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]>.