I think you're asking if Stein's result extends to $R^2$ for $p>2$. This was proved in

<cite authors="J. Bourgain" mrnumber="874045" cite="_J. Analyse Math._ **47** (1986), 69--85">_J. Bourgain_, MR 874045 [**Averages in the plane over convex curves and maximal operators**](http://dx.doi.org/10.1007/BF02792533), _J. Analyse Math._ **47** (1986), 69--85.</cite>

See

<cite authors="W. Schlag" mrnumber="1388870" cite="_J. Amer. Math. Soc._ **10** (1997), no. 1, 103--122">_W. Schlag_, MR 1388870 [**A generalization of Bourgain’s circular maximal theorem**](http://dx.doi.org/10.1090/S0894-0347-97-00217-8), _J. Amer. Math. Soc._ **10** (1997), no. 1, 103--122.</cite>
 
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 can 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][1], Invent. Math. 104 (1991), 349-376.


  [1]: http://link.springer.com/article/10.1007%2FBF01245080