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 geometric situation, and the elucidation of an attached notion of curvature (a kind of ``finite-type'' condition) which is crucial for our analysis.
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 $L\log\log L$.
We prove $\ell^p\big(\mathbb Z^d\big)$ bounds, for $p\in(1, \infty)$, of discrete maximal functions corresponding to averaging operators and truncated singular integrals of Radon type, and their applications to pointwise ergodic theory. Our new approach is based on a unified analysis of both types of operators, and also yields an extension to the vector-valued form of these results.
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.
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…
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.
We study a generalized spherical means operator, viz. generalized spherical mean Radon transform, acting on radial functions. As the main results, we find conditions for the associated maximal operator and its local variant to be bounded on power weighted Lebesgue spaces. This translates, in particular, into almost everywhere convergence to radial initial data results for solutions to certain Cauchy problems for classical Euler-Poisson-Darboux and wave equations. Moreover, our results shed some new light to the interesting and important question of optimality of the yet known $L^p$ boundedness…
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 aim of this paper is to develop bootstrap arguments to establish maximal, oscillation, variational and jump inequalities for the discrete averaging Radon operators on $\ell^p(\mathbb Z^d)$.
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…
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 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…
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 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 ope…
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.
The goal of this paper is to study operators of the form, \[ Tf(x)= ψ(x)\int f(γ_t(x))K(t)\: dt, \] where $γ$ is a real analytic function defined on a neighborhood of the origin in $(t,x)\in \R^N\times \R^n$, satisfying $γ_0(x)\equiv x$, $ψ$ is a cutoff function supported near $0\in \R^n$, and $K$ is a "multi-parameter singular kernel" supported near $0\in \R^N$. A main example is when $K$ is a "product kernel." We also study maximal operators of the form, \[ \mathcal{M} f(x) = ψ(x)\sup_{0<δ_1,..., δ_N<<1} \int_{|t|<1} |f(γ_{δ_1 t_1,...,δ_N t_N}(x))|\: dt. \] We show that $\mathcal{M}$ is boun…
We add to the literature the following observation. If $μ$ is a singular measure on $\mathbb{R}^n$ which assigns measure zero to every porous set and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lipschitz function which is non-differentiable $μ$-a.e. then for every $C^1$ function $g:\mathbb{R}^n\rightarrow\mathbb{R}$ it holds $$μ\{x\in\mathbb{R}^n: f(x)=g(x)\}=0.$$ In other words the Lusin type approximation property of Lipschitz functions with $C^1$ functions does not hold with respect to a general Radon measure.
The purpose of this paper is to study the $L^p$ boundedness of operators 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 $(t,x)\in \R^N\times \R^n$, satisfying $γ_0(x)\equiv x$, $ψ$ is a $C^\infty$ cutoff function supported on a small neighborhood of $0\in \R^n$, and $K$ is a "multi-parameter singular kernel" supported on a small neighborhood of $0\in \R^N$. We also study associated maximal operators. The goal is, given an appropriate class of kernels $K$, to give conditions on $γ$ such that every opera…
The purpose of this paper is to study the $L^2$ boundedness of operators 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 $(t,x)\in \R^N\times \R^n$, satisfying $γ_0(x)\equiv x$, $ψ$ is a $C^\infty$ cutoff function supported on a small neighborhood of $0\in \R^n$, and $K$ is a "multi-parameter singular kernel" supported on a small neighborhood of $0\in \R^N$. The goal is, given an appropriate class of kernels $K$, to give conditions on $γ$ such that every operator of the above form is bounded on $L^2$. Th…
This paper may be viewed as a companion paper to [G1]. In that paper, $L^2$ Sobolev estimates derived from a Newton polyhedron-based resolution of singularities method are combined with interpolation arguments to prove $L^p$ to $L^q_s$ estimates, some sharp up to endpoints, for translation invariant Radon transforms over hypersurfaces and related operators. Here $q \geq p$ and $s$ can be positive, negative, or zero. In this paper, we instead use $L^2$ Sobolev estimates derived from the resolution of singularities methods of [G2] and combine with analogous interpolation arguments, again resulti…
The purpose of this paper is to study the smoothing properties (in $L^p$ Sobolev spaces) of operators 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 $(t,x)\in\mathbb{R}^N\times \mathbb{R}^n$, 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 "multi-parameter fractional kernel" supported on a small neighborhood of $0\in \mathbb{R}^N$. When $K$ is a Calderón-Zygmund kernel these operators were studied by Christ, Nagel, St…
We prove $L^p$ boundedness results, $p > 2$, for local maximal averaging operators over a smooth 2D hypersurface $S$ with either a $C^1$ density function or a density function with a singularity that grows as $|(x,y)|^{-β}$ for $β< 2$. Suppose one is in coordinates such that the surface is localized near some $(x_0,y_0,z_0)$ at which $(0,0,1)$ is normal to the surface, and suppose the surface is represented as the graph of $z_0 + s(x - x_0, y - y_0)$ near $(x_0,y_0)$, with $s(0,0) = 0$. It is shown that as long as the Taylor series of the Hessian determinant of $s(x,y)$ at $(0,0)$ is not ident…
We extend the theorems of [G1] on $L^p$ to $L^p_s$ Sobolev improvement for translation invariant Radon and fractional singular Radon transforms over hypersurfaces, proving $L^p$ to $L^q_s$ boundedness results for such operators. Here $q \geq p$ but $s$ can be positive, negative, or zero. For many such operators we will have a triangle $Z \subset (0,1) \times (0,1) \times {\mathbb R} $ such that one has $L^p$ to $L^q_{s}$ boundedness for $({1 \over p}, {1 \over q}, s)$ beneath $Z$, and in the case of Radon transforms one does not have $L^p$ to $L^q_{s}$ boundedness for $({1 \over p}, {1 \over q…
Microlocal analysis of Radon transforms over quadric surfaces
University of Texas at Arlington（阿灵顿得克萨斯大学）；Rochester Institute of Technology（罗切斯特理工学院）；Tata Institute of Fundamental Research - Centre for Applicable Mathematics（塔塔基础研究所应用数学中心）；University of Limerick（利默里克大学）；Tufts University（塔夫茨大学）
We prove $\ell^p\big(\mathbb Z^d\big)$ bounds for $p\in(1, \infty)$, of $r$-variations $r\in(2, \infty)$, for discrete averaging operators and truncated singular integrals of Radon type. We shall present a new powerful method which allows us to deal with these operators in a unified way and obtain the range of parameters of $p$ and $r$ which coincide with the ranges of their continuous counterparts.
We prove sharp $L^p$ regularity results for a class of generalized Radon transforms for families of curves in a three-dimensional manifold associated to a canonical relation with fold and blowdown singularities. The proof relies on decoupling inequalities by Wolff and Bourgain-Demeter for plate decompositions of thin neighborhoods of cones and $L^2$ estimates for related oscillatory integrals.
Discrete analogues in harmonic analysis: Multi-parameter Radon averages
Instytut Matematyczny, Uniwersytet Wrocławski（弗罗茨瓦夫大学数学研究所）；Department of Mathematics, Yonsei University（延世大学数学系）；Department of Mathematics and Physical Sciences, Franciscan University of Steubenville（斯蒂布尼维尔方济各会大学数学与物理科学系）；Department of Mathematics, Rutgers University（罗格斯大学数学系）
Several estimates for singular integrals, maximal functions and the spherical summation operator are given in the spaces $L^p_{\text{rad}}L^2_{\text{ang}}(\mathbb{R}^n)$, $n\geq 2$.
We prove weak-type (1,1) estimates for compositions of maximal operators with singular integrals. Our main object of interest is the operator $Δ^*Ψ$ where $Δ^*$ is Bourgain's maximal multiplier operator and $Ψ$ is the sum of several modulated singular integrals; here our method yields a significantly improved bound for the $L^q$ operator norm when $1 < q < 2$. We also consider associated variation-norm estimates.
Lebesgue space bounds $L^{p_1}({\mathbb R}^1) \times L^{p_2}(^1) \to L^q({\mathbb R}^1)$ are established for certain maximal bilinear operators. The proof combines a trilinear smoothing inequality with Calderón-Zygmund theory. A reference to overlapping work of other authors on one observation has been added.
This is an expository paper on the characterization of the even (or odd) smooth homogeneous convolution Calderón-Zygmund operators in R^n such that the maximal singular integral can be controlled in the L^2 norm by the singular integral. We focuss attention on special cases of the general statements to convey the main ideas of the proofs in a transparent way, as free as possible of the technical complications inherent to the general case. Particular attention is devoted to higher Riesz transforms. The exposition is based on two papers by J.Mateu, J.Orobitg, C.Pérez and J.Verdera published in 2…
Discrete analogues in harmonic analysis: $TT^*$ methods
Franciscan University of Steubenville（斯蒂夫维尔方济各大学）；Rutgers University（罗格斯大学）；Instytut Matematyczny, Uniwersytet Wrocławski（弗罗茨瓦夫大学数学研究所）；University of Georgia（佐治亚大学）
Let 0<n<d be integers and let H denote the n-dimensional Hausdorff measure restricted to an n-dimensional Lipschitz graph in R^d with slope strictly less than 1. For r>2, we prove that the r-variation and oscillation for Calderón-Zygmund singular integrals with odd kernel are bounded operators in L^p(H) for 1<p finite, from L^1(H) to weak-L^1(H), and from the space of bounded H-measurable functions to BMO(H). Concerning the first endpoint estimate, we actually show that such operators are bounded from the space of finite complex Radon measures in R^d to weak-L^1(H).
Let $μ$ be a finite Radon measure in $\mathbb{R}^d$ with polynomial growth of degree $n$, although not necessarily $n$-AD-regular. We prove that under some geometric conditions on $μ$ that are closely related to rectifiability and involve the so-called $β$-numbers of Jones, David and Semmes, all singular integral operators with an odd and sufficiently smooth Calderón-Zygmund kernel are bounded in $L^2(μ)$. As a corollary, we obtain a lower bound for the Lipschitz harmonic capacity of a compact set in $\mathbb{R}^d$ only in terms of its metric and geometric properties.
In this paper it is shown that for $Ω\in L\log L(\mathbb{S}^{d-1})$, the rough maximal singular integral operator $T_Ω^*$ is of weak type $L\log\log L(\mathbb{R}^d)$. Furthermore, for $w\in A_1$ and $Ω\in L^\infty(\mathbb{S}^{d-1})$, it is shown that $T_Ω^*$ is of weak type $L\log\log L(w)$ with weight dependence $[w]_{A_1}[w]_{A_{\infty}}\log([w]_{A_{\infty}}+1),$ which is same as the best known constant for the singular integral $T_Ω$.
Let $K$ be a standard Hölder continuous Calderón--Zygmund kernel on $\mathbb{R}^{\mathbf{d}}$ whose truncations define $L^2$ bounded operators. We show that the maximal operator obtained by modulating $K$ by polynomial phases of a fixed degree is bounded on $L^p(\mathbb{R}^{\mathbf{d}})$ for $1 < p < \infty$. This extends Sjölin's multidimensional Carleson theorem and Lie's polynomial Carleson theorem.
We discuss the L^p-boundedness of maximal singular integrals in the plane over a finite set V of N directions. Logarithmic bounds are established for a set V of arbitrary structure in the 2<=p<infinity range. Sharp bounds are proved for lacunary and Vargas sets of directions. The latter include the case of uniformly distributed directions and the finite truncations of the Cantor set. We make use of both classical harmonic analysis methods and product-BMO based time-frequency analysis techniques. As a further application of the latter, we derive an L^p almost orthogonality principle for Fourier…
We control a broad class of singular (or "rough") Fourier multipliers by geometrically-defined maximal operators via general weighted $L^2(\mathbb{R})$ norm inequalities. The multipliers involved are related to those of Coifman--Rubio de Francia--Semmes, satisfying certain weak Marcinkiewicz-type conditions that permit highly oscillatory factors of the form $e^{i|ξ|^α}$ for both $α$ positive and negative. The maximal functions that arise are of some independent interest, involving fractional averages associated with tangential approach regions (related to those of Nagel and Stein), and more no…
We prove in this paper the following estimate for the maximal operator $T^*$ associated to the singular integral operator $T$: $ \|T^*f\|_{L^{1,\infty}(w)} \lesssim \frac{1}ε \int_{\mathbb{R}^n} |f(x)| M_{L(\log L)^ε} (w)(x)dx$, for $w\geq 0, 0<ε\leq 1.$ This follows from the sharp $L^p$ estimate $ \|T^*f \|_{ L^{p}(w) } \lesssim p' (\frac{1}δ)^{1/p'} \|f \|_{L^{p}(M_{ L(\log L)^{p-1+δ}} (w))}$, for $1<p<\infty, w\geq 0, 0<δ\leq 1. $ As as a consequence we deduce that $ \|T^*f\|_{L^{1,\infty}(w)} \lesssim [w]_{A_{1}} \log(e+ [w]_{A_{\infty}}) \int_{\mathbb{R}^n} |f| w dx, $ extending the endpo…
The following two inversion methods for Radon-like transforms are widely used in integral geometry and related harmonic analysis. The first method invokes mean value operators in accordance with the classical Funk-Radon-Helgason scheme. The second one employs integrals of the potential type and polynomials of the Beltrami-Laplace operator. Applicability of these methods to the horospherical transform in the hyperbolic space was an open problem. In the present paper we solve this problem for $L^p$ functions in the maximal range of the parameter $p$ and for compactly supported smooth functions, …
In this paper, we study the boundedness theory for maximal Calderón-Zygmund operators acting on noncommutative $L_p$-spaces. Our first result is a criterion for the weak type $(1,1)$ estimate of noncommutative maximal Calderón-Zygmund operators; as an application, we obtain the weak type $(1,1)$ estimates of operator-valued maximal singular integrals of convolution type under proper {regularity} conditions. These are the {\it first} noncommutative maximal inequalities for families of linear operators that can not be reduced to positive ones. For homogeneous singular integrals, the strong type …
The purpose of this paper is to describe the smooth homogeneous Calderon-Zygmund operators for which the maximal singular integral T*f may be controlled by the singular integral Tf. We consider two types of control. The first is the L2 estimate of T*f by Tf, namely the estimate of the L2 norm of T*f by a constant times the L2 norm of Tf. The second is the pointwise estimate of T*f(x) by a constant times M(Tf)(x), where M denotes the Hardy-Littlewood maximal operator. Notice that this is an improved variant of Cotlar's inequality, because the term Mf(x) is missing on the right hand side. Our ma…
We prove that the class of convolution-type kernels satisfying suitable decay conditions of the Fourier transform, appearing in the works of Christ, Christ-Rubio de Francia, and Duoandikoetxea-Rubio de Francia gives rise to maximally truncated singular integrals satisfying a sparse bound by $(1+\varepsilon,1+\varepsilon)$-averages for all $\varepsilon>0$, with linear growth in $\varepsilon^{-1}$. This is an extension of the sparse domination principle by Conde-Alonso, Culiuc, Ou and the first author to maximally truncated singular integrals. Our results cover the rough homogeneous singular int…
