Convolution and Fourier restriction estimates for measures on curves in R 2

We study convolution and Fourier restriction estimates for some degenerate curves in R . Notational comment: This note concerns certain operators de ned on functions on R . Thus f will always denote an appropriate function on R , L will usually mean the L space constructed with Lebesgue measure m2 on R , and k kp stands for the norm in L . The following two theorems are well known and are prototypical for many important results in harmonic analysis: Theorem 1. Suppose a < b and write Tf(x) = R b a f x (t; t) dt. Then there is a constant C such that kTfk3 C kfk 3 2 : Theorem 2. If 1 p < 4 3 and 1 q = 3(1 1 p ), there is a constant C = C(p) such that the estimate Z b a j b f(t; t)jdt 1 q C(p) kfkp holds. It is natural to wonder what happens to Theorems 1 and 2 when the curve (t; t) is replaced by a general t; (t) . Since the curvature of the parabola is key to the proofs of Theorems 1 and 2, a reasonable starting point for generalization is the hypothesis 00 Æ > 0. And it has been known for