We show how the techniques introduces by Christ can be employed to derive endpoint $L^p-L^q$ bounds for the X-ray transform associated to the line complex generated by the curve $t\to(t,t^2,...,t^{d-1}).$ Almost-sharp Lorentz space estimates are produced as well.Comment: 11 page

We obtain L 2 bounds for strongly singular integral operators along curves in ‫ޒ‬ d . Our results both generalize and extend to higher dimensions the planar results of Chandarana. In addition, we show that these operators are bounded from L log L to weak L 1 at the critical exponent α = 0.

Abstract. In this article we study the behavior of strongly singular integrals associated to three different, albeit equivalent, quasi-norms on Heisenberg groups; these quasi-norms give rise to phase functions whose mixed Hessians may or may not drop rank along suitable varieties. In the particular case of the Koranyi norm we improve on the arguments in [7] and obtain sharp L 2 estimates for the associated operators.

Abstract. We prove sharp L 2 regularity results for classes of strongly singular Radon transforms on the Heisenberg group by means of oscillatory integrals. We show that the problem in question can be effectively treated by establishing uniform estimates for certain oscillatory integrals whose canonical relations project with two-sided fold singularities; this new approach also allows us to treat operators which are not necessarily translation invariant.

We prove sharp L p → L q estimates for averaging operators along general polynomial curves in two and three dimensions. These operators are translation-invariant, given by convolution with the so-called affine arclength measure of the curve and we obtain universal bounds over the class of curves given by polynomials of bounded degree. Our method relies on a geometric inequality for general vector polynomials together with a combinatorial argument due to M. Christ. Almost sharp Lorentz space estimates are obtained as well.