An analogue of Hardy's theorem for the Dunkl transform

In this Note we give a generalization of Hardy's theorem for the Dunkl transform F-D on R-d. More precisely, for all a > 0; b > 0 and p, q is an element of [1; +infinity], we determine the measurable functions f such that e(a\\x\\2) f is an element of L-k(p)(R-d) and e(b\\y\\2) F-D(f) is an element of L-k(q)(R-d), where L-k(p)(R-d) are the L-p spaces associated with the Dunkl transform. (C) 2002 Academie des science/Edition scientifiques et medicales Elsevier SAS.