Andreas Seeger, Stephen Wainger
We prove variable coefficient versions of L^p boundedness results on Hilbert transforms and maximal functions along convex curves in the plane.
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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.
Maximal Polynomial Modulations of Singular Radon Transforms
该研究证明了环面上各向异性尺度Calderón-Zygmund算子的极大多项式调制的$L^2 \ o L^p$估计并优化了常数，还推导了沿抛物线的光滑化Hilbert变换的极大多项式调制的有界性结论。
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…
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…
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…
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 $\ell^p(\mathbb Z^d)$.
We use a variant of the technique in [Lac17a] to give sparse L^p(log(L))^4 bounds for a class of model singular and maximal Radon transforms
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…
Reconstruction of a function defined on a sphere using the Funk transform
本文针对球面上非偶连续函数，提出附加条件并研究双数据Funk变换的单射性，给出迭代反演公式，具有凸性理论、积分几何及球面断层扫描的理论意义，并应用于扩散加权磁共振成像。
In this paper we give an integral representation of an $n$-convex function $f$ in general case without additional assumptions on function $f$. We prove that any $n$-convex function can be represented as a sum of two $(n+1)$-times monotone functions and a polynomial of degree at most $n$. We obtain a decomposition of $n$-Wright-convex functions which generalizes and complements results of Maksa and Pales (2009). We define and study relative $n$-convexity of $n$-convex functions. We introduce a measure of $n$-convexity of $f$. We give a characterization of relative $n$-convexity in terms of this…
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…
Endpoint estimates and optimality for the generalized spherical maximal operator on radial functions
本文研究径向函数上广义球面平均 Radon 变换极大算子 $M^{\alpha,\beta}_t$，给出其在幂加权 Lebesgue 空间有界的尖锐条件及最优端点弱型估计。
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…
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…
Discrete analogues in harmonic analysis: Multi-parameter Radon averages
研究多参数离散拉东平均算子的极大值和振荡不等式，通过开发多参数圆法的稳健变体给出定量估计，解决了遍历理论中的多参数贝洛 - 弗斯滕伯格问题。
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…
Discrete analogues in harmonic analysis: $TT^*$ methods
本文展示如何利用基于 $TT^*$ 论证的几乎正交性方法研究 Radon 型离散算子的有界性，并给出 Bourgain 极大不等式的新证明。
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, …
On the resonant Carleson-Radon transform in all dimensions. The degree one resonant case
本文解决了所有维度上的一次共振情形，证明了对于特定子空间V，最大Carleson-Radon变换在L^p有界，其中1<p<∞。
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