An L p version of Hardy's theorem for the Jacobi–Dunkl transform

Abstract

In this paper, we give a generalization of Hardy's theorem for the Jacobi–Dunkl transform ℱ on ℝ. More precisely for all a > 0, b > 0 and p, q ∈ [1, +∞], we determine the measurable functions f on ℝ such that E 1/4a −1 f ∈ L α,β p (ℝ) and e bλ2 ℱf ∈ L σ q (ℝ), where E t , t > 0, L α,β p (ℝ), p ∈ [1, +∞], and L σ q (ℝ), q ∈ [1, +∞], are respectively the heat kernel and the Lebesgue spaces associated with the Jacobi–Dunkl operator. *E-mail: fredj.chouchane@ipeim.rnu.tn †E-mail: maher.mili@fsm.rnu.tn ‡E-mail: mohamed.sifi@fst.rnu.tn