A note on commutators of singular integrals with BMO and VMO functions in the Dunkl setting

On RN$\mathbb {R}^N$ equipped with a root system R, multiplicity function k≥0$k \ge 0$ , and the associated measure dw(x)=∏α∈R|⟨x,α⟩|k(α)dx$dw(\mathbf {x})=\prod _{\alpha \in R}|\langle \mathbf {x},\alpha \rangle |^{k(\alpha )}\,d\mathbf {x}$ , we consider a (nonradial) kernel K(x)${K}(\mathbf {x})$ , which has properties similar to those from the classical theory of singular integrals and the Dunkl convolution operator Tf=f∗K$\mathbf {T}f=f*K$ associated with K. Assuming that b belongs to the BMO space on the space of homogeneous type X=(RN,∥·∥,dw)$X=(\mathbb {R}^N,\Vert \cdot \Vert ,dw)$ , we prove that the commutator [b,T]f(x)=b(x)Tf(x)−T(bf)(x)$[b,\mathbf {T}]f(\mathbf {x})=b(\mathbf {x})\mathbf {T}f(\mathbf {x})-\mathbf {T}(bf)(\mathbf {x})$ is a bounded operator on Lp(dw)$L^p(dw)$ for all 1

1. Introduction 1.1.Introduction.Consider R N equipped with a root system R and a non-negative multiplicity function k ≥ 0. Let dw(x) = α∈R | α, x | k(α) dx be the associated measure.For f ∈ L 1 loc (dw) and a measurable bounded set E ⊂ R N , we denote (1.1) Let G be the Coxeter group generated by the reflections σ α , α ∈ R. For E ⊂ R N , we set In Han, Lee, Li, and Wick [14] the authors investigated two types of BMO  where the supremum is taken over all Euclidean balls B = B(y, r) = {z ∈ R N : y − z < r}.
The space BMO d is a proper subspace of BMO (see [16]) and In [14] commutators of BMO and BMO d functions with the Dunkl-Riesz transforms R j are studied.The Dunkl-Riesz transforms are Calderón-Zygmund type operators which are formally defined by R j = T e j (−∆ k ) −1/2 , where T e j are the Dunkl operators (see (2.6)) and T 2 e j is the Dunkl Laplacian.They were studied by Thangavelu and Xu [24] (in dimension 1 and in the product case) and by Amri and Sifi [1] (in higher dimensions) who proved their bounds on L p (dw) spaces.One of the main results of [14] asserts that if b ∈ BMO d , then the commutator [b, R j ]f (x) = b(x)R j f (x) − R j (b(•)f (•))(x) is a bounded operator on L p (dw) for 1 < p < ∞ and Conversely, if for b ∈ L 1 loc (dw), the commutator [b, R j ] is bounded on L p (dw) for some 1 < p < ∞, then b ∈ BMO and (1.3) b BMO [b, R j ] L p (dw)→L p (dw) .
The authors of [14] raised the question if the possible lower bound b BMO d [b, R j ] L p (dw)→L p (dw) holds true.
Our first goal in this note is to improve (1.2) by showing that it holds for b ∈ BMO, that is, there is a constant C p > 0 such that Let us point out that (1.4) gives a negative answer to the question formulated above, because otherwise we would get b BMO , which is impossible (see [16]).
An essential part of [14] is devoted for studying compactness of the commutators of VMO functions with the Dunkl-Riesz transforms.The VMO and VMO d spaces are defined as the closures of the sets of compactly supported functions from the Lipschitz spaces Λ and Λ d in the norms • BMO and • BMO d respectively.Then VMO d ⊂ VMO and b VMO b VMO d and, thanks to [5,Theorem 4.1], the dual space to VMO is the Hardy space H 1 considered in [2] and [10].Theorem 1.5 of [14] states that if b ∈ VMO d , then the commutator [b, R j ] is a compact operator on L p (dw) for all 1 < p < ∞.Our second aim is to extend this result for all b ∈ VMO (see Theorem 3.2).Actually we will prove (1.2) and the compactness result for commutators [b, T] of BMO and VMO functions with Dunkl singular integral operators T of convolution type (under certain regularity for the associated kernels K(x)).The Dunkl-Riesz transforms are the basic examples of such operators.
Let us remark that for the Riesz transforms, the lower bounds (1.3) proved in [14] together with the upper bounds (1.4) (see Theorem 3.1) generalize (to the Dunkl setting) the classical results of Coifman, Rochberg and Weiss [4] and Janson [15] obtained on the Euclidean spaces (R N , dx).As far as the compactness is concerned, the necessity result [14, Theorem 1.5] together with the sufficiency result (see Theorem 3.2) extend to the Dunkl theory the classical theorems of Uchiyama [26] about the characterization of the VMO functions by commutators with the Riesz transforms.
We consider the Euclidean space R N with the scalar product x, y = N j=1 x j y j , where x = (x 1 , ..., x N ), y = (y 1 , ..., y N ), and the norm The finite group G generated by the reflections σ α , α ∈ R, is called the Coxeter group (reflection group) of the root system.
A multiplicity function is a G-invariant function k : R → C which will be fixed and ≥ 0 throughout this paper.
The associated measure dw is defined by dw(x) = w(x) dx, where α) .
Observe that there is a constant C > 0 such that for all x ∈ R N and r > 0 we have so dw(x) is doubling.Moreover, there exists a constant C ≥ 1 such that, for every x ∈ R N and for all r 2 ≥ r 1 > 0, (2.5) For ξ ∈ R N , the Dunkl operators T ξ are the following k-deformations of the directional derivatives ∂ ξ by difference operators: The Dunkl operators T ξ , which were introduced in [7], commute and are skew-symmetric with respect to the G-invariant measure dw.
The function E(x, y), which generalizes the exponential function e x,y , has a unique extension to a holomorphic function E(z, w) on The Dunkl transform is a generalization of the Fourier transform on R N .It was introduced in [8] for k ≥ 0 and further studied in [6] in the more general context.It was proved in [8, Corollary 2.7] (see also [6,Theorem 4.26]) that it extends uniquely to an isometry on L 2 (dw).
We have also the following inversion theorem ([6, Theorem 4.20]): for all The Dunkl translation was introduced in [17].The definition can be extended to functions which are not necessary in S(R N ).For instance, thanks to the Plancherel's theorem, one can define the Dunkl translation of L 2 (dw) function f by (2.11) (see [17] of [23,Definition 3.1]).In particular, the operators f → τ x f are contractions on L 2 (dw).Here and subsequently, for a reasonable function g(x), we write g(x, y) := τ x g(−y).
We will need the following result concerning the support of the Dunkl translation of a compactly supported function.
equivalently, by Generalized convolution of f, g ∈ S(R N ) was considered in [17] and [25], the definition was extended to f, g ∈ L 2 (dw) in [23].
2.5.Singular integral kernels.Let us consider a (non-radial) kernel K(x) which has properties similar to those from the classical theory.Namely, let s 0 be an even positive integer larger than N, which will be fixed in the whole paper.Consider a function where φ is a fixed radial C ∞ -function supported by the unit ball B(0, 1) such that φ(x) = 1 for x < 1/2.The following theorem was proved in [11].

Statement of the results
3.1.Commutators.In order to define the commutator operator, we come back to the definition of the limit operator T. Let 0 < ε < min(1, s 0 − N).For any t > 0 let us denote [11, (3.1)]).In order to simplify the notation, we write [12, (4.24), (4.25)] that for all x, y ∈ R N and ℓ ∈ Z, we have Moreover, K ℓ (x, y) = K ℓ (−y, −x) and by [12, proof of Theorem 4.6], The number ε > 0 will be fixed in the whole paper.Thanks to (3.3) the function is well defined for d(x, y) > 0 and, by Theorem 2.2, it is the associated kernel to the operator T, that is, for f ∈ L p (dw) and x / ∈ supp f .Let us emphasize that the estimate for K(x, y) which are consequences of (3.3) and (3.4) turn out to be very useful in handling some harmonic analysis problems in the Dunkl setting (see [22]).
From now on we fix a kernel K ∈ C s 0 (R N \ {0}) satisfying (A), (L), and (D) for some s 0 > N. Let b ∈ BMO.For any compactly supported f ∈ L p (dw) for some p > 1, we define The existence of the limit in (3.7) in any L p 0 (dw)-norm, provided 1 < p 0 < p, is proved in Lemma 4.2.Then, Theorem 3.1 and its proof allows one to extend the definition for all f ∈ L p (dw).

3.2.
Statements of main theorems.Our main results are the following two theorems.Theorem 3.1.Let p > 1. Assume that a kernel K ∈ C s 0 (R N \ {0}) satisfies (A), (L), (D) for a certain even integer s 0 > N, and b ∈ BMO.Then there is a constant C > 0 independent of b such that for all f ∈ L p (dw) we have In order to formulate our second theorem, recall that VMO is the closure in BMO of compactly supported Lipschitz functions, i.e., functions f satisfying sup x =y To prove the theorems, we adapt the ideas of classical proofs (cf.e.g.[13]) to apply estimates for integral kernels of operators which are expressed in terms of the orbit distance d(x, y) and the Euclidean metric, see (3.3) and (3.4) (cf.also [14]).These require in some places much careful analysis.For example, we use the Coxeter group for decomposing L p (dw)functions (see (4.9)) or we split integration over R N onto the Weyl chambers (see (4.10)).

Proof of Theorem 3.1
Let us begin with the following lemma.Recall that b B(x,r) is defined by (1.1).
Hence, applying the John-Nirenberg inequality (4.4), we get We turn to analyse g 22 .Observe that for z / ∈ O(5B) and y ∈ B we have x 0 − y ≤ d(x 0 , z).Let Γ be a fixed closed Weyl chamber such that x 0 ∈ Γ, then by the estimates (3.4), In dealing with J σ (y) we shall use the inequalities: So, Further, by (4.3),We turn to considering J σ,2 (y).Applying Hölder's inequality and then the John-Nirenberg inequality (4.4) we obtain (4.12) Thus, by (4.11) and (4.12) we have got Finally we turn to estimate g σ j 2 .To this end we note that for z ∈ U j and y ∈ B we have )) 1/s .Finally we end up with the estimate Hence, using the L p 1 (dw)-boundedness of the Hardy-Littlewood maximal M function for all 1 < p 1 < ∞ and the fact that the measure dw is G-invariant, we conclude (4.8) from (4.14) and (4.9).

5 . 2 Lemma 5 . 1 .
Proof of Theorem 3.Let 1 < p < ∞.Assume that b is a compactly supported Lipschitz function.Then (5.1) lim m→∞ C − C m L p (dw) −→L p (dw) = 0. Proof.Let r b > 1 be such that supp b ⊂ B(0, r b ) and let L b > 0 be such that |b(x) − b(y)| ≤ L b x − y for all x, y ∈ R N .By (3.7), it is enough to prove that there is a constant C > 0 such that for all positive integers m such that 2 m ≥ 2r b , f ∈ L p (dw), and x ∈ R N , we have (5.2)
Dunkl transform.For fixed y ∈ R N , the Dunkl kernel x −→ E(x, y) is a unique solution to the system