Maximal functions associated to flat plane curves with Mitigating factors

TL;DR

This paper investigates the boundedness of maximal operators associated with flat plane curves with mitigating factors, establishing boundedness conditions in certain Lebesgue spaces based on the curve's curvature and mitigating factor.

Contribution

It provides new boundedness results for maximal operators linked to flat plane curves with mitigating factors, extending previous understanding in harmonic analysis.

Findings

Boundedness of the maximal operator in specified Lebesgue space regions.

Identification of the range of (p, q) for which the operator is bounded.

Extension of classical results to 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 $$\mathbb{M}_{\sigma}f(x) \, := \, \sup_{1 \leq t \leq 2} \left|\int_{0}^{1} f(x-t\Gamma(s)) \, (\kappa(s))^{\sigma} \, ds\right|,$$ 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)}. $