Noncommutative pointwise convergence of orthogonal expansions of several variables

In this paper, the boundedness of the maximal function for an operator‐valued weighted L 1 space on the unit sphere of R d + 1 , in which the weight functions are invariant under finite reflection groups, is established. We use it to prove the boundedness of orthogonal expansions in h ‐harmonics and this result applies to various methods of summability. Furthermore, we obtain the corresponding pointwise convergence theorems. At last, we give some results in the special reflection group Z 2 d .