$L^{p}$ boundedness of maximal averages over hypersurfaces in $\mathbb {R}^{3}$

Abstract:

Extending the methods developed in the author’s recent paper and using some techniques from a paper by Sogge and Stein in conjunction with various facts about adapted coordinate systems in two variables, an $L^p$ boundedness theorem is proven for maximal operators over hypersurfaces in $\mathbb {R}^3$ when $p > 2.$ When the best possible $p$ is greater than $2$, the theorem typically provides sharp estimates. This gives another approach to the subject of recent work of Ikromov, Kempe, and Müller (2010).