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 bounded on $L^p$ ($1<p\leq \infty$). We give conditions on $γ$ under which $T$ is bounded on $L^p$ ($1<p<\infty$); these conditions hold automatically when $K$ is a Calderón-Zygmund kernel. This is the final paper in a three part series. The first two papers consider the more general case when $γ$ is $C^\infty$.
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…
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 …
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.
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…
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 $L^p$ 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 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.
Geometric conditions for the $L^2$-boundedness of singular integral operators with odd kernels with respect to measures with polynomial growth in $\mathbb{R}^d$
We consider a class of multiparameter singular Radon integral operators on the Heisenberg group ${\mathbb H}^1$ where the underlying variety is the graph of a polynomial. A remarkable difference with the euclidean case, where Heisenberg convolution is replaced by euclidean convolution, is that the operators on the Heisenberg group are always $L^2$ bounded. This is not the case in the euclidean setting where $L^2$ boundedness depends on the polynomial defining the underlying surface. Here we uncover some new, interesting phenomena. For example, although the Heisenberg group operators are always…
