Nicola Arcozzi, Fausto Ferrari
We compute the horizontal Hessian of the signed Carnot-Charatheodory distance from a surface S in the Heisenberg group H. The expression for the Hessian is in terms of the surface's intrinsic curvatures. As an application, we compute the horizontal Hessian of the Carnot-Charatheodory distance from a point in H.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
A Liouville theorem for the $2$-Hessian equation on the Heisenberg group
本文通过选取检验函数和分部积分，证明了海森堡群上$2$-Hessian方程的Liouville定理。
We describe the precise structure of the distributional Hessian of the distance function from a point of a Riemannian manifold. In doing this we also discuss some geometrical properties of the cutlocus of a point and we compare some different weak notions of Hessian and Laplacian.
We prove the existence of solutions to Monge transport problem between two compactly supported Borel probability measures in the Heisenberg group equipped with its Carnot-Caratheodory distance assuming that the initial measure is absolutely continuous with respect to the Haar measure of the group.
In this paper we study various Hardy inequalities in the Heisenberg group $\mathbb H^n$, w.r.t. the Carnot-Carathéodory distance $δ$ from the origin. We firstly show that the optimal constant for the Hardy inequality is strictly smaller than $n^2 = (Q-2)^2/4$, where $Q$ is the homogenous dimension. Then, we prove that, independently of $n$, the Heisenberg group does not support a radial Hardy inequality, i.e., a Hardy inequality where the gradient term is replaced by its projection along $\nabla_{\mathbb H}δ$. This is in stark contrast with the Euclidean case, where the radial Hardy inequality…
Geometric Hardy inequalities on the Heisenberg groups via convexity
本文基于几何条件，在Heisenberg群上证明了带有边界距离的L^p-Hardy不等式，并讨论了不同距离定义下的凸域性质及Hardy不等式证明。
We study the boundary behavior of nonnegative p-harmonic functions which vanish on a portion of the boundary of a domain in the Heisenberg group H^n. Our main results are: 1) An estimate from above which shows that, under suitable geometric assumptions on the relevant domain, such a p-harmonic function vanishes at most linearly with respect to the sub-Riemannian distance to the boundary. 2) An estimate from below which shows that for a (Euclidean) C^{1,1} domain, away from the characteristic set, such a p-harmonic function vanishes exactly like the distance to the boundary. By combining 1) and…
We consider the multi-marginal optimal transport of aligning several compactly supported marginals on the Heisenberg group to minimize the total cost, which we take to be the sum of the squared Carnot-Carathéodory distances from the marginal points to their barycenter. Under certain technical hypotheses, we prove existence and uniqueness of optimal maps. We also point out several related open questions.
We prove a Koebe distortion theorem for the average derivative of a quasiconformal mapping between domains in the sub-Riemannian Heisenberg group $\mathbb{H}_1$. Several auxiliary properties of quasiconformal mappings between subdomains of $\mathbb{H}_1$ are proven, including distortion of balls estimates and local BMO-estimates for the logarithm of the Jacobian of a quasiconformal mapping. Applications of the Koebe theorem include diameter bounds for images of curves, comparison of integrals of the average derivative and the operator norm of the horizontal differential, as well as the study o…
A representation formula for regular functions on the characteristic plane of the second Heisenberg group
本文研究第二海森堡群特征平面上的拉普拉斯型算子及其基本解，引入共形拉普拉斯算子，证明外围空间嵌入诱导距离是其基本解的良好近似，给出光滑函数的表示公式且该公式在特征点内稳定。
We investigate observability and Lipschitz stability for the Heisenberg heat equation on the rectangular domain $$Ω= (-1,1)\times\mathbb{T}\times\mathbb{T}$$ taking as observation regions slices of the form $ω=(a,b) \times \mathbb{T} \times \mathbb{T}$ or tubes $ω= (a,b) \times ω_y \times \mathbb{T}$, with $-1<a<b<1$. We prove that observability fails for an arbitrary time $T>0$ but both observability and Lipschitz stability hold true after a positive minimal time, which depends on the distance between $ω$ and the boundary of $Ω$: $$T_{\min} \geqslant \frac{1}{8} \min\{(1+a)^2,(1-b)^2\}.$$ Our…
Uniform volume estimates and maximal functions on generalized Heisenberg-type groups
研究广义海森堡型群上Carnot-Carathéodory距离定义的球体的均匀体积估计，以及关联的中心Hardy-Littlewood最大函数的弱(1,1)估计，扩展了BLZ25的结果，并建立了Heisenberg群上左不变Riemannian度量的均匀体积倍增性质。
Unweighted Hardy Inequalities on the Heisenberg Group and in Step-Two Carnot Groups
本文在Heisenberg群和步二Carnot群上建立了无权Hardy不等式，并通过构造水平向量场得到最优常数的显式下界，同时推广到非各向同性结构中。
A semi-discrete scheme for horizontal mean curvature flow in the Heisenberg group
本文针对第一海森堡群的水平平均曲率流提出极小化运动半离散格式，证明了满足特定连续性条件的平坦流存在，适用于任意有限水平周长的有界集。
We give some estimates of the remainder terms for several conformally-invariant Sobolev-type inequalities on the Heisenberg group, in analogy with the Euclidean case. By considering the variation of associated functionals, we give a stability of two dual forms: the fractional Sobolev (Folland-Stein) and Hardy-Littlewood-Sobolev inequality, in terms of distance to the submanifold of extremizers. Then we compare their remainder terms to improve the inequalities in another way. We also compare, in the limit case s = Q (or $λ$ = 0), the remainder terms of Beckner-Onofri inequality and its dual Log…
In this paper, we present the geometric Hardy inequality for the sub-Laplacian in the half-spaces on the stratified groups. As a consequence, we obtain the following geometric Hardy inequality in a half-space on the Heisenberg group with a sharp constant \begin{equation*} \int_{\mathbb{H}^+} |\nabla_{H}u|^p dξ\geq \left(\frac{p-1}{p}\right)^p \int_{\mathbb{H}^+} \frac{\mathcal{W}(ξ)^p}{dist(ξ,\partial \mathbb{H}^+)^p} |u|^p dξ, \,\, p>1, \end{equation*} which solves the conjecture in the paper \cite{Larson}. Also, we obtain a version of the Hardy-Sobolev inequality in a half-space on the Heise…
The purpose of this work is twofold. First we study the solutions of a Hamilton-Jacobi equation of the form $u_t(t,x)+\mathcal{H}(t,x,\nabla_H u(t,x))=0$, where $\nabla_H u$ represents the horizontal gradient of a function $u$ defined on the Heisenberg group ${I\!\!H}$. Motivated by the recent paper by Liu, Manfredi and Zhou (\cite{LiMaZh2016}), we prove a Lipschitz continuity preserving property for $u$ with respect to the Korányi homogeneous distances $d_G$ in ${I\!\!H}$. Secondly, we are keenly interested in introducing the game theory in ${I\!\!H}$, taking into account its Sub-Riemannian s…
We prove geometric $L^p$ versions of Hardy's inequality for the sub-elliptic Laplacian on convex domains $Ω$ in the Heisenberg group $\mathbb{H}^n$, where convex is meant in the Euclidean sense. When $p=2$ and $Ω$ is the half-space given by $\langle ξ, ν\rangle > d$ this generalizes an inequality previously obtained by Luan and Yang. For such $p$ and $Ω$ the inequality is sharp and takes the form \begin{equation} \int_Ω|\nabla_{\mathbb{H}^n}u|^2 \, dξ\geq \frac{1}{4}\int_Ω \sum_{i=1}^n\frac{\langle X_i(ξ), ν\rangle^2+\langle Y_i(ξ), ν\rangle^2}{\textrm{dist}(ξ, \partial Ω)^2}|u|^2\, dξ, \end{e…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。