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This paper aims to show that there exists a triangulation of the Heisenberg group H n into singular simplexes with regularity properties on both the low-dimensional and high-dimensional layers. For low dimensions, we request our simplexes to be horizontal while, for high dimensions, we define a notion of straight simplexes using exponential and logarithmic maps, and we require our simplexes to have high-dimensional straight layers. A triangulation with such simplexes is first constructed on a general polyhedral structure and then extended to the whole Heisenberg group. In this paper, we also provide some explicit examples of grid and triangulations. [ABSTRACT FROM AUTHOR]

The sub-Riemannian Heisenberg group is the simplest nonabelian example of a Carnot group. Suppose (γ k) 0 ≤ k ≤ m is a collection of continuous function defined on a compact set K ⊂ R taking values in the sub-Riemannian Heisenberg group. When is there a horizontal C m curve Γ so that D k Γ | K = γ k for k = 0 , 1 , ⋯ , m ? Such extensions are known as “Whitney extensions” due to the original work of Whitney for real valued mappings. This question was answered by the authors together with Andrea Pinamonti. Suppose in addition that γ m is uniformly continuous with modulus of continuity ω . When, then, is there a horizontal C m , ω curve Γ so that D k Γ | K = γ k for k = 0 , 1 , ⋯ , m ? In this paper, we show that the hypotheses in the previous C m extension result are not sufficient in this case, and we provide new assumptions which are necessary and sufficient to guarantee the existence of such an extension. [ABSTRACT FROM AUTHOR]

We extend the celebrate De Giorgi-Nash-Moser theory to a wide class of nonlinear equations driven by nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional p-Laplacian operator on the Heisenberg-Weyl group H n . Among other results, we prove that the weak solutions to such a class of problems are bounded and Hölder continuous, by also establishing general estimates as fractional Caccioppoli-type estimates with tail and logarithmic-type estimates. [ABSTRACT FROM AUTHOR]

We study the isoperimetric problem for anisotropic perimeter measures on R 3 , endowed with the Heisenberg group structure. The perimeter is associated with a left-invariant norm ϕ on the horizontal distribution. In the case where ϕ is the standard norm in the plane, such isoperimetric problem is the subject of Pansu's conjecture, which is still unsolved. Assuming some regularity on ϕ and on its dual norm ϕ ∗ , we characterize C 2 -smooth isoperimetric sets as the sub-Finsler analogue of Pansu's bubbles. The argument is based on a fine study of the characteristic set of ϕ -isoperimetric sets and on establishing a foliation property by sub-Finsler geodesics. When ϕ is a crystalline norm, we show the existence of a partial foliation for constant ϕ -curvature surfaces by sub-Finsler geodesics. By an approximation procedure, we finally prove a conditional minimality property for the candidate solutions in the general case (including the case where ϕ is crystalline). [ABSTRACT FROM AUTHOR]

Timelike surfaces in the three-dimensional Heisenberg group with left-invariant semi-Riemannian metric are studied. In particular, non-vertical timelike minimal surfaces are characterized by the non-conformal Lorentz harmonic maps into the de Sitter two sphere. On the basis of the characterization, the generalized Weierstrass type representation will be established through the loop group decompositions. [ABSTRACT FROM AUTHOR]

Let H n = C n × R be the n-dimensional Heisenberg group, Q = 2 n + 2 be the homogeneous dimension of H n . We establish in this paper that the following sharpened Trudinger–Moser inequalities on the Heisenberg group H n under the homogeneous constraints of the Sobolev norm: sup ‖ ∇ H u ‖ Q Q + ‖ u ‖ Q Q ≤ 1 ∫ H n Φ Q α Q 1 + α ‖ u ‖ Q Q 1 Q - 1 | u | Q Q - 1 d ξ < + ∞ , holds if and only if α < 1 , where Φ Q (t) = e t - ∑ 0 Q - 2 t j j ! . Unlike all the proofs in the literature even in the Euclidean spaces, our proof avoids using the complicated blow-up analysis of the Euler–Lagrange equation associated with the Moser functional. In fact, our proof reveals a surprising fact that the known critical Trudinger–Moser inequality on the entire space is equivalent to seemingly much stronger sharpened Trudinger–Moser inequality on the entire space. Furthermore, we obtain the critical Trudinger–Moser inequality and the Concentration-Compactness Principle under the inhomogeneous constraints on the entire Heisenberg group. Finally, using the method of scaling again, we obtain improved Trudinger–Moser inequality under the inhomogeneous constraints. Our approach is surprisingly simple and general and can be easily applied to the all stratified nilpotent groups and other settings. In particular, our method also gives an alternative and much simpler proof of the corresponding results in the Euclidean space. [ABSTRACT FROM AUTHOR]

In this paper we investigate the property of engulfing for H-convex functions defined on the Heisenberg group H n . Starting from the horizontal sections introduced by Capogna and Maldonado (Proc Am Math Soc 134:3191–3199, 2006) , we consider a new notion of section, called H n -section, as well as a new condition of engulfing associated to the H n -sections, for an H-convex function defined in H n. These sections, that arise as suitable unions of horizontal sections, are dimensionally larger; as a matter of fact, the H n -sections, with their engulfing property, will lead to the definition of a quasi-distance in H n in a way similar to Aimar et al. in the Euclidean case (J Fourier Anal Appl 4:377–381, 1998). A key role is played by the property of round H-sections for an H-convex function, and by its connection with the engulfing properties. [ABSTRACT FROM AUTHOR]

Embedded symmetry within the Heisenberg group is used to couple geometric insight and analytic calculation to obtain a new sharp Stein–Weiss inequality with mixed homogeneity on the line of duality. SL(2,R) invariance and Riesz potentials define a natural bridge for encoded information that connects distinct geometric structures. The intrinsic character of the Heisenberg group makes it the natural playing field on which to explore the laws of symmetry and the interplay between analysis and geometry on a manifold. [ABSTRACT FROM AUTHOR]

We survey the work of Elias M. Stein in the field of analysis on the Heisenberg group and other nilpotent Lie groups, together with its applications to complex analysis in several variables and partial differential equations. [ABSTRACT FROM AUTHOR]

We classify all the translating solitons to the mean curvature flow in the three-dimensional Heisenberg group that are invariant under the action of some one-parameter group of isometries of the ambient manifold. We provide a complete classification for any canonical deformation of the standard Riemannian metric of the Heisenberg group. We highlight similarities and differences with the analogous Euclidean translators: we mention in particular that we describe the analogous of the tilted grim reaper cylinders, of the bowl solution and of translating catenoids, but some of them are not convex in contrast with a recent result of Spruck and Xiao (Complete translating solitons to the mean curvature flow in R 3 with non-negative mean curvature, arXiv:1703.01003v2, 2017) in the Euclidean space. Moreover we also prove some non-existence results. Finally we study the convergence of these surfaces as the ambient metric converges to the standard sub-Riemannian metric on the Heisenberg group. [ABSTRACT FROM AUTHOR]

In this paper we study Hardy inequalities in the Heisenberg group H n , with respect to the Carnot–Carathéodory distance δ from the origin. We firstly show that, letting Q be the homogenous dimension, the optimal constant in the (unweighted) Hardy inequality is strictly smaller than n 2 = (Q - 2) 2 / 4 . 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 ∇ H δ . This is in stark contrast with the Euclidean case, where the radial Hardy inequality is equivalent to the standard one, and has the same constant. Motivated by these results, we consider Hardy inequalities for non-radial directions, i.e., directions tangent to the Carnot–Carathéodory balls. In particular, we show that the associated constant is bounded on homogeneous cones C Σ with base Σ ⊂ S 2 n , even when Σ degenerates to a point. This is a genuinely sub-Riemannian behavior, as such constant is well known to explode for homogeneous cones in the Euclidean space. [ABSTRACT FROM AUTHOR]

In this paper, we study existence and asymptotic behavior of nontrivial solutions of a series of problems in general open subsets Ω of the Heisenberg group H n , possibly unbounded or even H n . The problems involve the p-Laplacian operator on H n , a Hardy coefficient and different critical nonlinearities. The main originality of the paper is to work in the Heisenberg group. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the existence of weak solutions for some classes of one-parameter subelliptic gradient-type systems involving a Sobolev–Hardy potential defined on an unbounded domain Ω ψ of the Heisenberg group H n = C n × R ( n ≥ 1 ) whose geometrical profile is determined by two real positive functions ψ 1 and ψ 2 that are bounded on bounded sets. The treated problems have a variational structure, and thanks to this, we are able to prove the existence of an open interval Λ ⊂ (0 , ∞) such that, for every parameter λ ∈ Λ , the system has at least two non-trivial symmetric weak solutions that are uniformly bounded with respect to the Sobolev H W 0 1 , 2 -norm. Moreover, the existence is stable under certain small subcritical perturbations of the nonlinear term. The main proof, crucially based on the Palais principle of symmetric criticality, is obtained by developing a group-theoretical procedure on the unitary group U (n) = U (n) × { 1 } and by exploiting some compactness embedding results into Lebesgue spaces, recently proved for suitable U (n) -invariant subspaces of the Folland–Stein space H W 0 1 , 2 (Ω ψ) . A key ingredient for our variational approach is a very general min–max argument valid for sufficiently smooth functionals defined on reflexive Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study some global properties of curves in the Heisenberg group H 1 . In particular, we obtain Fenchel-type theorem and Fáry–Milnor type theorem, together with Bray–Jauregui type theorem. We also prove the fundamental theorem of curves in the standard CR 3-sphere S 3 . As an application, we classify some horizontally regular curves in S 3 . [ABSTRACT FROM AUTHOR]

The Besicovitch covering property (BCP) is known to be one of the fundamental tools in measure theory, and more generally, a useful property for numerous purposes in analysis and geometry. We prove both sufficient and necessary criteria for the validity of BCP in the first Heisenberg group equipped with a homogeneous distance. Beyond recovering all previously known results about the validity or non-validity of BCP in this setting, we get simple descriptions of new large classes of homogeneous distances satisfying BCP. We also obtain a full characterization of rotationally invariant distances for which BCP holds in the first Heisenberg group under mild regularity assumptions about their unit sphere. [ABSTRACT FROM AUTHOR]

For the natural two-parameter filtration Fλ:λ∈P on the boundary of a triangle building, we define a maximal function and a square function and show their boundedness on Lp(Ω0) for p∈(1,∞). At the end, we consider Lp(Ω0) boundedness of martingale transforms. If the building is of GL(3,Qp), then Ω0 can be identified with p-adic Heisenberg group. [ABSTRACT FROM AUTHOR]

For a real-valued function defined on a compact set $$K \subset {\mathbb {R}}^m$$ , the classical Whitney Extension Theorem from 1934 gives necessary and sufficient conditions for the existence of a $$C^k$$ extension to $${\mathbb {R}}^m$$ . In this paper, we prove a version of the Whitney Extension Theorem in the case of $$C^1$$ , horizontal extensions for mappings defined on compact subsets of $${\mathbb {R}}$$ taking values in the sub-Riemannian Heisenberg group $$\mathbb {H}^n$$ . [ABSTRACT FROM AUTHOR]

We obtain area growth estimates for constant mean curvature graphs in $$\mathbb {E}(\kappa ,\tau )$$ -spaces with $$\kappa \le 0$$ , by finding sharp upper bounds for the volume of geodesic balls in $$\mathbb {E}(\kappa ,\tau )$$ . We focus on complete graphs and graphs with zero boundary values. For instance, we prove that entire graphs in $$\mathbb {E}(\kappa ,\tau )$$ with critical mean curvature have at most cubic intrinsic area growth. We also obtain sharp upper bounds for the extrinsic area growth of graphs with zero boundary values, and study distinguished examples in detail such as invariant surfaces, k-noids, and ideal Scherk graphs. Finally, we give a relation between height and area growth of minimal graphs in the Heisenberg space ( $$\kappa =0$$ ), and prove a Collin-Krust type estimate for such minimal graphs. [ABSTRACT FROM AUTHOR]

Recently, shearlet groups have received much attention in connection with shearlet transforms applied for orientation sensitive image analysis and restoration. The square integrable representations of the shearlet groups provide not only the basis for the shearlet transforms but also for a very natural definition of scales of smoothness spaces, called shearlet coorbit spaces.The aim of this paper is twofold: first we discover isomorphisms between shearlet groups and other well-knowngroups, namely extended Heisenberg groups and subgroups of the symplectic group. Interestingly, the connected shearlet group with positive dilations has an isomorphic copy in the symplectic group, while this is not true for the full shearlet group with all nonzero dilations. Indeed we prove the general result that there exist, up to adjoint action of the symplectic group, only one embedding of the extended Heisenberg algebra into the Lie algebra of the symplectic group. Having understood the various group isomorphisms it is natural to ask for the relations between coorbit spaces of isomorphic groups with equivalent representations. These connections are examined in the second part of the paper. We describe how isomorphic groups with equivalent representations lead to isomorphic coorbit spaces. In particular we apply this result to square integrable representations of the connected shearlet groups and metaplectic representations of subgroups of the symplectic group. This implies the definition of metaplectic coorbit spaces. Besides the usual full and connected shearlet groups we also deal with Toeplitz shearlet groups. [ABSTRACT FROM AUTHOR]

We use intersections with horizontal manifolds to show that high-dimensional cycles in the Heisenberg group can be approximated efficiently by simplicial cycles. This lets us calculate all of the higher-order Dehn functions of the Heisenberg groups, thus proving a conjecture of Gromov. [ABSTRACT FROM AUTHOR]

We establish a 'low rank property' for Sobolev mappings that pointwise solve a first-order nonlinear system of PDEs, whose smooth solutions have the so-called 'contact property'. As a consequence, Sobolev mappings from an open set of the plane, taking values in the first Heisenberg group $$\mathbb {H}^1$$ , and that have almost everywhere maximal rank must have images with positive 3-dimensional Hausdorff measure with respect to the sub-Riemannian distance of $$\mathbb {H}^1$$ . This provides a complete solution to a question raised in a paper by Balogh et al. (Ergodic Theory Dynam Syst 26(3):621-651, ). Our approach differs from the previous ones. Its technical aspect consists in performing an 'exterior differentiation by blow-up', where the standard distributional exterior differentiation is not possible. This method extends to higher-dimensional Sobolev mappings, taking values in higher-dimensional Heisenberg groups. [ABSTRACT FROM AUTHOR]

We prove several rearrangement theorems in the setting of a metric measure space. We adapt the general scheme of the argument to the Heisenberg group, where we study Steiner and circular rearrangement for functions and sets having a suitable symmetry. [ABSTRACT FROM AUTHOR]

We introduce the Besov space $\dot{B}^{0,L}_{1,1}$ associated with the Schrödinger operator L with a nonnegative potential satisfying a reverse Hölder inequality on the Heisenberg group, and obtain the molecular decomposition. We also develop the Hardy space $H_{L}^{1}$ associated with the Schrödinger operator via the Littlewood-Paley area function and give equivalent characterizations via atoms, molecules, and the maximal function. Moreover, using the molecular decomposition, we prove that $\dot{B}^{0,L}_{1,1}$ is a subspace of $H_{L}^{1}$. [ABSTRACT FROM AUTHOR]

The purpose of this article is to expand the number of examples for which the complex Green operator, that is, the fundamental solution to the Kohn Laplacian, can be computed. We use the Lie group structure of quadric submanifolds of ℂ n×ℂ m and the group Fourier transform to reduce the □ b equation to ones that can be solved using modified Hermite functions. We use Mehler’s formula and investigate (1) quadric hypersurfaces, where the eigenvalues of the Levi form are not identical (including possibly zero eigenvalues), and (2) the canonical quadrics in ℂ 4 of codimension two. [ABSTRACT FROM AUTHOR]

We give an explicit, and geometrical formula for the fundamental solution for higher order sub-Laplacians on a model step two nilpotent Lie group. [ABSTRACT FROM AUTHOR]

The primary purpose of this paper is to obtain explicit, coordinate-based descriptions of Ricci flow solutions-especially those corresponding to Ricci solitons-on two classes of nilpotent Lie groups. On the odd-dimensional classical Heisenberg groups, we determine the asymptotics of Ricci flow starting at any metric, and use Lott's blowdown method to demonstrate convergence to soliton metrics. On the groups of real unitriangular matrices, which are more complicated, we describe the solitons and corresponding solutions using a suitable ansatz. [ABSTRACT FROM AUTHOR]

We show the existence of a non-injective uniformly quasiregular mapping acting on the one-point compactification $\bar{ {\mathbb{H}}}^{1}={\mathbb{H}}^{1}\cup\{\infty\}$ of the Heisenberg group ℍ equipped with a sub-Riemannian metric. The corresponding statement for arbitrary quasiregular mappings acting on sphere ${\mathbb{S}}^{n} $ was proven by Martin (Conform. Geom. Dyn. 1:24-27, ). Moreover, we construct uniformly quasiregular mappings on $\bar{ {\mathbb{H}}}^{1}$ with large-dimensional branch sets. We prove that for any uniformly quasiregular map g on $\bar{ {\mathbb{H}}}^{1}$ there exists a measurable CR structure μ which is equivariant under the semigroup Γ generated by g. This is equivalent to the existence of an equivariant horizontal conformal structure. [ABSTRACT FROM AUTHOR]

In this article, we give an explicit calculation of the partial Fourier transform of the fundamental solution to the □-heat equation on quadric submanifolds M⊂ℂ×ℂ. As a consequence, we can also compute the heat kernel associated with the weighted $\overline{\partial}$-equation in ℂ when the weight is given by exp (− φ( z, z)⋅ λ) where φ:ℂ×ℂ→ℂ is a quadratic, sesquilinear form and λ∈ℝ. Our method involves the representation theory of the Lie group M and the group Fourier transform. [ABSTRACT FROM AUTHOR]

We investigate certain singular integral operators with Riesz-type kernels on s-dimensional Ahlfors-David regular subsets of Heisenberg groups. We show that L-boundedness, and even a little less, implies that s must be an integer and the set can be approximated at some arbitrarily small scales by homogeneous subgroups. It follows that the operators cannot be bounded on many self-similar fractal subsets of Heisenberg groups. [ABSTRACT FROM AUTHOR]

We study the ℍ-regular surfaces, a class of intrinsic regular hypersurfaces in the setting of the Heisenberg group ℍ n = ℂ n × ℝ = ℝ2n+1 endowed with a leftinvariant metric d∞ equivalent to its Carnot-Carathéodory (CC) metric. Here hypersurface simply means topological codimension 1 surface and by the words “intrinsic” and “regular” we mean, respectively notions involving the group structure of ℍ n and its differential structure as CC manifold. In particular, we characterize these surfaces as intrinsic regular graphs inside ℍ n by studying the intrinsic regularity of the parameterizations and giving an areatype formula for their intrinsic surface measure. [ABSTRACT FROM AUTHOR]

In this article we study the singular integral operators along the curve on the Heisenberg group. It is a variable coefficient extension of the singular integrals along the odd curves on the Euclidean space ℝ2. The proof is based on the generalized Calderon-Zygmund theory on the space of homogeneous type. [ABSTRACT FROM AUTHOR]

In this article we prove that bounded Hua-harmonic functions on tube domains that satisfy some boundary regularity condition are necessarily pluriharmonic. In doing so, we show that a similar theorem is true on one-dimensional extensions of the Heisenberg group or equivalently on the Siegel upper half-plane. [ABSTRACT FROM AUTHOR]

Let M be the quotient of the Heisenberg group by a discrete co-compact subgroup, with the natural strongly pseudoconvex CR structure. We identify the eigenvalues and eigenforms of the Kohn Laplacians on M and show how to realize M as the boundary of a bounded domain in a line bundle over an Abelian variety. [ABSTRACT FROM AUTHOR]

In this article the following class of partial differential operators is examined for local solvability: Let P(X, Y) be a homogeneous polynomial of degree n ≥ 2 in the non-commuting variables X and Y. Suppose that the complex polynomial P(iz, 1) has distinct roots and that P(z, 0) = zn. The operators which we investigate are of the form P(X, Y) where X = δx and Y = δy + xδw for variables (x, y, w) ∈ ℝ3. We find that the operators P (X, Y) are locally solvable if and only if the kernels of the ordinary differential operators P(iδx, ± x)* contain no Schwartz-class functions other than the zero function. The proof of this theorem involves the construction of a parametrix along with invariance properties of Heisenberg group operators and the application of Sobolev-space inequalities by Hörmander as necessary conditions for local solvability. [ABSTRACT FROM AUTHOR]

In this paper we initiate the study of quasiregular maps in a sub-Riemannian geometry of general Carnot groups. We suggest an analytic definition for quasiregularity and then show that nonconstant quasiregular maps are open and discrete maps on Carnot groups which are two-step nilpotent and of Heisenberg type; we further establish, under the same assumption, that the branch set of a nonconstant quasiregular map has Haar measure zero and, consequently, that quasiregular maps are almost everywhere differentiable in the sense of Pansu. Our method is that of nonlinear potential theory. We have aimed at an exposition accessible to readers of varied background. [ABSTRACT FROM AUTHOR]