q (t H )



1

≤ B(α, q)H (., 1)k, p +

t

α

0



1

q ∂s H (., s)k, p ds

t

The result announced arises from Hardy inequality.

dt t

1 q

.  

Remark 6.16 Observe that, for α > 0, the tempered distribution Bαk is a function in L 1 (R, |x|2k d x). For α = 0 it is the Dirac delta δ0 and for −α ∈]0, 2[ k k G kt (Bαk )(x) = G kt (Bα+2 )(x) − ∂t G kt (Bα+2 )(x)

which is easily verified by taking the Dunkl transform Fk . Similarly, we may construct G kt (Bαk ) for all α < 0 and find in particular that for each t > 0, G kt (Bαk ) ∈ L 1 (R, |x|2k d x) and is uniformly bounded in L 1 (R, |x|2k d x) in each proper sub-half space of R2+ . Lemma 6.17 Let α be real number, T ∈ L 1 (R) and n ∈ N, n ≥ ( α2 ). Then the α− 2 ,k norm p

α

n− 2 n k T k, p,α− 1 + Ak,∗ ∂t G t (T )) p,q (t 2

is equivalent to the norm with n = ( α2 ). p (R), α− 12 ,k

Proof If T ∈ L

from Proposition 3.1(iv), we have

α ∂tn G k1 (T )k, p ≤ B(k, n, α)T k, p,α− 1 , n > l = ( ). 2 2

S. Kallel

Therefore by Lemma 6.15, we obtain  α    l− 2 l k n− α2 n k ∂t G t (T ) ≤ B(k, α, n) Ak,∗ ∂t G t (T )) + T k, p,α− 1 . Ak,∗ p,q t p,q (t 2

Conversely, a direct check shows that   β+1 β k Ak,∗ ∂t G kt (T ) ≤ B(k, β)Ak,∗ p,q t p,q (t G t (T )), β > 0. Thus α

α

n− 2 n k l− 2 l k Ak,∗ ∂t G t (T )) ≤ B(k, α, n)Ak,∗ ∂t G t (T )), where n > l = p,q (t p,q (t

α  , 2  

which proves the results.

Lemma 6.18 Let α be real, n = ( α2 ) and 1 ≤ p, q ≤ ∞. Then the set of tempered p distributions T ∈ L 1 (R) for which α− 2 ,k

α

n− 2 n k Ak,∗ ∂t G t (T )) < ∞, p,q (t

normed with

α

n− 2 n k Ak,∗ ∂t G t (T )) + T k, p,α− 1 p,q (t

(26)

2

is topologically and algebraically equal to ∧kα, p,q (R). Proof By definition of ∧kα, p,q (R), one only needs to consider the case α > 0. Assume p that T ∈ L 1 (R) and (26) is finite. It is easily seen that α− 2 ,k

  α n− α2 n k Akp,q (t n− 2 ∂tn G kt (T )) ≤ B(k, α, q) Ak,∗ (t ∂ G (T )) + T  1 p,q t t k, p,α− 2 , α > 0. (27) If α ≥ 21 , thus T ∈ L p (R, |x|2k d x) is obvious. On the other hand, if 0 < α < 21 , then for t ≥ 1, G kt (T )k, p ≤ B(k, α)T k, p,α− 1 . By the relation (27) and Lemma 6.8, 2

there exists a function ψ ∈ ∧kα, p,q (R) such that G kt (T ) = G kt (ψ) and   n− α2 n k ∂t G t (T )) + T k, p,α− 1 . ψ∧kα, p,q ≤ B(k, α, q) Ak,∗ p,q (t 2

Now T and ψ have the same k-heat transform and thus are equal as distributions. This implies that T is a function and is in L p (R, |x|2k d x), when α ∈]0, 21 ]. Summarizing, the above two cases show that T ∈ ∧kα, p,q (R) and

Generalized Dunkl-Lipschitz spaces

  n− α2 n k T ∧kα, p,q ≤ B(k, α, q) Ak,∗ ∂t G t (T )) + T k, p,α− 1 , α > 0. p,q (t 2

Conversely, let T ∈ ∧kα, p,q (R) and T ∧kα, p,q is finite. Note that α

α

n− 2 n k ∂t G t (T )) ≤ Akp,q (t n− 2 ∂tn G kt (T )) < ∞, α > 0. Ak,∗ p,q (t p (R) α− 21 ,k

If α ∈]0, 21 ], then T ∈ L obtain

is obvious. If α > 21 , thus from Theorem 6.11, we

k k k J−(α− (R) ⊂ L p (R, |x|2k d x) and J−(α− 1 (T ) ∈ ∧ 1 1 (T )k, p ) , p,q ) 2

2

2

≤ B(k, α)T ∧kα, p,q . Since T k, p,α− 1 = J k

−(α− 12 )

2

(T )k, p and T ∧kα, p,q is finite, the proof is finished.  

Remark 6.19 From Lemmas 6.7 and 6.18 for β > 0, Remark 6.16 for β < 0 and Proposition 3.1(iv) for β = 0, we get β

∂tn G kt (Bβk )k,1 ≤ B(k, β)t 2 −n , where n −

β > 0 and t > 0. 2

We can now state the main result of this section. Theorem 6.20 Let α, β be real and 1 ≤ p, q ≤ ∞. Then Jβk is a topological isomorphism from ∧kα, p,q (R) onto ∧kα+β, p,q (R). Proof Suppose f ∈ ∧kα, p,q (R), by Remark 6.19, we obtain β

β

∂tl G kt (Jβk ( f ))k, p ≤ B(k, β)t 2 −( 2 ) ∂ts G kt ( f )k, p , 2

where l = ( α2 ) + ( β2 ) and s = ( α2 ). As a consequence, we deduce l− Ak,∗ p,q (t

α+β 2

α

s− 2 s k ∂tl G kt (Jβk ( f ))) ≤ B(k, β)Ak,∗ ∂t G t ( f )). p,q (t

From Lemmas 6.17 and 6.18, we conclude that Jβk f ∈ ∧kα+β, p,q (R) and Jβk f ∧k

α+β, p,q

≤ B(k, α, β) f ∧kα, p,q .

Moreover, the following relation k G kt1 (Bβk ) ∗k G kt2 (B−β ) = Ftk1 +t2 , t1 , t2 > 0,

S. Kallel k (J k ( f )) = f as a distribution. Similar provide that if f ∈ ∧kα, p,q (R) then J−β β k ( f )) = f as a distribution. conclusions show that if f ∈ ∧kα+β, p,q (R) then Jβk (J−β The announced statement arises.  

Theorem 6.21 Let T ∈ S  (R). Then for each integer n > ( α2 ) and real number β < α, the norm n− α2 n k ∂t G t (T )) + T k, p,β (28) Ak,∗ p,q (t is equivalent to T ∧kα, p,q , where 1 ≤ p, q ≤ ∞. Proof Suppose T ∈ ∧kα, p,q (R). Since α − β > 0 and by Theorem 6.20, we have k k T k, p ≤ J−β T ∧k T k, p,β = J−β

α−β, p,q

≤ B(k, α, β)T ∧kα, p,q .

Then, Lemmas 6.18 and 6.17 ensure that relation (28) is finite. Conversely, if relation (28) is finite and let l > ( α−β 2 ). By Lemmas 6.15 and 6.17, Remark 6.19 and change of variables, we have l− Ak,∗ p,q (t

α−β 2

  k n− α2 n k ∂tl G kt (J−β (T ))) ≤ B(k, n, α, β) Ak,∗ (t ∂ G (T )) + T  k, p,β p,q t t

k T and J−β k, p = T k, p,β . Note that k T ∧k J−β

α−β, p,q

  l− α−β 2 ∂ l G k (J k (T ))) + J k T k, p , ≤ B(k, α, β) Ak,∗ (t p,q t t −β −β

hence from Theorem 6.20, we obtain k T ∧kα, p,q ≤ B(k, α, β)J−β T ∧k α−β, p,q   α k,∗ n− 2 n k ≤ B(k, n, α, β) A p,q (t ∂t G t (T )) + T k, p,β

 

which prove the theorem. Note We are essentially defining ∧k−α, p,q (R) to be J k

−α− 12

(∧k1

2 , p,q

(R)), α > 0. The

choice of 21 is arbitrary. Any β > 0, would work as well. The remainder of this section is devoted to some properties and embedding theorems for the spaces ∧kα, p,q (R). Theorem 6.22 Let f in ∧kα0 , p0 ,q0 (R) ∩ ∧kα1 , p1 ,q1 (R), then f belongs to ∧kα, p,q (R) and we have  f ∧kα, p,q ≤ B(k, α0 , α1 ) f 1−θ ∧k

α0 , p0 ,q0

where α = (1 − θ )α0 + θ α1 , particular

 f θ∧α

1 , p1 ,q1

,

1−θ 1−θ 1 θ 1 θ = + , = + , and θ ∈ [0, 1]. In p p0 p1 q q0 q1

Generalized Dunkl-Lipschitz spaces θ (a)  f k, p,β ≤  f 1−θ k, p0 ,β  f k, p1 ,β , β < min(α0 , α1 ).

1−θ

θ α0 α1 α (b) Akp,q (t n− 2 ∂tn G kt ( f ))≤ Akp0 ,q0 (t n− 2 ∂tn G kt ( f )) Akp1 ,q1 (t n− 2 ∂tn G kt ( f )) ,

where n > max( α20 , α21 ).

Proof This can be proved from Theorem 6.21 and the Logarithmic convexity of the p   L k -norms. Let us study some inclusions among the generalized Dunkl-Lipschitz spaces : Lemma 6.23 The continuous embedding ∧kα1 , p,q1 (R) → ∧kα2 , p,q2 (R) holds if either (i) if α1 > α2 ( then q1 and q2 need not be related), or (ii) if α1 = α2 and q1 ≤ q2 . Proof We give the argument for q = ∞. The case q = ∞ is done similarly. We may suppose 0 < α2 < α1 < 1. Let f ∈ ∧kα1 , p,q1 (R) and consider first the case q1 = q2 . In the one hand, it is easily to see that 1−α2 Ak,∗ ∂t Ptk ( f )) ≤  f ∧kα p,q1 (t

1 , p,q1

.

In the other hand, using the fact that ∂t Ptk ( f )k, p ≤ B(k)t −1  f k, p , we get 

∞

t 1

1−α2

∂t Ptk ( f )k, p

q1 dt  q11 ≤ B(k, α2 , q1 ) f ∧kα , p,q 1 1 t

which proves that ∧kα1 , p,q2 (R) → ∧kα2 , p,q2 (R). Moreover, if q1 < q2 , Lemma 5.2 of [16] and Lemma 1.2 of [15] show that ∧kα1 , p,q1 (R) → ∧kα1 , p,q2 (R). Hence ∧kα1 , p,q1 (R) → ∧kα1 , p,q2 (R) → ∧kα2 , p,q2 (R). If q1 > q2 , let 1s = q12 − q11 . Applying Hölder’s inequality and analogous reasoning as before finish the Proof of the Lemma.   Lemma 6.24 If 1 ≤ p1 ≤ p2 and α1 − embedding

2k+1 p1

= α2 −

2k+1 p2 ,

we have the continuous

∧kα1 , p1 ,q (R) → ∧kα2 , p2 ,q (R). Proof We may assume that 0 < α1 , α2 < 1. If f ∈ ∧kα1 , p1 ,q (R), Young’s inequality yields that ∂t Ptk ( f )k, p2 ≤ ∂t P kt ( f )k, p1 P kt k,s 2

≤ B(k, p1 , p2 )t

2

(− p1 + p1 )(2k+1) 1

2

∂t P kt ( f )k, p1 , 2

S. Kallel

where 1s = p12 − p11 + 1. Hence Akp2 ,q (t 1−α2 ∂t Ptk ( f )) ≤ B(k, α1 , p1 , p2 ) Akp1 ,q (t 1−α1 ∂t Ptk ( f )). On the other hand, for t ≥ 1, Ptk ( f )k, p2 ≤ B(k, p1 , p2 )  f k, p1 and therefore by Lemma 6.8, we can deduce that f ∈ ∧kα2 , p2 ,q (R) and    f ∧kα , p ,q ≤ B(k, α1 , p1 , p2 ) f ∧kα , p ,q which end the proof. 2

2

1

1

As consequence of Lemmas 6.23 and 6.24, we deduce the following theorem : Theorem 6.25 Let α1 , α2 ∈ R and 1 ≤ p1 ≤ p2 ≤ ∞, then we have the continuous embedding ∧kα1 , p1 ,q1 (R) → ∧kα2 , p2 ,q2 (R) if α1 −

2k+1 p1

> α2 −

2k+1 p2

or if α1 −

2k+1 p1

= α2 −

2k+1 p2

and 1 ≤ q1 ≤ q2 ≤ ∞.

The action of Dunkl derivatives on Dunkl-Lipschitz spaces is as follows : Proposition 6.26 Let α > 0, 1 ≤ p, q ≤ ∞ and 0 ≤ n ≤ α. Then the norm  f k, p + Dkn f ∧k is equivalent to  f ∧kα, p,q . α−n, p,q

Proof If  f ∧k is finite, then according to the Proposition 6.1(v) and Remark (5.14) k, p,q of [16], it is easy to see that Akp,q (t α−(α−n) ∂tα Dkn Ptk ( f )) ≤ B(k, α, n) f ∧kα, p,q , and Dkn Ptk ( f )k, p ≤ B(k, n) f k, p , t ≥ 1. Thus by Lemma 6.8, we deduce that there exists g ∈ ∧kα−n, p,q (R) such that ≤ B(k, α, n) f ∧kα, p,q . On the other hand, since Dkn Ptk ( f ) = Ptk (g) and g∧k α−n, p,q

Dkn Ptk ( f ) = Ptk (Dkn f ) (in the distribution sense), we have Ptk (g) = Ptk (Dkn f ). Let  ting t −→ 0 yields that g = Dkn f . An easy check shows the converse result. Lemma 6.27 If f ∈ ∧kα,∞,q (R), α ∈]0, 1[, then f is uniformly continuous. Proof It suffices to show that  y,k f k,∞ → 0 as y → 0. By Theorem 6.25, f ∈   ∧kα,∞,∞ (R), so  y,k f k,∞ ≤ A|y|α and thus tends to zero as y → 0. Theorem 6.28 ∧kα, p,q (R) is complete if 1 ≤ p, q ≤ ∞ and α ∈ R. Proof By Theorem 6.20, we may suppose α ∈]0, 1[. If ( f n ) is a Cauchy sequence in p ∧kα, p,q (R), then ( f n ) is obviously Cauchy sequence in L k , and therefore converges in p L k to a function f . Hence ∂t Ptk ( f s )k, p → ∂t Ptk ( f )k, p as s → ∞ and for m = 1, 2, . . ., ∂t (Ptk f m − Ptk f s )k, p → ∂t (Ptk f m − Ptk f )k, p as s → ∞. Consequently, by Fatou’s Lemma, we have −−−−→ 0, Akp,q (t 1−α ∂t (Ptk f m − Ptk f )) ≤ m = lim inf Akp,q (t 1−α ∂t (Ptk f m − Ptk f s ))m→∞ s→∞

Generalized Dunkl-Lipschitz spaces

and Akp,q (t 1−α ∂t Ptk ( f )) ≤ lims→∞ inf  f s ∧kα, p,q < ∞. So that f ∈ ∧kα, p,q (R) and f m → f , as m → ∞, in ∧kα, p,q (R) which conclude the proof.

 

The object of the next section will be to derive a similar result for k-temperatures on R2+ .

7 Dunkl-Lipschitz spaces of k-temperatures We shall define a generalized Dunkl-Lipschitz space of k-temperatures on R2+ which will be denoted by T ∧kα, p,q (R2+ ) and prove that various norms are equivalent to our original definition. Finally, the isomorphism of T ∧kα, p,q (R2+ ) and ∧kα, p,q (R) is established. We begin this section by stating the following standard Lemmas. Definition 7.1 Let α be a real number. For any k-temperature U in k (R2+ ), 1 ≤ p ≤ ∞ and 1 ≤ q ≤ ∞, let ⎧  ⎪ ⎪ ⎨

+∞

k t q−1 e−t J−α−2 U(., t)k, p dt k,α E p,q (U) := 0  ⎪ k ⎪ ⎩ sup te−t J−α−2 U(., t)k, p q

 q1

(1 ≤ q < ∞), (q = ∞),

t>0

with infinite values being allowed. Lemma 7.2 Let α, U, p, q be as in the above definition and let γ be a real number. Then k,α+γ

k,α E p,q (U) = E p,q

(Jγk U).

k,α k k k Proof By Theorem 5.5, J−α−2 U = J−α−γ −2 (Jγ U) which implies that E p,q (U) = k,α+γ

E p,q

(Jγk U).

 

Definition 7.3 Let 1 ≤ p, q ≤ ∞, let α, β be real numbers such that β > α. For any k-temperature U in k (R2+ ), let

k,α,β E p,q (U) :=

⎧  ⎪ ⎪ ⎨

+∞

1 2 q(β−α)−1

−t

q k J−β U(., t)k, p dt

t e 0  1 ⎪ k ⎪ ⎩ sup t 2 (β−α) e−t J−β U(., t)k, p t>0

and Lkp (U) := sup U(., t)k, p . t≥ 21

 q1

(1 ≤ q < ∞), (q = ∞),

S. Kallel k,α Remark 7.4 Let 1 ≤ p, q ≤ ∞, and γ be real number. If U ∈ k (R2+ ) and E p,q (U) < ∞, where α is real, so that Theorem 5.9 and Corollary 5.8 yield that for each a > 0 there exists a positive constant B such that for all t ≥ a k,α Jγk U(., t)k, p ≤ B(k, α, γ , q, a)E p,q (U).

Lemma 7.5 Let α, β, U, p, q be as in Definition 7.3. Then k,α,β

k,α (i) E p,q (U) is equivalent to E p,q (U).  1 k,α (β−α) k (ii) E p,q (U) is equivalent to Ak,∗ J−β U + Lkp (U). p,q t 2

Proof The proof is a simple consequence of Remark 7.4, Theorem 5.10 and Corollary 5.8.   Lemma 7.6 Let α be real number, U ∈ k (R2+ ), 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞, k,α (U) is equivalent to and n be a non-negative integer greater than α2 . Then E p,q  1  k,∗ n− 2 α n k A p,q t ∂t U + L p (U). Proof If n = 0, the result will be obtained from Lemma 7.5(ii). First suppose that k,α (U) < ∞. For i = 0, 1, . . . , n − 1, we have E p,q k k U(., t)k, p ≤ J−2n U(., t)k, p J−2i k U(., t), . . . , J k U(., t), and since ∂tn U(., t) is a linear combination of U(., t), J−2 −2n it follows that k U(., t)k, p (29) ∂tn U(., t)k, p ≤ B(k, n)J−2n

and therefore by Lemma 7.5(ii), we obtained  1  1   n− 2 α n k k,∗ n− 2 α k Lkp (U) + Ak,∗ t t ∂ U ≤ L (U) + A J U p,q t p p,q −2n k,α (U ). ≤ B(k, n, α, q)E p,q

 1  n− 2 α n Conversely, suppose Lkp (U) + Ak,∗ ∂t U . From Theorem 4.4, Minkowski’s p,q t integral inequality, relation (5) and Proposition 3.1(iv), we deduce that for i = 1, 2 . . . n sup ∂ti U(., t)k, p ≤ B(k, i)Lkp (U) t≥1

and ∂ti U(., t)k, p ≤ B(k, n)Lkp (U) + ∂tn U(., t)k, p .

(30)

Thus  1  1    n− 2 α k k k,∗ n− 2 α n Ak,∗ t t J U ≤ B(k, n, α, q) L (U) + A ∂ U . p,q p p,q t −2n Again Lemma 7.5(ii) shows the desired result.

 

Generalized Dunkl-Lipschitz spaces

Now we turn to the definitions of the generalized Dunkl-Lipschitz space of ktemperatures on R2+ . Definition 7.7 Let α be a real number, 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞. We define   k,α T ∧kα, p,q (R2+ ) := U ∈ k (R2+ ) : E p,q (U) < ∞ ;  k T λkα, p,∞ (R2+ ) := U ∈ T ∧kα, p,∞ (R2+ ) : J−α−2 U(., t)k, p  = ◦(t −1 ) as t −→ 0+ . k,α Then, E p,q is a norm on T ∧kα, p,q (R2+ ).

First we give : Lemma 7.8 Let 1 ≤ p, q ≤ ∞, α and γ be real numbers. Then Jγk is an isometric isomorphism of T ∧kα, p,q (R2+ ) (T λkα, p,∞ (R2+ ) resp.) onto T ∧kα+γ , p,q (R2+ ) k . (T λkα+γ , p,∞ (R2+ ) resp.) with inverse J−γ   k k k U , then Corollary 5.6 proves the result. J Proof Since J−α−2 U = J−α−γ γ −2

 

The basic properties of the spaces T ∧kα, p,q (R2+ ) lie in the following theorem : Theorem 7.9 Let 1 ≤ p, q ≤ ∞ and α be a real number. (i) If 1 ≤ q1 < q2 < ∞, we have the continuous embedding T ∧kα, p,q1 (R2+ ) → T ∧kα, p,q2 (R2+ ) → T λkα, p,∞ (R2+ ) → T ∧kα, p,∞ (R2+ ). k,α,β

(ii) If β is a real number such that β > α, then E p,q is an equivalent norm on T ∧kα, p,q (R2+ ); moreover U ∈ T λkα, p,∞ (R2+ ) if and only if U ∈ T ∧kα, p,∞ (R2+ ) 1

− (β−α) k U(., t) and J−β ) as t −→ 0+ . k, p = ◦(t 2

 1  n− 2 α n (iii) If n is a non-negative integer greater than 21 α, then Ak,∗ ∂t U + Lkp (U) p,q t is an equivalent norm on T ∧kα, p,q (R2+ ). (iv) The spaces T ∧kα, p,q (R2+ ), where p, q are fixed and α varies, are isomorphic to one another. The same conclusion holds for the spaces T λkα, p,∞ (R2+ ). Proof (i) follows easily from Theorem 5.9. (ii) is an easy consequence of Lemma 7.5 and Theorem 5.10(iii). (iii) is derived from Lemma 7.6. To prove (iv), let δ be another real number. It then follows from k is an isometric isomorphism of T ∧k k 2 2 Lemma 7.8 that J−n δ, p,q (R+ ) (T λδ, p,∞ (R+ ) k k k 2 2 −1 resp.) onto T ∧δ−n, p,q (R+ ) (T λδ−n, p,∞ (R+ ) resp.); denote its inverse by (J−n ) . k is an isometric isomorphism of T ∧kα, p,q (R2+ ) This Lemma again implies that Jδ−α−n (T λkα, p,∞ (R2+ ) resp.) onto T ∧kδ−n, p,q (R2+ ) (T λkδ−n, p,∞ (R2+ ) resp.). Consequently,

S. Kallel k )−1 ◦ J k k 2 k 2 (J−n δ−α−n is an isometric isomorphism of T ∧α, p,q (R+ ) (T λα, p,∞ (R+ )   resp.) onto T ∧kδ, p,q (R2+ ) (T λkδ, p,∞ (R2+ ) resp.).

The following theorem establish the relation between ∧kα, p,q (R) and T ∧kα, p,q

(R2+ ).

Theorem 7.10 If 1 ≤ p, q ≤ ∞ and α is real, then the k-heat transform is a topological isomorphism from ∧kα, p,q (R) onto T ∧kα, p,q (R2+ ). Moreover if f ∈ ∧kα, p,q (R),

k,α then G kt ( f ) ∈ T ∧kα, p,q (R2+ ) and E p,q (G kt ( f )) ≤ B(k, α) f ∧kα, p,q . Conversely, if

U ∈ T ∧kα, p,q (R2+ ), then there exists f ∈ ∧kα, p,q (R) such that

k,α U(., t) = G kt ( f )(.), t > 0, and  f ∧kα, p,q ≤ B(k, α)E p,q (U).

Proof Let f ∈ ∧kα, p,q (R), by Theorem 3.4, Lemmas 6.14 and 7.6, we deduce that k,α G kt ( f ) ∈ T ∧kα, p,q (R) and E p,q (G kt ( f )) ≤ B(k, α) f ∧kα, p,q .

To prove the converse we proceed first in case α > 0. For U ∈ T ∧kα, p,q (R2+ ), let k U(., t), t > 0, then for s > 0 V(., t) = J−α−2 U(x, s) =

1 α ( 2 + 1)



+∞

α

ξ 2 e−ξ V(x, ξ + s)dξ.

0

Moreover, by Theorem 5.9 yields k k,α J−α−2 U(., t)k, p ≤ B(q)(t −1 + 1)E p,q (U)

(31)

which together with Minkowski’s integral inequality, we find that  k,α U(., s)k, p ≤ B(q, α)E p,q (U)

+∞

α

ξ 2 e−ξ (ξ −1 + 1)dξ

0

k,α = B(q, α)E p,q (U), if 1 ≤ p ≤ ∞.

On the one hand, for p = 1 and  > 0, from inequality (31), we can find δ satis1 fying 0 < δ < 1 such that V(., t)k,1 ≤ t −1− 4 α for 0 < t ≤ δ. On the other hand, by a simple verification yields U(., s) − U(., s  )k,1 → 0 as s, s  → 0. Summarizing the above two cases show that from Remark 6.4, there exists a function f ∈ L p (R, |x|2k d x), 1 ≤ p ≤ ∞, such that U(., t) = G kt ( f )(.). Next, in case α ≤ 0, then using Lemma 7.8, we have k, 1

k k k k,α J−α+ (R2+ ) and E p,q2 (J−α+ 1 U ∈ T ∧1 1 U) ≤ BE p,q (U). , p,q 2

2

2

Generalized Dunkl-Lipschitz spaces

Applying the above case α > 0, then there exists g ∈ L p (R, |x|2k d x), p ∈ [1, ∞], k,α (U). Due to Theorem 3.12 such that J k 1 U(., t) = G kt (g)(.), and gk, p ≤ BE p,q −α+ 2

for [6],

k k,α U(., t) = G kt ( f )(.), f = Jα− 1 (g) and  f k, p,α− 1 = gk, p ≤ BE p,q (U). 2

2

By Proposition 3.1(iv), we obtain for α > 0  1  α α n− 2 α n Akp,q (t n− 2 ∂tn U) ≤ Ak,∗ ∂t U + B(k, α)Lkp (U), n = ( ). p,q t 2 k,α Therefore by Lemma 7.6, we obtain  f ∧kα, p,q ≤ BE p,q (U), α ∈ R, and the theorem is proved.  

Theorem 7.11 Let 1 ≤ p < r ≤ ∞, 1 ≤ q ≤ ∞, α be a real number and δ = Then

1 p

− r1 .

(i) T ∧kα, p,q (R2+ ) → T ∧kα−δ(2k+1),r,q (R2+ ), (ii) T λkα, p,∞ (R2+ ) → T λkα−δ(2k+1),r,∞ (R2+ ). Proof Let h such that r1 = 1p + h1 − 1, ( h1 = 1 − δ). We give the argument for q = ∞. The case q = ∞ is done similarly. Let U be in T ∧kα, p,q (R2+ ) and β be a real number k,α,β

k,α (U), β > α. greater than α. Theorem 7.9(ii) implies that E p,q (U) is equivalent to E p,q k Then t → J−β U(., t)k, p is locally integrable on ]0, ∞[, so the semi-group formula k U. By Theorem 4.4 and Young’s inequality (Proposition 7.2 of [26]), holds for J−β we have

t k k J−β U(., t)k,r ≤ J−β U(., )k, p F kt k,h . 2 2 1

By a simple verification, we deduce that F kt k,h ≤ B(k, p, r )t −(k+ 2 )δ . Hence, we 2 obtain 1 t k k U(., t)k,r ≤ B(k, p, r )t −(k+ 2 )δ J−β U(., )k, p . J−β 2

Therefore  k,α−2kδ−δ,β (U) Er,q

=

+∞

t

1 2 q(β−α+2kδ+δ)−1

0



≤ B(k, p, r )

+∞

t

e

−t

q k J−β U(., t)k,r dt

1 2 q(β−α)−1

0 k,α,β ≤ B(k, p, α, β, r )E p,q (U),

e

−t

k J−β U(.,

 q1

t q ) dt 2 k, p

 q1

S. Kallel

from which we obtain the part (i) after making use of Theorem 7.9(ii) again. We proceed in the same way to prove the assertion (ii).   Remark 7.12 In view of the isometry between ∧kα, p,q (R) and T ∧kα, p,q (R2+ ), the same result of Theorem 7.11 holds for spaces ∧kα, p,q (R). Acknowledgments comments.

The author thanks the referee for his careful reading of the manuscript and insightful

References 1. Abdelkefi, C., Anker, J.P., Sassi, F., Sifi, M.: Besov-type spaces on IR d and integrability for the Dunkl transform. Symmetry Integr. Geom. Methods Appl. 5, Paper 019, p. 15 (2009) 2. Abdelkefi, C., Sassi, F.: Various characterizations of Besov-Dunkl spaces. Int. J. Pure Appl. Math. 39(4), 475–488 (2007) 3. Abdelkefi, C., Sifi, M.: Characterization of Besov spaces for the Dunkl operator on the real line. J. Inequal. Pure Appl. Math. 8(3), p. 11, Article 73 (2007) 4. Aronszajn, N., Smith, K.T.: Theory of Bessel potentials. I. Ann. de L’Inst. Fourier 11, 385–475 (1961) 5. Bateman, H.: Tables of integral transforms, vol. II. New York (1954) 6. Ben Salem, N., El Garna, A., Kallel, S.: Bessel and Flett potentials associated with Dunkl operators on IR d . Methods Appl. Anal. 15(4), 477–494 (2008) 7. Ben Salem, N., Kallel, S.: Mean-periodic functions associated with the Dunkl operators. Integral Transforms Spec. Funct. 15(2), 155–179 (2004) 8. Bouguila, R., Lazhari, M.N., Assal, M.: Besov spaces associated with Dunkl’s operator. Integral Transforms Spec. Funct. 18(8), 545–557 (2007) 9. de Jeu, M.F.E.: The Dunkl transform. Invent. Math. 113, 147–162 (1993) 10. Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc. 311, 167–183 (1989) 11. Dunkl, C.F.: Integral kernels with reflection group invariance. Can. J. Math. 43, 1213–1227 (1991) 12. Dunkl, C.F.: Hankel transforms associated to finite reflection groups. Contemp. Math. 138, 123–138 (1992) 13. Flett, T.M.: Temperatures, Bessel potentials and Lipschitz spaces. Proc. Lond. Math. Soc. 22(3), 385– 451 (1971) 14. Friedman, A.: Partial differential equations of parabolic type. Englewood Cliffs (1964) 15. Johnson, R.: Temperatures, Riesz potentials, and the Lipschitz spaces of Hertz. Proc. Lond. Math. Soc. 27(3), 290–316 (1973) 16. Kallel, S.: Characterization of function spaces for the Dunkl type operator on the real line. Potential Anal. 41(1), 143–169 (2014) 17. Kamoun, L.: Besov-type spaces for the Dunkl operator on the real line. J. Comput. Appl. Math. 199, 56–67 (2007) 18. Kamoun, L., Nagzaoui, S.: Lipschitz spaces associated with reflection group Z 2d . Commun. Math. Anal. 7(1), 21–36 (2009) 19. Kawazoe, T., Mejjaoli, H.: Generalized Besov spaces and their applications. Tokyo J. Math. 35(2), 297–320 (2012) 20. Li, Zh, Liao, J.: Harmonic analysis associated with the one-dimensional Dunkl transform. Constr. Approx. 37, 233–281 (2013) 21. Mejjaoli, H.: Dunkl-heat semigroup and applications. Appl. Anal. 92(9), 1980–2007 (2013) 22. Mourou, M.A.: Taylor series associated with a differential-difference operator on the real line. J. Comput. Appl. Math. 153, 343–354 (2003) 23. Rösler, M.: Bessel-type signed hypergroups on IR. In: Heyer, H., Mukherjea, A. (eds.) Probability measures on groups and related structures XI (Oberwolfach, 1994), pp. 292–304. World Sci. Publ. (1995) 24. Rösler, M.: Generalized Hermite polynomials and the heat equation for Dunkl operators. Commun. Math. Phys. 192, 519–541 (1998)

Generalized Dunkl-Lipschitz spaces 25. Taibleson, M.H.: On the theory of Lipschitz spaces of distributions on Euclidean n-spaces. I. Principal properties. J. Math. Mech. 13(3), 407–479 (1964) 26. Thangavelu, S., Xu, Y.: Convolution operator and maximal function for the Dunkl transform. J. Anal. Math. 97, 25–55 (2005) 27. Thangavelu, S., Xu, Y.: Riesz transforms and Riesz potentials for the Dunkl transform. J. Comput. Appl. Math. 199, 181–195 (2007) 28. Trimèche, K.: The Dunkl intertwining operator on spaces of functions and distributions and integral representations of its dual. Integral Transforms Spec. Funct. 12, 349–374 (2001)