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In Dunkl theory on \(\mathbb R ^d\) which generalizes classical Fourier analysis, we prove first weighted inequalities for certain Hardy-type averaging operators. In particular, we deduce for specific choices of the weights the \(d\)-dimensional Hardy inequalities whose constants are sharp and independent of \(d\). Second, we use the weight characterization of the Hardy operator to prove weighted Dunkl transform inequalities. As consequence, we obtain Pitt’s inequality which gives an integrability theorem for this transform on radial Besov spaces.

In this paper, we prove the Hardy-Littlewood-Paley inequality for the generalized Fourier transform on Chebli-Trimeche hypergroups and we study in the particular case of the Jacobi hypergroup the integrability of this transform on Besov-type spaces.