  The goal of this article is to establish $L^{p} \rightarrow L^{q}$ estimates
for maximal functions associated with nonisotropic dilations
$\delta_t(x)=(t^{a_1}x_1,t^{a_2}x_2,t^{a_3}x_3)$ of hypersurfaces $(x_{1},
x_{2},\Phi(x_1,x_2))$ in $\mathbb{R}^3$, where the Gaussian curvatures of the
hypersurfaces are allowed to vanish. When $2 \alpha_{2} = \alpha_{3}$, this
problem is reduced to study of the $L^{p} \rightarrow L^{q}$ estimates for
maximal functions along the curve $\gamma(x)=(x,x^2(1+\phi(x)))$ and associated
dilations $\delta_t(x)=(tx_1,t^2x_2)$. The corresponding maximal function shows
features related to the Bourgain circular maximal function, whose $L^{p}
\rightarrow L^{q}$ estimate has been considered by [Schlag, JAMS, 1997],
[Schlag-Sogge, MRL, 1997] and [Lee, PAMS, 2003]. However, in the study of the
maximal function related to the mentioned curve $\gamma(x)$ and associated
dilations, we get the $L^{p} \rightarrow L^{q}$ regularity properties for a
family of corresponding Fourier integral operators which fail to satisfy the
"cinematic curvature condition" uniformly, which means that classical local
smoothing estimates could not be directly applied to our problem. What's more,
the $L^{p} \rightarrow L^{q}$ estimates are also new for maximal functions
associated with isotropic dilations of hypersurfaces $(x_{1},
x_{2},\Phi(x_1,x_2))$ mentioned before.
