An Lp-Lq-version of Morgan's theorem for the Dunkl-Bessel transform

In this paper, we give an L-p - L-q - version of Morgan's theorem for the Dunkl-Bessel transform F-D,F-B on R-+(d+1).. More precisely, we prove that for all 1 <= p, q <= +infinity, alpha > 2, eta = alpha/(alpha - 1) and a > 0, b > 0, then for all measurable function f on R-+(d+1), the conditions e(a parallel to x parallel to alpha) f is an element of L-k,beta(p) (R-+(d+1)) and e(b parallel to y parallel to eta) F-D,F-B(f) is an element of L-k,beta(q) (R-+(d+1)) imply f = 0, if and only if (a alpha)(1/alpha)(b eta)(1/eta) > (sin(pi/2)(eta- 1))(1/eta), where L-k,beta(p) (R-+(d+1)), are the Lebesgue spaces associated with the Dunkl - Bessel transform.