Weak type estimates for the maximal operators associated to plane curves

Abstract

For the plane curves Γ, the maximal operator M associated to it is defined by

$$Mf(x) = \mathop {\sup |}\limits_{r > 0} \int {f(x - \Gamma (t))\varphi (r^{ - 1} t)r^{ - 1} dt|} $$

\] where ϕ is a Schwartz function. For a certain class of curves in$\mathbb{R}^2$, M is shown to bounded on$(H(\mathbb{R}^2 )$, Weak$L^1 (\mathbb{R}^2 )$. This extends the theorem of Stein & Wainger^[1] and the theorem of Weinberg^[2],