I am working on some results related with a paper of Elias Stein (on the almost every where convergence of wavelet summation methods), and I have the following questions:

1. The maximal function operator of $f$ in the Elias Stein paper (1976) is bounded on $L^p({\mathbb R}^ n)$, whenever $p > n/(n - 1)$, and
$n \ge 3$. Does it stay bounded when we define the maximal function on the new space $L^2(S^2)$?

and

2. How can I define the maximal function operator when the function is in $L^2(S^2)$?