Let S S be a smooth compact hypersurface in R n {{\mathbf {R}}^n} , and let μ \mu be a measure on S S , absolutely continuous with respect to surface measure. For t t in R + , μ t {{\mathbf {R}}^ + },{\mu _t} denotes the dilate of μ \mu by t t , normalised to have the same total variation as μ \mu : for f f in S ( R n ) , μ # f \mathcal {S}({{\mathbf {R}}^n}),{\mu ^\# }f denotes the maximal function sup t > 0 | μ t ∗ f | {\sup _{t > 0}}|{\mu _t}\ast f| . We seek conditions on μ \mu which guarantee that the a priori estimate \[ ‖ μ # f ‖ p ≤ C ‖ f ‖ p , f ∈ S ( R n ) , \left \| \mu ^\# f\right \|_p \leq C\left \| f \right \|_p, \quad f \in S(\mathbf {R}^n), \] holds; this estimate entails that the sublinear operator μ # {\mu ^\# } extends to a bounded operator on the Lebesgue space L p ( R n ) {L^p}({{\mathbf {R}}^n}) . Our methods generalise E. M. Stein’s treatment of the "spherical maximal function" [5]: a study of "Riesz operators", g g -functions, and analytic families of measures reduces the problem to that of obtaining decay estimates for the Fourier transform of μ \mu . These depend on the geometry of S S and the relation between μ \mu and surface measure on S S . In particular, we find that there are natural geometric maximal operators limited on L p ( R n ) {L^p}({{\mathbf {R}}^n}) if and only if p ∈ ( q , ∞ ] ; q p \in (q,\infty ];q is some number in ( 1 , ∞ ) (1,\infty ) , and may be greater than 2 2 . This answers a question of S. Wainger posed by Stein [6]>.