Maximal operators and Hilbert transforms along variable non-flat homogeneous curves

We prove that the maximal operator associated with variable homogeneous planar curves (t,ut ^α ) _t∈ℝ , α≠1 positive, is bounded on L ^p (ℝ ² ) for each p>1, under the assumption that u:ℝ ² → ℝ is a Lipschitz function. Furthermore, we prove that the Hilbert transform associated with (t,ut ^α ) _t∈ℝ , α≠1 positive, is bounded on L ^p (ℝ ² ) for each p>1, under the assumption that u:ℝ ² →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.