Strongly singular Radon transforms on the Heisenberg group and folding singularities

$L^p$-Sobolev estimates for a class of integral operators with folding canonical relations

We prove a sharp $L^p$-Sobolev regularity results for a class of generalized Radon transforms for families of curves in a three dimensional manifold, with folding canonical relations. The proof relies on decoupling inequalities by Wolff and by Bourgain-Demeter for plate decompositions of thin neighborhoods of cones.

Heisenberg群中的Finner型不等式

Finner-like inequalities in the Heisenberg group

研究Heisenberg群中涉及垂直投影的多线性Radon型变换的L^p有界性范围

通过广义 Radon 变换计数 Korányi 球面附近的格点

Counting Lattice Points near Korányi Spheres via Generalized Radon Transforms

本文利用 Heisenberg 群平移平均和广义 Radon 变换研究 Korányi 球面邻域内格点计数问题，克服极点处曲率退化，在低维情形获得对数级改进上界，并推广到一般 Heisenberg 齐次范数球面。

Heisenberg群上的Brascamp-Lieb型不等式

Inequalities of Brascamp-Lieb type on the Heisenberg group

本文研究Heisenberg群上的Brascamp–Lieb型不等式，刻画相关多重线性广义Radon变换的$L^p$有界性，并指出其充要条件在部分重要情形一致、其余情形存在差异。

Multiparameter singular integrals on the Heisenberg group: uniform estimates

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…

Marstrand's density theorem in the Heisenberg group

We prove that if $μ$ is a Radon measure on the Heisenberg group $\mathbb{H}^n$ such that the density $Θ^s(μ,\cdot)$, computed with respect to the Korányi metric $d_H$, exists and is positive and finite on a set of positive $μ$ measure, then $s$ is an integer. The proof relies on an analysis of uniformly distributed measures on $(\mathbb{H}^n,d_H)$. We provide a number of examples of such measures, illustrating both the similarities and the striking differences of this sub-Riemannian setting from its Euclidean counterpart.