1 results
Maximal operators along flat curves with one variable vector field
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 67 / Issue 4 / November 2024
- Published online by Cambridge University Press:
- 13 November 2024, pp. 1212-1228
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- Article
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We study a maximal average along a family of curves
$\{(t,m(x_1)\gamma(t)):t\in [-r,r]\}$, where 
$\gamma|_{[0,\infty)}$ is a convex function and m is a measurable function. Under the assumption of the doubling property of 
$\gamma'$ and 
$1\leqslant m(x_1)\leqslant 2$, we prove the 
$L^p(\mathbb{R}^2)$ boundedness of the maximal average. As a corollary, we obtain the pointwise convergence of the average in r > 0 without any size assumption for a measurable m.
