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.