Antoine Julia, Sebastiano Nicolussi Golo
We give a geometric criterion for a topological surface in the first Heisenberg group to be an intrinsic Lipschitz graph, using planar cones instead of the usual open cones.
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We introduce a notion of rectifiability modeled on Carnot groups. Precisely, we say that a subset E of a Carnot group M and N is a subgroup of M, we say E is N-rectifiable if it is the Lipschitz image of a positive measure subset of N. First, we discuss the implications of N-rectifiability, where N is a Carnot group (not merely a subgroup of a Carnot group), which include N-approximability and the existence of approximate tangent cones isometric to N almost everywhere in E. Second, we prove that, under a stronger condition concerning the existence of approximate tangent cones isomorphic to N a…
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。
Let $E$ be an $1$-Ahlfors regular subset of the Heisenberg group $\mathbb{H}$. We prove that there exists a $-1$-homogeneous kernel $K_1$ such that if $E$ is contained in a $1$-regular curve the corresponding singular integral is bounded in $L^2(E)$. Conversely, we prove that there exists another $-1$-homogeneous kernel $K_2$, such that the $L^2(E)$-boundedness of its corresponding singular integral implies that $E$ is contained in an $1$-regular curve. These are the first non-Euclidean examples of kernels with such properties. Both $K_1$ and $K_2$ are weighted versions of the Riesz kernel cor…
Cones, rectifiability, and singular integral operators
本文定义并研究了锥形能量，用于刻画测度的可求长性与Lipschitz图像的大块性质，并在测度具有多项式增长时，给出了卷积型光滑奇核奇异积分算子 $L^2(μ)$ 有界性的充分条件。
This paper is devoted to show that the flatness of tangents of $1$-codimensional measures in Carnot Groups implies $C^1_\mathbb{G}$-rectifiability. As applications we prove that measures with $(2n+1)$-density in the Heisenberg groups $\mathbb{H}^n$ are $C^1_{\mathbb{H}^n}$-rectifiable, providing the first non-Euclidean extension of Preiss's rectifiability theorem and a criterion for intrinsic Lipschitz rectifiability of finite perimeter sets in general Carnot groups.
Two definitions for the rectfiability of hypersurfaces in Heisenberg groups $\mathbb{H}^n$ have been proposed: one based on $\mathbb{H}$-regular surfaces, and the other on Lipschitz images of subsets of codimension-$1$ vertical subgroups. The equivalence between these notions remains an open problem. Recent partial results are due to Cole-Pauls, Bigolin-Vittone, and Antonelli-Le Donne. This paper makes progress in one direction: the metric Lipschitz rectifiability of $\mathbb{H}$-regular surfaces. We prove that $\mathbb{H}$-regular surfaces in $\mathbb{H}^{n}$ with $α$-Hölder continuous horizo…
Lipschitz graphs and currents in Heisenberg groups
本文证明子黎曼海森伯格群中余维数k≤n的内蕴Lipschitz图的Rademacher型定理，并给出延拓、逼近、常值定理及Lusin型结果、可修正性等价和面积公式等应用。
In this article, we prove that a cone is a Heisenberg uniqueness pair corresponding to sphere as long as the cone does not completely recline on the level surface of any homogeneous harmonic polynomial on $\mathbb R^n.$ We derive that $\left(S^2, \text{ paraboloid}\right)$ and $\left(S^2, \text{ geodesic of } S_r(o)\right)$ are Heisenberg uniqueness pairs for a class of certain symmetric finite Borel measures in $\mathbb R^3.$ Further, we correlate the problem of Heisenberg uniqueness pairs to the sets of injectivity for the spherical mean operator.
We prove the $L^p$ boundedness of the circular maximal function on the Heisenberg group $\mathbb{H}^1$ for $2<p\le \infty$. The proof is based on the square sum estimate associated with the $2\times 2$ cone $|(ξ_1',ξ_2')|= |(ξ_3',ξ_4')| $ of the phase space arising from the vector fields $X_1,X_2,tX_3,\partial/\partial t$ on the Heisenberg group, rather than the $2\times 1$ cone $ |(ξ_1,ξ_2)|= |ξ_3|$ of the frequency space arising from $\partial/\partial x_1, \partial/\partial x_2, \partial/\partial t$ on the Euclidean space.
The purpose of this paper is to introduce and study some basic concepts of quantitative rectifiability in the first Heisenberg group $\mathbb{H}$. In particular, we aim to demonstrate that new phenomena arise compared to the Euclidean theory, founded by G. David and S. Semmes in the 90's. The theory in $\mathbb{H}$ has an apparent connection to certain nonlinear PDEs, which do not play a role with similar questions in $\mathbb{R}^{3}$. Our main object of study are the intrinsic Lipschitz graphs in $\mathbb{H}$, introduced by B. Franchi, R. Serapioni and F. Serra Cassano in 2006. We claim that …
We prove $L^p_{comp}\to L^p_{s}$ boundedness for averaging operators associated to a class of curves in the Heisenberg group $\mathbb{H}^1$ via $L^2$ estimates for related oscillatory integrals and Bourgain-Demeter decoupling inequalities on the cone. We also construct a Sobolev space adapted to translations on the Heisenberg group to which these averaging operators map all $L^p$ functions boundedly.
We prove local solvability for large classes of operators of the form $$ L=\sum_{j,k=1}^{2n}a_{jk}V_jV_k+iαU,$$ where the $V_j$ are left-invariant vector fields on the Heisenberg group satisfying the commutation relations $[V_j,V_{j+n}]=U$ for $1\le j\le n$, and where $A=(a_{jk})$ is a complex symmetric matrix with semi-definite real part. Our results widely extend all of the results for the case of non-real, semi-definite matrices $A$ known to date, in particular those obtained recently jointly with F. Ricci under Sjöstrand's cone condition.
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。
Let $p,q$ positive integers. The groups $U_p(\b C)$ and $U_p(\b C)\times U_q(\b C) $ act on the Heisenberg group $H_{p,q}:=M_{p,q}(\b C)\times \b R$ canonically as groups of automorphisms where $M_{p,q}(\b C)$ is the vector space of all complex $p\times q$-matrices. The associated orbit spaces may be identified with $Π_q\times \b R$ and $Ξ_q\times \b R$ respectively with the cone $Π_q$ of positive semidefinite matrices and the Weyl chamber $Ξ_q={x\in\b R^q: x_1\ge...\ge x_q\ge 0}$. In this paper we compute the associated convolutions on $Π_q\times \b R$ and $Ξ_q\times \b R$ explicitly dependin…
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