A method for proving Lp-boundedness of singular Radon transforms in codimension 1

𝐿^{𝑝} improving bounds for averages along curves

“…• It seems likely that one could extend these results (with the aid of the results in [8] or [12]) to obtain good (L p1 , L p 2 ) mapping properties for fractional with k odd, then one easily sees (from a simple change of variables) that T α falls within the class of operators described by (2) where the corresponding vector fields X 1 and X 2 are now linearly dependent at the origin.…”

“…While algebraic varieties do have some structure which can be exploited (for instance, their intersections with other algebraic sets have a bounded number of connected 5 Our argument follows the iteration ideas of Christ [6] (which in turn were inspired by the work of Bourgain, Wolff, and Schlag on Kakeya-type problems) adapted to the general vector field setting. Of course, the idea of iterating an operator supported mostly on one-dimensional sets to increase the dimension of the support is hardly new (see, e.g., [4], [9], [5], [12]); our main innovation is the utilization of the concept of width introduced in the previous section.…”

Singular Radon transforms and maximal functions under convexity assumptions

“…Under very general finite type conditions the L p boundedness of M and H has been proved by Christ, Nagel, Stein and Wainger [7] (see also Greenblatt [11]). Thus we are mainly interested in the flat case.…”

A T(1) theorem for singular Radon transforms

“…We will thereby prove an analogue for singular Radon transforms to the T (1) theorem of David and Journe [4]. As in [6,7], we will associate a singular integral operator to a singular Radon transform. The crux of this paper consists of showing that the difference of the singular integral operator and the singular Radon transform is bounded on L 2 , analogous to [6,7] and their predecessors.…”

“…The purpose of this paper is to extend and clarify the methods of the papers [6,7], using the methods of the author [8] and developing ideas in Carnot-Caratheodory geometry including some based on the fundamental paper [14]. We will thereby prove an analogue for singular Radon transforms to the T (1) theorem of David and Journe [4].…”