Maximal functions associated to flat plane curves with Mitigating factors

Abstract

We study the boundedness problem for maximal operators ${\mathbb{M}}_{\sigma }$ associated to flat plane curves with Mitigating factors, defined by $${\mathbb{M}}_{\sigma }f(x):=\underset{1\le t\le 2}{\sup }\left|{\int }_0^{1}f(x-t\Gamma (s))(\kappa (s){)}^{\sigma }ds\right|,$$ 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 $△$ 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 $(\frac{1}{p},\frac{1}{q})\in \left[(\frac{1}{p},\frac{1}{q}):(\frac{1}{p},\frac{1}{q})\in △\setminus \{P,Q\}\right]\cap \left[(\frac{1}{p},\frac{1}{q}):q>max\{{\sigma }^{-1},2\}\right]$, there is a constant $B$ such that $\Vert \mathbb{M}f{\Vert }_{L^q({\mathbb{R}}^2)}\le B\Vert f{\Vert }_{L^p({\mathbb{R}}^2)}.$

16 pages