Mixed-norm estimates for a restricted X-ray transform in ℝ4 and ℝ5

Abstract

The Radon transform found important applications in integral geometry and in the study of PDE’s. R1,d is often called the X-ray transform due to its applications in radiology, we denote it by Xfull. It is well-known [19], [12] that the sharp mixed norm estimates for the full X-ray transform implies the Kakeya conjecture and it is related to some of the main problems in the summability of Fourier transform, Fourier restriction and more generally to oscillatory integrals, non-linear P.D.E’s and number theory [7], [1], [2], [20], [3], [14]. For some mapping properties of Xfull, see, e.g., [6], [5], [19] and [12]. Note that G1,d is a 2d− 2-dimensional manifold, thus Xfull is overdetermined for d ≥ 3, and it is of interest to consider its restrictions to lower dimensional subspaces of G1,d. For the definition of the restricted X-ray transforms as part of a more general class of transformations and some of its properties, see [11]. One particular example is the restriction of Xfull to the space of light rays (lines in R making a 45 degree angle with the plane xd = 0). Recently, Wolff [21] obtained mixed norm estimates for this operator (almost sharp in R) and used this information to prove almost sharp bilinear cone restriction estimates in all dimensions. We are interested in the restriction of Xfull to d dimensional line complexes in R. Let d ≥ 3; the subspace Gd of G1,d we are interested in is defined as follows: Let γd be the curve {γd(t) : γd(t) = (1, t, t, ..., td−1), t ∈ (−1, 1)} in R. Let l(t, x) denote