$L^p$-boundedness for the Hilbert transform and maximal operator along a class of nonconvex curves

Some sufficient conditions on a real polynomial $P$ and a convex function $\gamma$ are given in order for the Hilbert transform and maximal operator along $(t,P(\gamma(t)))$ to be bounded on $L^p$, for all $p$ in $(1,\infty)$, with bounds independent of the coefficients of $P$. The same conclusion is shown to hold for the corresponding hypersurface in $\mathbb{R}^{d+1}$ $(d \geq 2)$ under weaker hypotheses on $\gamma$.