Two questions on Elias Stein paper (1976) 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:


*

*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


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

 A: I think you're asking if Stein's result extends to $R^2$ for $p>2$. This was proved in
J. Bourgain, MR 874045 Averages in the plane over convex curves and maximal operators, J. Analyse Math. 47 (1986), 69--85.
See
W. Schlag, MR 1388870 A generalization of Bourgain’s circular maximal theorem, J. Amer. Math. Soc. 10 (1997), no. 1, 103--122.
for a discussion and more precise results.
Edit: Following Terry's comment/interpretation, it may be that the OP is asking about the operator that maps functions on the sphere $S^2$ to the sphere by taking the maximal average of circles on the sphere centered at a given point.
This should be be bounded for $p>2$. Heuristically one would like to stereographically project the problem from (a cap on) $S^2$ back to $R^2$ and then apply Bourgain's theorem. The complication is that while a stereographic projection takes circles to circles, it will not preserve the property of a circle being centered at a given point. However the family of circles centered at a given point under the preimage of the stereographic projection should satisfy Sogge's cinematic curvature condition, and thus the result should follow from C.D.Sogge, Propagation of singularities and maximal functions in the plane, Invent. Math. 104 (1991), 349-376.
