Stability of Sublevel Set Estimates and Sharp L Regularity of Radon Transforms in the Plane

Thus Tf(x) is the average of f over a curve ”centered at x”. The condition ∂γ ∂t (x, t) 6= 0 ensures that the averaging is smooth; T doesn’t degenerate into a fractional or singular Radon transform. Our goal will be to prove sharp L 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 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].