In this paper we prove that every collection of measurable functions $f_α$, $|α|=m$ coincides a.e. with $m$th order derivatives of a function $g\in C^{m-1}$ whose derivatives of order $m-1$ may have any modulus of continuity weaker than that of a Lipschitz function. This is a stronger version of earlier results of Lusin, Moonens-Pfeffer and Francos. As an application we construct surfaces in the Heisenberg group with tangent spaces being horizontal a.e.
A Carnot group $\mathbb{G}$ admits Lusin approximation for horizontal curves if for any absolutely continuous horizontal curve $γ$ in $\mathbb{G}$ and $\varepsilon>0$, there is a $C^1$ horizontal curve $Γ$ such that $Γ=γ$ and $Γ'=γ'$ outside a set of measure at most $\varepsilon$. We verify this property for free Carnot groups of step 2 and show that it is preserved by images of Lie group homomorphisms preserving the horizontal layer. Consequently, all step 2 Carnot groups admit Lusin approximation for horizontal curves.
We prove a $C^m$ Lusin approximation theorem for horizontal curves in the Heisenberg group. This states that every absolutely continuous horizontal curve whose horizontal velocity is $m-1$ times $L^1$ differentiable almost everywhere coincides with a $C^m$ horizontal curve except on a set of small measure. Conversely, we show that the result no longer holds if $L^1$ differentiability is replaced by approximate differentiability. This shows our result is optimal and highlights differences between the Heisenberg and Euclidean settings.
Let $\mathcal L=-Δ_{\mathbb H^n}+V$ be a Schrödinger operator on the Heisenberg group $\mathbb H^n$, where $Δ_{\mathbb H^n}$ is the sublaplacian on $\mathbb H^n$ and the nonnegative potential $V$ belongs to the reverse Hölder class $RH_q$ with $q\geq Q/2$. Here $Q=2n+2$ is the homogeneous dimension of $\mathbb H^n$. Assume that $\{e^{-s\mathcal L}\}_{s>0}$ is the heat semigroup generated by $\mathcal L$. The Littlewood-Paley function $\mathfrak{g}_{\mathcal L}$ and the Lusin area integral $\mathcal{S}_{\mathcal L}$ associated with the Schrödinger operator $\mathcal L$ are defined, respectively…
Let ${\mathcal X}$ be a doubling metric measure space. If ${\mathcal X}$ has the $δ$-annular decay property for some $δ\in (0,\,1]$, the authors then establish the boundedness of the Lusin-area function, which is defined via kernels modeled on the semigroup generated by the Schrödinger operator, from localized spaces ${\rm BMO}_ρ({\mathcal X})$ to ${\rm BLO}_ρ({\mathcal X})$ without invoking any regularity of considered kernels. The same is true for the $g^\ast_\labda$ function and unlike the Lusin-area function, in this case, ${\mathcal X}$ is not necessary to have the $δ$-annular decay prope…
Unweighted Hardy Inequalities on the Heisenberg Group and in Step-Two Carnot Groups
Sapienza University of Rome（罗马第一大学）；Ikerbasque, Universidad del País Vasco / Euskal Herriko Unibertsitatea & BCAM（伊克拉巴斯科，巴斯克大学/BCAM）；Università degli Studi di Padova（帕多瓦大学）；Université Paris-Saclay, CentraleSupélec, CNRS（巴黎萨克雷大学，中央理工-苏佩莱克学院，法国国家科学研究中心）
We characterize which mappings from a compact subset of $\mathbb{R}$ into the Heisenberg group can be extended to a $C^{m,ω}$ horizontal curve for a given modulus of continuity $ω$. We motivate our characterization by showing that the $C^{m,ω}$ extension property fails if we instead use a more direct analogue of the conditions from the $C^{m}$ case.
We show that the Heisenberg group $\mathbb{H}^n$ contains a measure zero set $N$ such that every Lipschitz function $f\colon \mathbb{H}^n \to \mathbb{R}$ is Pansu differentiable at a point of $N$. The proof adapts the construction of small 'universal differentiability sets' in the Euclidean setting: we find a point of $N$ and a horizontal direction where the directional derivative in horizontal directions is almost locally maximal, then deduce Pansu differentiability at such a point.
In this paper we investigate the property of engulfing for $H$-convex functions defined on the Heisenberg group ${\mathbb{H}}^n$. Starting from the horizontal sections introduced by Capogna and Maldonado, we consider a new notion of section, called ${\mathbb{H}}^n$-section, as well as a new condition of engulfing associated to the ${\mathbb{H}}^n$-sections, for an $H$-convex function defined in ${\mathbb{H}}^n.$ These sections, that arise as suitable unions of horizontal sections, are dimensionally larger; as a matter of fact, the ${\mathbb{H}}^n$-sections, with their engulfing property, will …
In this paper we extend some classical results of Convex Analysis to the sub-Riemannian setting of the Heisenberg group. In particular, we provide a horizontal version of Minty's theorem concerning maximal H-monotone operators defined in the Heisenberg group with values in the first layer of its Lie algebra.
Let $\H= < a,b | a[a,b]=[a,b]a \wedge b[a,b]=[a,b]b>$ be the discrete Heisenberg group, equipped with the left-invariant word metric $d_W(\cdot,\cdot)$ associated to the generating set ${a,b,a^{-1},b^{-1}}$. Letting $B_n= {x\in \H: d_W(x,e_\H)\le n}$ denote the corresponding closed ball of radius $n\in \N$, and writing $c=[a,b]=aba^{-1}b^{-1}$, we prove that if $(X,|\cdot|_X)$ is a Banach space whose modulus of uniform convexity has power type $q\in [2,\infty)$ then there exists $K\in (0,\infty)$ such that every $f:\H\to X$ satisfies {multline*} \sum_{k=1}^{n^2}\sum_{x\in B_n}\frac{|f(xc^k)-f(…
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…
We investigate metric properties of level sets of horizontally differentiable maps defined on the first Heisenberg group $(\Bbb{H}^1,d_{cc})$ equipped with the standard sub-Riemannian structure. In particular, we present an exhaustive analysis in a new case of a map $F\in C^1_H(\Bbb{H}^1, \Bbb{R}^2)$ with surjective horizontal differential (an analogue of the classical implicit function theorem). Among other results, we show that a level set of such map is locally a simple curve of Hausdorff sub-Riemannian dimension 2, but, surprisingly, in general its two-dimensional Hausdorff measure can be …
We prove a rank-one theorem à la G. Alberti for the derivatives of vector-valued maps with bounded variation in a class of Carnot groups that includes Heisenberg groups $\mathbb H^n$ for $n\geq 2$. The main tools are properties relating the horizontal derivatives of a real-valued function with bounded variation and its subgraph.
We study the question: when are Lipschitz mappings dense in the Sobolev space $W^{1,p}(M,\mathbf{H}^n)$? Here $M$ denotes a compact Riemannian manifold with or without boundary, while $\mathbf{H}^n$ denotes the $n$th Heisenberg group equipped with a sub-Riemannian metric. We show that Lipschitz maps are dense in $W^{1,p}(M,\mathbf{H}^n)$ for all $1\le p<\infty$ if $\dim M \le n$, but that Lipschitz maps are not dense in $W^{1,p}(M,\mathbf{H}^n)$ if $\dim M \ge n+1$ and $n\le p<n+1$. The proofs rely on the construction of smooth horizontal embeddings of the sphere $S^n$ into $\mathbf{H}^n$. We …
Let $α\geq 0$, $1 < p < \infty$, and let $\mathbb{H}^{n}$ be the Heisenberg group. Folland in 1975 showed that if $f \colon \mathbb{H}^{n} \to \mathbb{R}$ is a function in the horizontal Sobolev space $S^{p}_{2α}(\mathbb{H}^{n})$, then $φf$ belongs to the Euclidean Sobolev space $S^{p}_α(\mathbb{R}^{2n + 1})$ for any test function $φ$. In short, $S^{p}_{2α}(\mathbb{H}^{n}) \subset S^{p}_{α,\mathrm{loc}}(\mathbb{R}^{2n + 1})$. We show that the localisation can be omitted if one only cares for Sobolev regularity in the vertical direction: the horizontal Sobolev space $S_{2α}^{p}(\mathbb{H}^{n})$…
The discrete Heisenberg group $\mathbb{H}_{\mathbb{Z}}^{2k+1}$ is the group generated by $a_1,b_1,\ldots,a_k,b_k,c$, subject to the relations $[a_1,b_1]=\ldots=[a_k,b_k]=c$ and $[a_i,a_j]=[b_i,b_j]=[a_i,b_j]=[a_i,c]=[b_i,c]=1$ for every distinct $i,j\in \{1,\ldots,k\}$. Denote $S=\{a_1^{\pm 1},b_1^{\pm 1},\ldots,a_k^{\pm 1},b_k^{\pm 1}\}$. The horizontal boundary of $Ω\subset \mathbb{H}_{\mathbb{Z}}^{2k+1}$, denoted $\partial_{h}Ω$, is the set of all $(x,y)\in Ω\times (\mathbb{H}_{\mathbb{Z}}^{2k+1}\setminus Ω)$ such that $x^{-1}y\in S$. The horizontal perimeter of $Ω$ is $|\partial_{h}Ω|$. Fo…
