LOCAL RESTRICTION THEOREM AND MAXIMAL BOCHNER-RIESZ OPERATORS FOR THE DUNKL TRANSFORMS

For the Dunkl transforms associated with the weight functions h(k)(2)(x) = Pi(d)(j=1) vertical bar x(j)vertical bar(2kj), k(1), ... , k(d) >= 0 on R-d, it is proved that if p >= 2 + 1/lambda(k) and lambda(k) := d-1/2 + Sigma(d)(j=1) k(j), the maximal Bochner-Riesz operator B-*(delta) (h(k)(2); f) order delta > 0 is bounded on the space L-p(R-d; h(k)(2)dx) if and only if delta > delta(k)(p) := max{(2 lambda(k) + 1)(1/2 - 1/p) - 1/2, 0}. This extends a well known result of M. Christ for the classical Fourier transforms (Proc. Amer. Math. Soc. 95 (1985), 16-20). The proof relies on a new local restriction theorem for the Dunkl transforms, which is stronger than the corresponding global restriction theorem, but significantly more difficult to prove.