Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userRay transform on Sobolev spaces of symmetric tensor fields, I: Higher order Reshetnyak formulas
143
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
For an integer $rge0$, we prove the $r$th order Reshetnyak formula for the ray transform of rank $m$ symmetric tensor fields on $mathbb{R}^n$. Certain differential operators $A^{(m,r,l)} (0le lle r)$ on the sphere $mathbb{S}^{n-1}$ are main ingredients of the formula. The operators are defined by an algorithm that can be applied for any $r$ although the volume of calculations grows fast with $r$. The algorithm is realized for small values of $r$ and Reshetnyak formulas of orders $0,1,2$ are presented in an explicit form.
rate research
Read More
By using, among other things, the Fourier analysis techniques on hyperbolic and symmetric spaces, we establish the Hardy-Sobolev-Mazya inequalities for higher order derivatives on half spaces. The proof relies on a Hardy-Littlewood-Sobolev inequality on hyperbolic spaces which is of its independent interest. We also give an alternative proof of Benguria, Frank and Loss work concerning the sharp constant in the Hardy-Sobolev-Mazya inequality in the three dimensional upper half space. Finally, we show the sharp constant in the Hardy-Sobolev-Mazya inequality for bi-Laplacian in the upper half space of dimension five coincides with the Sobolev constant.
We study the problem of inverting a restricted transverse ray transform to recover a symmetric $m$-tensor field in $mathbb{R}^3$ using microlocal analysis techniques. More precisely, we prove that a symmetric $m$-tensor field can be recovered up to a known singular term and a smoothing term if its transverse ray transform is known along all lines intersecting a fixed smooth curve satisfying the Kirillov-Tuy condition.
We study {em $ abla$-Sobolev spaces} and {em $ abla$-differential operators} with coefficients in general Hermitian vector bundles on Riemannian manifolds, stressing a coordinate free approach that uses connections (which are typically denoted $ abla$). These concepts arise naturally from Partial Differential Equations, including some that are formulated on plain Euclidean domains, such as the weighted Sobolev spaces used to study PDEs on singular domains. We prove several basic properties of the $ abla$-Sobolev spaces and of the $ abla$-differential operators on general manifolds. For instance, we prove mapping properties for our differential operators and independence of the $ abla$-Sobolev spaces on the choices of the connection $ abla$ with respect to totally bounded perturbations. We introduce a {em Frechet finiteness condition} (FFC) for totally bounded vector fields, which is satisfied, for instance, by open subsets of manifolds with bounded geometry. When (FFC) is satisfied, we provide several equivalent definitions of our $ abla$-Sobolev spaces and of our $ abla$-differential operators. We examine in more detail the particular case of domains in the Euclidean space, including the case of weighted Sobolev spaces. We also introduce and study the notion of a {em $ abla$-bidifferential} operator (a bilinear version of differential operators), obtaining results similar to those obtained for $ abla$-differential operators. Bilinear differential operators are necessary for a global, geometric discussion of variational problems. We tried to write the paper so that it is accessible to a large audience.
Motivated by a recent work of Ache and Chang concerning the sharp Sobolev trace inequality and Lebedev-Milin inequalities of order four on the Euclidean unit ball, we derive such inequalities on the Euclidean unit ball for higher order derivatives. By using, among other things, the scattering theory on hyperbolic spaces and the generalized Poisson kernel, we obtain the explicit formulas of extremal functions of such inequations. Moreover, we also derive the sharp trace Sobolev inequalities on half spaces for higher order derivatives. Finally, we compute the explicit formulas of adapted metric, introduced by Case and Chang, on the Euclidean unit ball, which is of independent interest.
In this paper we study the attenuated $X$-ray transform of 2-tensors supported in strictly convex bounded subsets in the Euclidean plane. We characterize its range and reconstruct all possible 2-tensors yielding identical $X$-ray data. The characterization is in terms of a Hilbert-transform associated with $A$-analytic maps in the sense of Bukhgeim.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
