Maximal operators associated to families of flat curves in the plane

Bourgain showed that if C has non-vanishing curvature, the inequality (1.1) holds for p > 2 (see [B]). In this paper, we shall consider a situation when the curvature is allowed to vanish of finite order on a finite set of isolated points. We shall need the following definition: Definition1.1. Let C : I → R, where I is a compact interval in R, and C is smooth. We say that C is finite type if 〈(C(x)−C(x0)), μ〉 does not vanish of infinite order for any x0 ∈ I, and any unit vector μ. We shall also need a more precise definition which would specify the order of vanishing at each point. Let a0 denote a point in the compact interval I. We can always find a smooth function γ, such that in a small neighborhood of a0, C(s) = (s, γ(s)), where s ∈ I. Definition1.2. Let C be defined as before. Let C(s) = (s, γ(s)) in a small neighborhood of a0. We say that C is finite type m at a0 if γ(a0) = 0 for 1 ≤ k < m, and γ(a0) 6= 0. Our main result is the following: Theorem 1.1. Let C be a finite type curve which is finite type m at a0. Let Mtf(x) = ∫

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