  We consider, for a class of functions $\varphi : \mathbb{R}^{2} \setminus \{
{\bf 0} \} \to \mathbb{R}^{2}$ satisfying a nonisotropic homogeneity condition,
the Fourier transform $\hat{\mu}$ of the Borel measure on $\mathbb{R}^{4}$
defined by \[ \mu(E) = \int_{U} \chi_{E}(x, \varphi(x)) \, dx \] where $E$ is a
Borel set of $\mathbb{R}^{4}$ and $U = \{ (t^{\alpha_1}, t^{\alpha_2}s) : c < s
< d, \, 0 < t < 1 \}$. The aim of this article is to give a decay estimate for
$\hat{\mu}$, for the case where the set of nonelliptic points of $\varphi$ is a
curve in $\bar{U} \setminus \{ {\bf 0} \}$. From this estimate we obtain a
restriction theorem for the usual Fourier transform to the graph of
$\varphi_{U} : U \to \mathbb{R}^{2}$. We also give $L^{p}$-improving properties
for the convolution operator $T_{\mu} f = \mu \ast f$.
