$L^p$ estimates for Hilbert transform and maximal operator associated to variable polynomial

We investigate the Hilbert transform and the maximal operator along a class of variable non-flat polynomial curves $(P(t),u(x)t)$ with measurable $u(x)$, and prove uniform $L^p$ estimates for $1<p<\infty$. In particular, via the change of variable, these uniform estimates are equal to the ones for the curves $(P(v(x)t),t)$ with measurable $v(x)$. To obtain the desired bound, we make full use of time-frequency techniques and establish a crucial $\epsilon$-improving estimate for some special separate sets.