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