Lp-improving properties of certain singular measures on the Heisenberg group

Hausdorff dimension of limsup sets of isotropic rectangles in Heisenberg groups

A formula for the Hausdorff dimension of typical limsup sets generated by randomly distributed isotropic rectangles in Heisenberg groups is derived in terms of directed singular value functions.

关于抛物空间和海森堡群中的奇异积分与定量可测性

On Singular Integrals and Quantitative Rectifiability in Parabolic Space and the Heisenberg Group

本文将David和Semmes关于奇异积分与定量可测性的定理扩展到抛物空间和海森堡群。

Hausdorff dimension of limsup sets of rectangles in the Heisenberg group

The almost sure value of the Hausdorff dimension of limsup sets generated by randomly distributed rectangles in the Heisenberg group is computed in terms of directed singular value functions.

一类两步幂零李群上的奇异球极大算子

Singular spherical maximal operators on a class of two step nilpotent Lie groups

令$H^n\cong \Bbb R^{2n}\ltimes \Bbb R$是海森堡群，并令$μ_t$是$\Bbb R^{2n}$中半径为$t$的球面的归一化表面测度。考虑由$Mf=\sup_{t>0} |f*μ_t|$定义的极大函数。我们证明了$n\ge 2$，$M$定义了一个有界的操作$L^p(H^n)$提供$p>2n/(2n-1)$。这改进了Nevo和Thangavelu的早期结果，并且$L^p$有界的范围是最优的。我们还将结果扩展到更一般的设置的表面和满足非退化条件的群体，这些包括海森堡型的群体。

On the lacunary spherical maximal function on the Heisenberg group

In this paper we investigate the $L^p$ boundedness of the lacunary maximal function $ M_{\Ha}^{lac} $ associated to the spherical means $ A_r f$ taken over Koranyi spheres on the Heisenberg group. Closely following an approach used by M. Lacey in the Euclidean case, we obtain sparse bounds for these maximal functions leading to new unweighted and weighted estimates. The key ingredients in the proof are the $L^p$ improving property of the operator $A_rf$ and a continuity property of the difference $A_rf-τ_y A_rf$, where $τ_yf(x)=f(xy^{-1})$ is the right translation operator.

The Riesz tranform on intrinsic Lipschitz graphs in the Heisenberg group

We prove that the Heisenberg Riesz transform is $L_2$--unbounded on a family of intrinsic Lipschitz graphs in the first Heisenberg group $\mathbb{H}$. We construct this family by combining a method from \cite{NY2} with a stopping time argument, and we establish the $L_2$--unboundedness of the Riesz transform by introducing several new techniques to analyze singular integrals on intrinsic Lipschitz graphs. These include a formula for the Riesz transform in terms of a singular integral on a vertical plane and bounds on the flow of singular integrals that arises from a perturbation of a graph. On…

Singular integrals on regular curves in the Heisenberg group

Let $\mathbb{H}$ be the first Heisenberg group, and let $k \in C^{\infty}(\mathbb{H} \, \setminus \, \{0\})$ be a kernel which is either odd or horizontally odd, and satisfies $$|\nabla_{\mathbb{H}}^{n}k(p)| \leq C_{n}\|p\|^{-1 - n}, \qquad p \in \mathbb{H} \, \setminus \, \{0\}, \, n \geq 0.$$ The simplest examples include certain Riesz-type kernels first considered by Chousionis and Mattila, and the horizontally odd kernel $k(p) = \nabla_{\mathbb{H}} \log \|p\|$. We prove that convolution with $k$, as above, yields an $L^{2}$-bounded operator on regular curves in $\mathbb{H}$. This extends a…

关于Heisenberg群中非负核的奇异积分

On singular integrals with non-negative kernels in the Heisenberg group

本文研究了Heisenberg群中非负核的奇异积分，证明了在特定条件下1-阿赫尔福斯正则集的性质，并给出了对Fässler和Orponen问题的否定回答，同时展示了纯1-不可测集的奇异积分在L^2空间中的有界性。

Heisenberg群中$C^{1,α}$内蕴图上奇异积分的有界性

Boundedness of singular integrals on $C^{1,α}$ intrinsic graphs in the Heisenberg group

本文证明Heisenberg群中3维C-Z奇异积分在垂直平面上的一致$L^2$有界性可推广至紧支撑$C^{1,α}$内蕴图，并据此得到次Laplacian相关算子有界性及首批非垂直平面的3维不可去集。

The local symmetry condition in the Heisenberg group

I propose an analogue in the first Heisenberg group $\mathbb{H}$ of David and Semmes' local symmetry condition (LSC). For closed $3$-regular sets $E \subset \mathbb{H}$, I show that the (LSC) is implied by the $L^{2}(\mathcal{H}^{3}|_{E})$ boundedness of $3$-dimensional singular integrals with horizontally antisymmetric kernels, and that the (LSC) implies the weak geometric lemma for vertical $β$-numbers.

Harmonic intrinsic graphs in the Heisenberg group

Minimal surfaces in $\mathbb{R}^n$ can be locally approximated by graphs of harmonic functions, i.e., functions that are critical points of the Dirichlet energy, but no analogous theorem is known for $H$-minimal surfaces in the three-dimensional Heisenberg group $\mathbb{H}$, which are known to have singularities. In this paper, we introduce a definition of intrinsic Dirichlet energy for surfaces in $\mathbb{H}$ and study the critical points of this energy, which we call contact harmonic graphs. Nearly flat regions of $H$-minimal surfaces can often be approximated by such graphs. We give a cal…

Heisenberg群中的非负核与$1$ -可求正性

Nonnegative kernels and $1$-rectifiability in the Heisenberg group

设$E$是海森堡群$\mathbb{H}$的一个$1$ -Ahlfors正则子集.证明了存在$-1$ -齐次核$K_1$，使得若$E$包含在$1$ -正则曲线中，则相应的奇异积分在$L^2(E)$中有界.反过来，我们证明了存在另一个$-1$ -齐次核$K_2$，使得其相应的奇异积分的$L^2(E)$ -有界性蕴涵着$E$包含在$1$ -正则曲线中.这是第一个非欧几里德核的例子。$K_1$和$K_2$都是里斯核的加权版本，对应于$\mathbb{H}$的垂直分量。与欧几里德情形不同，在欧几里德情形中，与可求正性相关的所有已知核都是反对称的，核$K_1$和$K_2$是偶数和非负的。

利用反向Hölder不等式改进$A_{\infty}$奇异算子的界

Improving bounds for singular operators via Sharp Reverse Hölder Inequality for $A_{\infty}$

在这篇文章中，我们收集并讨论了最近的一些结果的不同后果的夏普反向赫尔德不等式的$A_{\infty}$重量。对于两个给定的算子$T$和$S$，我们研究了Coifman-Festerman型算子的$L^p(w)$界. $ \|Tf\|_{L^p(w)}\le c_{n,w,p} \|Sf\|_{L^p(w)}, $可以理解为通过$S$控制$T$的方式。我们将专注于\emph{quantitative}分析所涉及的常数，并表明，我们可以改善经典的结果依赖于重量$w$威尔逊的$A_{\infty}$常数$ [w]_{A_{\infty}}:=\sup_Q\frac{1}{w(Q)}\int_Q M(wχ_Q).$我们还将展示最近的改进问题上找到尖锐的常数加权范数不等式涉及几个奇异算子。我们得到了交换子$[b,T]$及其高阶模拟$T^k_{b}$的混合$A_{1}$-$A_{\infty}$估计。一个共同的成分在这里提出的证明是最近改进的反向赫尔德不等式的$A_{\infty}$重量涉及威尔逊常数。

适用于含旋转的奇异积分算子的齐性空间

Homogeneous spaces adapted to singular integral operators involving rotations

函数的Calderón-Zygmund分解被用来证明奇异积分算子的弱型（1，1）有界性。在许多示例中，分解是关于对应于某个膨胀族的球族来完成的。我们研究奇异积分运营商$T$，需要更特殊的家庭的球，提供新的空间的齐型。旋转在这些球的构造中起着决定性的作用。$T$的有界性可以通过关于这个齐型空间的Calderón-Zygmund分解来证明。证明了算子$T$作用于$\LP^p(G)$的弱型（1，1）和$\LP^p$估计，其中$G$是齐次李群.我们的结果适用于设置的基础组是海森堡群和旋转是辛自同构。它们也适用于由涉及旋转的流体力学问题引起的算子。

海森堡群上的Flag Hardy空间和Marcinkiewicz乘子：一个扩展的版本

Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group: an expanded version

Marcinkiewicz multipliers are L^{p} bounded for 1<p< \infty on the Heisenberg group H^{n} \simeqC^{n} \timesR (D. Muller, F. Ricci and E. M. Stein) despite the lack of a two parameter group of automorphic dilations on H^{n}. This lack of dilations underlies the inability of classical one or two parameter Hardy space theory to handle Marcinkiewicz multipliers on H^{n} when 0<p \leq 1. We address this deficiency by developing a theory of flag Hardy spaces H_{flag} ^{p} on the Heisenberg group, 0<p \leq 1, that is in a sense `intermediate' between the classical Hardy spaces H^{p} and the product …