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Abstract: We consider Dunkl theory associated to a general Coxeter group G. A new characterization of the regularity of the weight k is given and a new construction, devoid of Kozul complex theory, of the Dunkl intertwining operator is established. We apply our results to derive sharp estimates of the Dunkl kernel. We give explicit formula in the case of orthogonal positive root systems. [Copyright &y& Elsevier]

Abstract: We study the boundedness of the multiplier of the interval for the Dunkl transform of order on weighted spaces, with . In particular, we get that it is bounded from into itself if and only if . [Copyright &y& Elsevier]

Abstract: We consider a vector version of -multipliers for the Dunkl transform on and we prove -inequalities for the Littlewood–Paley operators. [Copyright &y& Elsevier]

We consider Fourier multipliers for Lp associated with the Dunkl operator on R and establish a version of Ho¨rmander's multiplier theorem. In applying this version, we come up with some results regarding the oscillating multipliers, partial sum operators and generalized Bessel potentials. [Copyright &y& Elsevier]

In this paper we study the Sobolev inequality in the Dunkl setting using two new approaches which provide a simpler elementary proof of the classical case p = 2 , as well as an extension to the coefficient p = 1 that was previously unknown. We also find estimates of the sharp constants for the Sobolev inequality for Dunkl gradient. Related inequalities and some improvements are also considered (Nash inequality, Besov space embeddings). [ABSTRACT FROM AUTHOR]