$L^p$-improving properties of X-ray like transforms

Abstract

The purpose of this paper is to prove essentially sharp L-L estimates for nondegenerate one-dimensional averaging operators which generalize the classical X-ray transform. Let X and Y be C∞ manifolds with dimX =: dX and dimY =: dY ; we assume that X and Y are equipped with measures of smooth density and that dY > dX . Now let M be a smooth (dY + 1)-dimensional submanifold of X × Y (again equipped with a measure) such that the natural projections πX : M → X and πY : M → Y have everywhere surjective differential maps. For y ∈ Y , the set γy := {x ∈ X | (x, y) ∈ M } is a curve in X. As will be shown in the next section, there is an induced Radon-like operator R which averages functions of X over the curves γy. The focus of this paper is to study the L-boundedness of that operator. For simplicity, the question is posed as a bilinear one: for which p, q′ does there exist a finite constant Cp,q′ such ∣∣∣∣∫ fX(πX(m))fY (πY (m))dm∣∣∣∣ ≤ Cp,q′ (∫ |fX(x)|pdx) 1 p (∫ |fY (y)|q′dy) 1 q′

Locations

• Mathematical Research Letters - View - PDF