Related papers: The Light ray transform for pseudo-Euclidean metri…

For the light ray transform on globally hyperbolic Lorentzian manifolds of dimension $n+1 \geq 3$ acting on compactly supported distributions, we show that the Schwartz kernel of the normal operator is a paired Lagrangian distribution with…

We study the weighted light ray transform $L$ of integrating functions on a Lorentzian manifold over lightlike geodesics. We analyze $L$ as a Fourier Integral Operator and show that if there are no conjugate points, one can recover the…

We study the light-cone Siegel transform, transforming functions on the light cone of a rational indefinite quadratic form $Q$ to a function on the homogenous space $\text{SO}^+_Q(\mathbb{Z})\backslash \text{SO}^+_Q(\mathbb{R})$. In…

We consider light-ray operators $\mathcal{L}_{2n} = \int\mathrm{d} x^+ (x^+)^{2n}T_{++}$, where $x^+$ is a null coordinate and $n$ a positive integer, in QFT in Minkowski spacetime in arbitrary dimensions. These operators are…

We study a particular broken ray transform on the Euclidean unit square and establish injectivity and stability for $C_{0}^{2}$ perturbations of the constant unit weight. Given an open subset $E$ of the boundary, we measure the attenuation…

We study a family of convolution operators whose symbols and kernels have singularity on the light-cone in $\mathbb{R}^{n+1}$. First, we prove a desired ${\bf L}^p\to {\bf L}^q$ norm inequality which has been left open. Moreover, we obtain…

In this article, we study the properties of the geodesic X-ray transform for asymptotically Euclidean or conic Riemannian metrics and show injectivity under non-trapping and no conjugate point assumptions. We also define a notion of lens…

We study the problem of recovering a function on a pseudo-Riemannian manifold from its integrals over all null geodesics in three geometries: pseudo-Riemannian products of Riemannian manifolds, Minkowski spaces and tori. We give proofs of…

In this paper, we establish the core of singular integral theory and pseudodifferential calculus over the archetypal algebras of noncommutative geometry: quantum forms of Euclidean spaces and tori. Our results go beyond Connes'…

We study the Light-Ray transform of integrating vector fields on the Minkowski time-space R^{1+n}, n bigger than equal 2, with the Minkowski metric. We prove a support theorem for vector fields vanishing on an open set of light-like…

We study light ray transform of symmetric 2-tensor fields defined on a bounded time-space domain in $\mathbb{R}^{1+n}$ for $n\geq 3$. We prove a uniqueness result for such light ray transforms. More precisely, we characterize the kernel of…

We study left-invariant pseudo-Riemannian metrics on Lie groups using the bracket flow of the corresponding Lie algebra. We focus on metrics where the Lie algebra is in the null cone of the $G=O(p,q)$-action; i.e., Lie algebras $\mu$ where…

In this article, we study the microlocal properties of the geodesic ray transform of symmetric $m$-tensor fields on 2-dimensional Riemannian manifolds with boundary allowing the possibility of conjugate points. As is known from an earlier…

We consider the normal operator of the X-ray transform, weighted with Gaussian weights, in Euclidean space with dimension at least 3. We show the eigenfunctions of the normal operator are joint eigenfunctions of the harmonic oscillator and…

Spanners for low dimensional spaces (e.g. Euclidean space of constant dimension, or doubling metrics) are well understood. This lies in contrast to the situation in high dimensional spaces, where except for the work of Har-Peled, Indyk and…

In this article, we define Weyl transform on second countable type - $I$ locally compact group $G,$ and as an operator on $L^2(G),$ we prove that the Weyl transform is compact when the symbol lies in $L^p(G\times \hat{G})$ with $1\leq p\leq…

Given a Lorentzian manifold, the light ray transform of a function is its integrals along null geodesics. This paper is concerned with the injectivity of the light ray transform on functions and tensors, up to the natural gauge for the…

We consider restricted light ray transforms arising from an inverse problem of finding cosmic strings. We construct a relative left parametrix for the transform on two tensors, which recovers the space-like and some light-like singularities…

Let $L$ be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with $L$, such as the heat semigroup and Riesz transform, are not, in general, of…

The metaplectic transform (MT), also known as the linear canonical transform, is a unitary integral mapping which is widely used in signal processing and can be viewed as a generalization of the Fourier transform. For a given function…