Abstract. Our main goal in this article is to study mixed norm estimates for the Cesàro means associated with Dunkl-Hermite expansions on R d . These expansions arise when one considers the Dunkl-Hermite operator (or Dunkl harmonic oscillator) Hκ := −∆κ + |x| 2 , where ∆κ stands for the Dunkl-Laplacian. It is shown that the desired mixed norm estimates are equivalent to vectorvalued inequalities for a sequence of Cesàro means for Laguerre expansions with shifted parameter. In order to obtain such vector-valued inequalities, we develop an argument to extend these Laguerre operators for complex values of the parameters involved and apply a version of three lines lemma.

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We obtain generalised trace Hardy inequalities for fractional powers of general operators given by sums of squares of vector fields. Such inequalities are derived by means of particular solutions of an extended equation associated to the above-mentioned operators. As a consequence, Hardy inequalities are also deduced. Particular cases include Laplacians on stratified groups, Euclidean motion groups and special Hermite operators. Fairly explicit expressions for the constants are provided. Moreover, we show several characterisations of the solutions of the extension problems associated to operators with discrete spectrum, namely Laplacians on compact Lie groups, Hermite and special Hermite operators.

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In this paper we study weighted mixed norm estimates for Riesz transforms associated to Dunkl harmonic oscillators. The idea is to show that the required inequalities are equivalent to certain vector valued inequalities for operator defined in terms of Laguerre expansions. In certain cases the main result can be deduced from the corresponding result for Hermite Riesz transforms.