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

Let BRδ(hκ2;f), (R>0R>0) denote the Bochner–Riesz means of order δ>−1δ>−1 for the Dunkl transform of f∈L1(Rd;hκ2dx) associated with the weight function hκ2(x):=∏j=1d|xj|2κj on RdRd, where κ:=(κ1,⋯,κd)∈[0,∞)dκ:=(κ1,⋯,κd)∈[0,∞)d. This paper shows that if κ≠0κ≠0, then the Bochner–Riesz mean BRδ(hκ2;f)(x) of each function f∈L1(Rd;hκ2dx) converges almost everywhere to f(x)f(x) on RdRd at the critical index δ=λκ:=d−12+∑j=1dκj as R→∞R→∞. As is well-known in classical analysis, this result is no longer true in the unweighted case where κ=0κ=0, hκ(x)≡1hκ(x)≡1, and BRδ(hκ2;f) is the Bochner–Riesz mean of the Fourier transform.