I think you're asking if Stein's result extends to $R^2$ for $p>2$. This was proved by Bourgain in 1986 (see: J. Bourgain. Averages in the plane over convex curves and maximal operators. J. Analyse Math. 47 (1986), 69–85.) 

See [this paper of Schlag][1] for a discussion and more precise results.


  [1]: http://www.ams.org/journals/jams/1997-10-01/S0894-0347-97-00217-8/S0894-0347-97-00217-8.pdfi