Almost everywhere convergence of Bochner-Riesz means with critical index for Dunkl transforms

Let B-R(delta)(h(K)(2); f), (R > 0) denote the Bochner Riesz means of order delta > -1 for the Dunkl transform of f is an element of L-1 (R-d; h(k)(2)dx) associated with the weight function h(k)(2)(x) := Pi(d)(j=1) vertical bar x(j)vertical bar(2kj) on R-d, where K := (K-1, ... , K-d) is an element of [0, infinity)(d). This paper shows that if K not equal 0, then the Bochner Riesz mean B-R(delta) (h(K)(2); f) (x) of each function f is an element of L-1(R-d; h(k)(2)dx) converges almost everywhere to f (x) on R-d at the critical index delta = lambda(K) :=d-1/2 + Sigma(d)(j=1) K-j as R -> infinity. As is well-known in classical analysis, this result is no longer true in the unweighted case where K = 0, h(K) (x) = 1, and B-R(delta) (h(K)2: f) is the Bochner Riesz mean of the Fourier transform. 2015 Elsevier Inc. All rights reserved.