In this paper, we consider operators of the following form, acting on functions on R 2 : (1.1) Here φ(x, t) is a smooth function supported on a small neighborhood of the origin in R 2 × R with φ(0, 0) = 0, and γ(x, t) is a smooth function defined on the support of φ(x, t) satisfying is the average of f over a curve "centered at x". The condition ∂γ ∂t (x, t) = 0 ensures that the averaging is smooth; T doesn't degenerate into a fractional or singular Radon transform. Our goal will be to prove

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... rp L 2 estimates for T . In the semitranslationinvariant case, this has been done for real-analytic γ(x, t) by Phong and Stein [PS], and for general γ(x, t) (not just semi-translation invariant) this was done up to derivatives by Seeger [Se]. In this paper we will relate L 2 regularity of T to uniform sublevel set estimates for a certain determinant function that arises. The estimates will be sharp for a significant class of T many of which are not semitranslation-invariant; for such operators the results of this paper are not covered by [Se] or [PS]. There has also been quite a bit of work on sharp L 2 estimates for Radon transforms along curves in higher dimensions, such as that of Greenleaf and Seeger [GrSe1]-[GrSe5], and Comech and Cuccagna [CoCu1]-[CoCu2]. For L p to L q estimates there have also been a number of papers written; the author is most familiar with those of Christ [Ch], Oberlin [O1]-[O3], and [G2]. We refer the reader to [GrSe5] for more details on these and various other papers on related topics.