Maximal functions associated to flat plane curves with Mitigating factors

We study the boundedness problem for maximal operators $\mathbb{M}_{\sigma}$ associated to flat plane curves with Mitigating factors, defined by $$\mathbb{M}_{\sigma}f(x) \, := \, \sup_{1 \leq t \leq 2} \left|\int_{0}^{1} f(x-t\Gamma(s)) \, (\kappa(s))^{\sigma} \, ds\right|,$$ where $\kappa(s)$ denotes the curvature of the curve $\Gamma(s)=(s, g(s)+1), ~g(s) \in C^5[0,1]$ in $\mathbb{R}^2$. Let $\triangle$ be the closed triangle with vertices $P=(\frac{2}{5}, \frac{1}{5}), ~ Q=(\frac{1}{2}, \frac{1}{2}), ~ R=(0, 0).$ In this paper, we prove that for $(\frac{1}{p}, \frac{1}{q}) \in \left[(\frac{1}{p}, \frac{1}{q}) :(\frac{1}{p}, \frac{1}{q}) \in \triangle \setminus \{P, Q\} \right] \cap \left[(\frac{1}{p}, \frac{1}{q}) :q > max\{\sigma^{-1},2\} \right]$, there is a constant $B$ such that $\|\mathbb{M}f\|_{L^q(\mathbb{R}^2)} \leq \, B \, \|f\|_{L^p(\mathbb{R}^2)}.$