The purpose of this announcement is to describe a development given in a series of forthcoming papers by the authors that concern operators of the form \[ f\mapsto ψ(x) \int f(γ_t(x)) K(t)\: dt, \] where $γ_t(x)=γ(t,x)$ is a $C^\infty$ function defined on a neighborhood of the origin in $(t,x)\in \mathbb{R}^N\times \mathbb{R}^n$ satisfying $γ_0(x)\equiv x$, $K(t)$ is a "multi-parameter singular kernel" supported near $t=0$, and $ψ$ is a cutoff function supported near $x=0$. This note concerns the case when $K$ is a "product kernel". The goal is to give conditions on $γ$ such that the above operator is bounded on $L^p$ for $1<p<\infty$. Associated maximal functions are also discussed. The "single-parameter" case when $K$ is a Calderón-Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger. The theory here extends these results to the multi-parameter context and also deals effectively with the case when $γ$ is real-analytic.
We prove sharp $L^2$ regularity results for classes of strongly singular Radon transfoms on the Heisenberg group by means of oscillatory integrals. We show that the problem in question can be effectively treated by establishing uniform estimates for certain oscillatory integrals whose canonical relations project with two-sided fold singularities; this new approach also allows us to treat operators which are not necessarily translation invariant.
Singularities of the Radon transform of a piecewise smooth function $f(x)$, $x\in R^n$, $n\geq 2$, are calculated. If the singularities of the Radon transform are known, then the equations of the surfaces of discontinuity of $f(x)$ are calculated by applying the Legendre transform to the functions, which appear in the equations of the discontinuity surfaces of the Radon transform of $f(x)$; examples are given. Numerical aspects of the problem of finding discontinuities of $f(x)$, given the discontinuities of its Radon transform, are discussed.
We consider a class of operators defined by taking averages along polynomial sequences in discrete nilpotent groups. In this paper we prove $L^2$ boundedness of discrete singular Radon transforms along general polynomial sequences in discrete nilpotent groups of step 2.
We show that some singular maximal functions and singular Radon transforms satisfy a weak type $L\log\log L$ inequality. Examples include the maximal function and Hilbert transform associated to averages along a parabola. The weak type inequality yields pointwise convergence results for functions which are locally in $L\log\log L$.
In this paper we consider three types of discrete operators stemming from singular Radon transforms. We first extend an $\ell^p$ result for translation invariant discrete singular Radon transforms to a class of twisted operators including an additional oscillatory component, via a simple method of descent argument. Second, we note an $\ell^2$ bound for quasi-translation invariant discrete twisted Radon transforms. Finally, we extend an existing $\ell^2$ bound for a closely related non-translation invariant discrete oscillatory integral operator with singular kernel to an $\ell^p$ bound for all…
We show that discrete singular Radon transforms along a certain class of polynomial mappings $P:\mathbb{Z}^d\to \mathbb{Z}^n$ satisfy sparse bounds. For $n=d=1$ we can handle all polynomials. In higher dimensions, we pose restrictions on the admissible polynomial mappings stemming from a combination of interacting geometric, analytic and number-theoretic obstacles.
In this paper we prove uniform oscillation estimates on $L^p$, with $p\in(1,\infty)$, for truncated singular integrals of the Radon type associated with Calderón-Zygmund kernel, both in continuous and discrete settings. In the discrete case we use the Ionescu-Wainger multiplier theorem and the Rademacher-Menshov inequality to handle the number-theoretic nature of the discrete singular integral. The result we obtained in the continuous setting can be seen as a generalisation of the results of Campbell, Jones, Reinhold and Wierdl for the continuous singular integrals of the Calderón-Zygmund type…
The purpose of this paper is to prove the L^p boundedness of singular Radon transforms and their maximal analogues. These operators differ from the traditional singular integrals and maximal functions in that their definition at any point x in R^n involves integration over a k-dimensional submanifold of R^n, depending on x, with k < n. The role of the underlying geometric data which determines the submanifolds and how they depend on x, makes the analysis of these operators quite different from their standard analogues. In fact, much of our work is involved in the examination of the resulting g…
In this paper we investigate the mapping properties in Lebesgue-type spaces of certain generalized Radon transforms defined by integration over curves.
In this paper, we consider the conical Radon transform on all cones with horizontal central axis whose vertices are on a straight line. We derive an explicit inversion formula for such transform. The inversion makes use of the vertical slice transform on a sphere and V-line transform on a plane.
We prove a Calderón-Zygmund type estimate which can be applied to sharpen known regularity results on spherical means, Fourier integral operators and generalized Radon transforms.
The Radon transform is a bounded operator from L^p of Euclidean space R^d to L^q of the Grassmann manifold of all affine hyperplanes in R^d, for certain exponents. We identify all extremizers of the associated inequality for the endpoint case p=(d+1)/d and q=d+1.
In the paper [G1] the author proved $L^p$ Sobolev regularity results for averaging operators over hypersurfaces and connected them to associated Newton polyhedra. In this paper, we use rather different resolution of singularities techniques to prove $L^p$ Sobolev regularity results for a class of averaging operators over surfaces which can be of any codimension.
We study integral transforms mapping a function on the Euclidean space to the family of its integration on some hypersurfaces, that is, a function of hypersurfaces. The hypersurfaces are given by the graphs of functions with fixed axes of the independent variables, and are imposed some symmetry with respect to the axes. These transforms are higher dimensional version of generalization of the parabolic Radon transform and the hyperbolic Radon transform arising from seismology. We prove the inversion formulas for these transforms under some vanishing and symmetry conditions of functions.
A set in the Euclidean plane is constructed whose image under the classical Radon transform is Lipschitz in every direction. It is also shown that, under mild hypotheses, for any such set the function which maps a direction to the corresponding Lipschitz constant cannot be bounded.
The monogenic Hua-Radon transform is defined as an orthogonal projection on holomorphic functions in the Lie sphere. Extending the work of Sabadini and Sommen, J. Geom. Anal. 29 (2019), 2709-2737, we determine its reproducing kernel. Integrating this kernel over the Stiefel manifold yields a linear combination of the zonal spherical monogenics. Using the reproducing properties of those monogenics we obtain an inversion for the monogenic Hua-Radon transform.
In this article we study the spherical mean Radon transform in $\mathbf R^3$ with detectors centered on a plane. We use the consistency method suggested by the author of this article for the inversion of the transform in 3D. A new iterative inversion formula is presented. This formula has the benefit of being local and is suitable for practical reconstructions. The inversion of the spherical mean Radon transform is required in mathematical models in thermo- and photo-acoustic tomography, radar imaging, and others.
We establish a mixed norm estimate for the Radon transform in the plane when the set of directions has fractional dimension. This estimate is used to prove a result about an exceptional set of directions connected with projections of planar sets. That leads to a conjecture analogous to a well-known conjecture of Furstenberg.
In this article, we characterize the strength of the reconstructed singularities and artifacts in a reconstruction formula for limited data spherical Radon transform. Namely, we assume that the data is only available on a closed subset $Γ$ of a hyperplane in $\mathbb{R}^n$ ($n=2,3$). We consider a reconstruction formula studied in some previous works, under the assumption that the data is only smoothened out to a finite order $k$ near the boundary. For the problem in the two dimensional space and $Γ$ is a line segment, the artifacts are generated by rotating a boundary singularity along a circ…
We characterize (up to endpoints) the $k$-tuples $(p_1,\ldots,p_k)$ for which certain $k$-linear generalized Radon transforms map $L^{p_1} \times \cdots \times L^{p_k}$ boundedly into $\mathbb R$. This generalizes a result of Tao and Wright.
Convolution with an appropriate surface measure on a paraboloid in R^d defines a bounded operator T from L^p(R^d) to L^q(R^d) for certain exponents p,q. In this article it is proved that there exist functions which extremize the associated inequality.
We present the abstract framework and some applications of interpolation theory. The main new result concerns interpolation between H^1 and L^p estimates for analytic families of operators acting on Schwartz functions.
We study operators of the form $$ Tf(x)= ψ(x) \int f(γ_t(x))K(t)\,dt, $$ where $γ_t(x)$ is a real analytic function of $(t,x)$ mapping from a neighborhood of $(0,0)$ in $\mathbb{R}^N \times \mathbb{R}^n$ into $\mathbb{R}^n$ satisfying $γ_0(x)\equiv x$, $ψ(x) \in C_c^\infty(\mathbb{R}^n)$, and $K(t)$ is a "multi-parameter singular kernel" with compact support in $\mathbb{R}^N$; for example when $K(t)$ is a product singular kernel. The celebrated work of Christ, Nagel, Stein, and Wainger studied such operators with smooth $γ_t(x)$, in the single-parameter case when $K(t)$ is a Calderón-Zygmund k…
This paper gives a complete geometric characterization in all dimensions and codimensions of those Radon-like transforms which, up to endpoints, satisfy the largest possible range of local $L^p \rightarrow L^q$ inequalities permitted by quadratic-type scaling. The necessary and sufficient curvature-type criterion is phrased in terms of an associated Newton-like diagram. In the case of averages over families of polynomial graphs, the curvature condition implies sharp endpoint estimates as well. The proof relies on the recently-developed multilinear Radon-Brascamp-Lieb testing criterion and a re…
We show $\ell^p\big(\mathbb Z^d\big)$ boundedness, for $p\in(1, \infty)$, of discrete singular integrals of Radon type with the aid of appropriate square function estimates, which can be thought as a discrete counterpart of the Littlewood--Paley theory. It is a very powerful approach which allows us to proceed as in the continuous case.
If the Radon transform of a compactly supported distribution $f \ne 0$ in $\mathbb R^n$ is supported on the set of tangent planes to the boundary $\partial D$ of a bounded convex domain $D$, then $\partial D$ must be an ellipsoid. As a corollary of this result we get a new proof of a recent theorem of Koldobsky, Merkurjev, and Yaskin, which settled a special case of a conjecture of Arnold that was motivated by a famous lemma of Newton.
We prove several variations on the results of Ricci and Travaglini concerning bounds for convolution with all rotations of a measure supported by a fixed convex curve in the plane. Estimates are obtained for averages over higher-dimensional convex hypersurfaces, smooth k-dimensional surfaces and non-translation invariant families of surfaces.
The purpose of this paper is to study the sparse bound of the operator of the form $f \mapsto ψ(x) \int f(γ_t(x))K(t)dt$, where $γ_t(x)$ is a $C^\infty$ function defined on a neighborhood of the origin in $(x, t) \in \mathbb R^n \times \mathbb R^k$, satisfying $γ_0(x) \equiv x$, $ψ$ is a $C^\infty$ cut-off function supported on a small neighborhood of $0 \in \mathbb R^n$ and $K$ is a Calderón-Zygmund kernel suppported on a small neighborhood of $0 \in \mathbb R^k$. Christ, Nagel, Stein and Wainger gave conditions on $γ$ under which $T: L^p \mapsto L^p (1<p<\infty)$ is bounded. Under the these …
The Hua-Radon and polarized Hua-Radon transform are two orthogonal projections defined on holomorphic functions in the Lie sphere. Both transformations can be written as integral transforms with respect to a suitable reproducing kernel. Integrating both kernels over a Stiefel manifold yields a linear combination of zonal spherical monogenics. Using an Almansi type decomposition of holomorphic functions and reproducing properties of the zonal monogenics, we obtain an inversion formula for both the Hua-Radon and the polarized Hua-Radon transform.
We establish new relations which connect Euclidean sonar transforms (integrals taken over spheres with centers in a hyperplane) with classical Radon transforms. The relations, stated as operator identities, allow us to reduce the inversion of sonar transforms to classical Radon inversion.
Generalizing Lemma 28 from Newton's ``Principia", Arnold asked for a complete characterization of algebraically integrable domains. In this paper we describe the current state of Arnold's problems. We also consider closely related problems about the Radon transform of indicator functions.
We study integral transforms mapping a function on the Euclidean plane to the family of its integration on plane curves, that is, a function of plane curves. The plane curves we consider in the present paper are given by the graphs of functions with a fixed axis of the independent variable, and are imposed some symmetry with respect to the axes. These transforms contain the parabolic Radon transform and the hyperbolic Radon transform arising from seismology. We prove the inversion formulas for these transforms under some vanishing and symmetry conditions of functions.
The spherical Radon-Dunkl transform $R_κ$, associated to weight functions invariant under a finite reflection group, is introduced, and some elementary properties are obtained in terms of $h$-harmonics. Several inversion formulas of $R_κ$ are given with the aid of spherical Riesz-Dunkl potentials, the Dunkl operators, and some appropriate wavelet transforms.
In this article, we prove a stability estimate going from the Radon transform of a function with limited angle-distance data to the $L^p$ norm of the function itself, under some conditions on the support of the function. We apply this theorem to obtain stability estimates for an inverse boundary value problem with partial data.
We suggest new modifications of Helgason's support theorems and descriptions of the kernels for several projectively equivalent transforms of integral geometry. The paper deals with the hyperplane Radon transform and its dual, the totally geodesic transforms on the sphere and the hyperbolic space, the spherical slice transform, and the spherical mean transform for spheres through the origin. The assumptions for functions are formulated in integral terms. The proofs rely on the properties of the Gegenbauer-Chebyshev integrals which generalize Abel type fractional integrals on the positive half-…
Let $\mR$ be the restriction of the spherical Radon transform to the set of spheres centered on a hypersurface $\mS$. We study the inversion of $\mR$ by a closed-form formula. We approach the problem by studying an oscillatory integral, which depends on the observation surface $\mS$ as a parameter. We then derive various microlocal analytic properties of the associated closed-form inversion formula.
