For a normalized root system R in R N and a multiplicity function k ≥ 0 let N = N + α∈R k(α). Denote by dw(x) = α∈R | x, α | k(α) dx the associated measure in R N . Let F stands for the Dunkl transform. Given a bounded function m on R N , we prove that if there is s > N such that m satisfies the classical Hörmander condition with the smoothness s, then the multiplier operator T m f = F −1 (mF f ) is of weak type (1, 1), strong type (p, p) for 1 < p < ∞, and bounded on a relevant Hardy space H 1 . To this end we study the Dunkl translations and the Dunkl convolution operators and prove that if F is sufficiently regular, for example its certain Schwartz class seminorm is finite, then the Dunkl convolution operator with the function F is bounded on L p (dw) for 1 ≤ p ≤ ∞. We also consider boundedness of maximal operators associated with the Dunkl convolutions with Schwartz class functions.2000 Mathematics Subject Classification. Primary: 42B15, 42B20, 42B35. Secondary: 42B30, 42B25, 47D03.

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In this work we extend the theory of the classical Hardy space H 1 to the rational Dunkl setting. Specifically, let be the Dunkl Laplacian on a Euclidean space R N. On the half-space R + × R N , we consider systems of conjugate (∂ 2 t + x)-harmonic functions satisfying an appropriate uniform L 1 condition. We prove that the boundary values of such harmonic functions, which constitute the real Hardy space H 1 , can be characterized in several different ways, namely by means of atoms, Riesz transforms, maximal functions or Littlewood-Paley square functions.

On R N equipped with a root system R, multiplicity function k ≥ 0, and the associated measure dw(x) = α∈R | x, α | k(α) dx, we consider a (non-radial) kernel K(x) which has properties similar to those from the classical theory of singular integrals and the Dunkl convolution operator Tf = f * K associated with K. Assuming that b belongs to the BMO space on the space of homogeneous type X = (R N , • , dw), we prove that the commutatorThe paper extents results of Han, Lee, Li and Wick.

On R N equipped with a normalized root system R, a multiplicity function k(α) > 0, and the associated measurelet h t (x, y) denote the heat kernel of the semigroup generated by the Dunkl Laplace operator ∆ k . Let d(x, y) = min σ∈G x− σ(y) , where G is the reflection group associated with R. We derive the following upper and lower bounds for h t (x, y): for all c l > 1/4 and 0 < c u < 1/4 there are constants C l , C u > 0 such thatwhere Λ(x, y, t) can be expressed by means of some rational functions of x − σ(y) / √ t. An exact formula for Λ(x, y, t) is provided.