An Inversion Formula for Horizontal Conical Radon Transform

A note on Kakeya sets of horizontal and $SL(2)$ lines

We consider unions of $SL(2)$ lines in $\mathbb{R}^{3}$. These are lines of the form $$L = (a,b,0) + \mathrm{span}(c,d,1),$$ where $ad - bc = 1$. We show that if $\mathcal{L}$ is a Kakeya set of $SL(2)$ lines, then the union $\cup \mathcal{L}$ has Hausdorff dimension $3$. This answers a question of Wang and Zahl. The $SL(2)$ lines can be identified with horizontal lines in the first Heisenberg group, and we obtain the main result as a corollary of a more general statement concerning unions of horizontal lines. This statement is established via a point-line duality principle between horizontal …

The V-line transform with some generalizations and cone differentiation

The paper studies various properties of the V-line transform (VLT) in the plane and conical Radon transform (CRT) in $\mathbb{R}^n$. VLT maps a function to a family of its integrals along trajectories made of two rays emanating from a common point. The CRT considered in this paper maps a function to a set of its integrals over surfaces of polyhedral cones. These types of operators appear in mathematical models of single scattering optical tomography, Compton camera imaging and other applications. We derive new explicit inversion formulae for VLT and CRT, as well as proving some previously know…

锥、可求长性与奇异积分算子

Cones, rectifiability, and singular integral operators

本文定义并研究了锥形能量，用于刻画测度的可求长性与Lipschitz图像的大块性质，并在测度具有多项式增长时，给出了卷积型光滑奇核奇异积分算子 $L^2(μ)$ 有界性的充分条件。