  We prove that the maximal operator associated with variable homogeneous
planar curves $(t, u t^{\alpha})_{t\in \mathbb{R}}$, $\alpha\not=1$ positive,
is bounded on $L^p(\mathbb{R}^2)$ for each $p>1$, under the assumption that
$u:\mathbb{R}^2 \to \mathbb{R}$ is a Lipschitz function. Furthermore, we prove
that the Hilbert transform associated with $(t, ut^{\alpha})_{t\in
\mathbb{R}}$, $\alpha\not=1$ positive, is bounded on $L^p(\mathbb{R}^2)$ for
each $p>1$, under the assumption that $u:\mathbb{R}^2\to \mathbb{R}$ is a
measurable function and is constant in the second variable. Our proofs rely on
stationary phase methods, $TT^*$ arguments, local smoothing estimates and a
pointwise estimate for taking averages along curves.
