Professor 

I am Raghad Shamsah , researcher in  harmonic analysis and some related fields (wavelets, Fourier series). Now I am working on some results related with your paper (on the almost every where convergence of wavelet summation methods). I stopped with the following questions.

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

and
How can I define the maximal function operator when the function define on $L^2(S^2)$ ?

I hope that  you  can answer on my questions.

thanks

Raghad Shamsah
MATHEMATICAL DEPARTMENT/INSPEM/ UNIVERSITY PUTRA MALAYSIA