Consider the hypersurface
x
n
+
1
=
Π
1
n
x
i
α
i
{x_{n + 1}} = \Pi _1^nx_i^{{\alpha _i}}
in
R
n
+
1
{{\mathbf {R}}^{n + 1}}
. The associated maximal function operator is defined as the supremum of means taken over those parts of the surface lying above the rectangles
{
0
⩽
x
i
⩽
h
i
,
i
=
1
,
…
,
n
}
\{ 0 \leqslant {x_i} \leqslant {h_i},\;i = 1, \ldots ,n\}
. We prove that this operator is bounded on
L
p
{L^p}
for
p
>
1
p > 1
. An analogous result is proved for a quadratic surface in
R
3
{{\mathbf {R}}^3}
.