$L^ 2$-boundedness of spherical maximal operators with multidimensional parameter sets

Abstract:

For $s > 0$, let ${M_s}f(x) = \int _{|y| = 1} {f(x - sy)d\sigma (y)}$ be the spherical mean operator on ${R^n}$. For a certain class of surfaces $S$ in $R_ + ^{n + 1}$ with $\dim S = n - 2$ or $\dim S = n - 1$ with an additional condition, the maximal operator \[ \mathcal {M}f(x) = \sup \limits _{(u,s) \in S} |{M_s}f(x - u)|\] is shown to be bounded on ${L^2}({R^n})$. This extends (on ${L^2}({R^n})$) the theorem of Stein [7], where $S = \{ (0,s):s > 0\}$, and its generalizations to $\dim S = 1$ in Greenleaf [2] and Sogge and Stein [6].