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

We prove that the maximal operator associated with variable homogeneous planar curves $(t,u{t}^{\alpha }{)}_{t\in \mathbb{R}}$, $\alpha ≠1$ 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$positive,isboundedon$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, $T{T}^{\ast }$ arguments, local smoothing estimates and a pointwise estimate for taking averages along curves.