Local restriction theorem and maximal Bochner-Riesz operators for the Dunkl transforms

Abstract:

For the Dunkl transforms associated with the weight functions $h_\kappa ^2(x)=\prod _{j=1}^d |x_j|^{2\kappa _j}$, $\kappa _1,\cdots , \kappa _d\ge 0$ on $\mathbb {R}^d$, it is proved that if $p\ge 2+\frac 1{\lambda _\kappa }$ and $\lambda _\kappa :=\frac {d-1}2+\sum _{j=1}^d\kappa _j$, the maximal Bochner-Riesz operator $B_\ast ^\delta (h_\kappa ^2; f)$ order $\delta >0$ is bounded on the space $L^p(\mathbb {R}^d; h_\kappa ^2dx)$ if and only if $\delta >\delta _\kappa (p):=\max \{(2\lambda _\kappa +1) (\frac 12-\frac 1p)-\frac 12,0\}$. 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.