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This note concernsLoomis–Whitney inequalitiesin Heisenberg groups$$\mathbb {H}^n$$Hn:$$\begin{aligned} |K| \lesssim \prod _{j=1}^{2n}|\pi _j(K)|^{\frac{n+1}{n(2n+1)}}, \qquad K \subset \mathbb {H}^n. \end{aligned}$$|K|≲∏j=12n|πj(K)|n+1n(2n+1),K⊂Hn.Here$$\pi _{j}$$πj,$$j=1,\ldots ,2n$$j=1,…,2n, are thevertical Heisenberg projectionsto the hyperplanes$$\{x_j=0\}$${xj=0}, respectively, and$$|\cdot |$$|·|refers to a natural Haar measure on either$$\mathbb {H}^n$$Hn, or one of the hyperplanes. The Loomis–Whitney inequality in the first Heisenberg group$$\mathbb {H}^1$$H1is a direct consequence of known$$L^p$$Lpimproving properties of the standard Radon transform in$$\mathbb {R}^2$$R2. In this note, we show how the Loomis–Whitney inequalities in higher dimensional Heisenberg groups can be deduced by an elementary inductive argument from the inequality in$$\mathbb {H}^1$$H1. The same approach, combined with multilinear interpolation, also yields the following strong type bound:$$\begin{aligned} \int _{\mathbb {H}^n} \prod _{j=1}^{2n} f_j(\pi _j(p))\;dp\lesssim \prod _{j=1}^{2n} \Vert f_j\Vert _{\frac{n(2n+1)}{n+1}} \end{aligned}$$∫Hn∏j=12nfj(πj(p))dp≲∏j=12n‖fj‖n(2n+1)n+1for all nonnegative measurable functions$$f_1,\ldots ,f_{2n}$$f1,…,f2non$$\mathbb {R}^{2n}$$R2n. These inequalities and their geometric corollaries are thus ultimately based on planar geometry. Among the applications of Loomis–Whitney inequalities in$$\mathbb {H}^n$$Hn, we mention the following sharper version of the classical geometric Sobolev inequality in$$\mathbb {H}^n$$Hn:$$\begin{aligned} \Vert u\Vert _{\frac{2n+2}{2n+1}} \lesssim \prod _{j=1}^{2n}\Vert X_ju\Vert ^{\frac{1}{2n}}, \qquad u \in BV(\mathbb {H}^n), \end{aligned}$$‖u‖2n+22n+1≲∏j=12n‖Xju‖12n,u∈BV(Hn),where$$X_j$$Xj,$$j=1,\ldots ,2n$$j=1,…,2n, are the standard horizontal vector fields in$$\mathbb {H}^n$$Hn. Finally, we also establish an extension of the Loomis–Whitney inequality in$$\mathbb {H}^n$$Hn, where the Heisenberg vertical coordinate projections$$\pi _1,\ldots ,\pi _{2n}$$π1,…,π2nare replaced by more general families of mappings that allow us to apply the same inductive approach based on the$$L^{3/2}$$L3/2-$$L^3$$L3boundedness of an operator in the plane.

Since, the quantum modular variables are encoded in terms of one-parameter unitary groups, we are led to a careful re-evaluation of the Heisenberg group, from where the commutation relations emerge from. In particular, the re-evaluation pertains to the discrete Heisenberg group from the perspective of its genuine descent from a more fundamental layer of structure targeting the origin of the non-commutativity of quantum observables. Due to the fact that the modular variables commute, they give rise to an integrality condition inherent to the structure of the discrete Heisenberg group. In this manner, this group should mediate in the structural transition from non-commutativity to its integral Abelian shadow, which is qualified symplectically via the non-squeezing theorem. In particular, the integrality of symplectic area pertains to the global topology of the torus, meaning that the phase space of the 2-d Abelian shadow is topologically toroidal and isomorphic to the modular lattice $$\frac{\mathbb R^2}{\mathbb Z^2}$$ , such that it is universally covered by $$\mathbb R^2$$ , where $$\mathbb Z^2$$ is the free Abelian group in two generators acting by integer translations. The main conclusion of this work is that the structural transition from the non-Abelian-free fundamental group of based loops $$\Theta _2$$ , expressing the synchronization condition of the joint action of modular variables, to the Abelian-free homology group of cycles $$\mathbb Z^2$$ , where the entangled symplectic area pertains, factorizes though the discrete Heisenberg group. This elucidates the role and status of modular variables in quantum mechanics and constitutes a viable theoretical explanation of the nature and appearance of quantum interference phenomena underlying the significance of the notion of a geometric phase.

Let $$G_n:=U(n)\ltimes {\mathbb {H}}_n$$ be the semidirect product of the unitary group acting by automorphisms on the Heisenberg group $${\mathbb {H}}_n.$$ It was pointed out in [5], that the unitary dual $$\widehat{G_n}$$ of $$G_n$$ is homeomorphic to the space of admissible coadjoint orbits $${\mathfrak {g}}_n^\ddagger /G_n$$ of $$G_n.$$ One of the important subset of $$\widehat{G_n}$$ is what is called the cortex of $$G_n,$$ cor $$(G_n),$$ which is the set of all $$\pi \in \widehat{G_n}$$ that cannot be Hausdorff separated from the identity representation $$1_{G_n}$$ of $$G_n.$$ In the present paper, we use the orbit method of Lipsman to determine the cortex of $$G_n.$$

We study the dynamics of magnetic flows on Heisenberg groups. Let $H$ denote the three-dimensional simply connected Heisenberg Lie group endowed with a left-invariant Riemannian metric and an exact, left-invariant magnetic field. Let $\Gamma$ be a lattice subgroup of $H,$ so that $\Gamma\backslash H$ is a closed nilmanifold. We first find an explicit description of magnetic geodesics on $H$, then determine all closed magnetic geodesics and their lengths for $\Gamma \backslash H$. We then consider two applications of these results: the density of periodic magnetic geodesics and marked magnetic length spectrum rigidity., Comment: 32 pages

Global interference phenomena start to manifest in quantum transitions involving at least two projection filters which cannot be realized simultaneously, like in the case of the double-slit experiment. In terms of the 3-vertex invariants of the projective space of rays, thought of as a symplectic phase space, we encounter products involving the complex-valued $${\mathbf {I}}_3 (\varPsi _{in}, \varPsi _b, \varPsi _{fin})$$ with the complex conjugate $$\mathbf {I_3}^*(\varPsi _{in}, \varPsi _b', \varPsi _{fin})$$ , where the projection operators $$|\psi _b \rangle \langle \psi _b |$$ and $$|\psi _b' \rangle \langle \psi _b' |$$ are not simultaneously realizable. In this way, interference can be expressed in terms of products of this form defined over squares in the space of rays, which can be triangulated. Triangles in this space encode the invariant information of the geometric phase factor. From this perspective, we qualify the proposal of Aharonov and collaborators pertaining to the consideration of commutative modular variables evaluated in $$\mathbb R/ {\mathbb {Z}}$$ in deciphering the quantum interference pattern of the double-slit experiment. The quantum modular variables pertaining to conjugate observables are encoded in terms of one-parameter unitary groups acting jointly on the phase space, thus modeled through the continuous group action of $${\mathbb {R}}^2$$ . We show that $$\frac{h}{2}$$ expresses the minimal indistinguishable invariant area of the 2D symplectic Abelian shadow of the symplectic ball of radius $$R=\sqrt{\hbar }$$ in the 2n-phase space of the conjugate position and momenta. The above conclusion leads to a re-estimation of Weyl’s view of the quantum kinematical space in terms of an Abelian group of unitary ray rotations, and in particular the role that the discrete Heisenberg group plays in this conundrum.

Let $${\mathbb {H}}^{n}={\mathbb {C}}^{n}\times {\mathbb {R}}$$ be the n-dimensional Heisenberg group, $$Q=2n+2$$ be the homogeneous dimension of $${\mathbb {H}}^{n}$$ . We establish in this paper that the following sharpened Trudinger–Moser inequalities on the Heisenberg group $${\mathbb {H}}^{n} $$ under the homogeneous constraints of the Sobolev norm: $$\begin{aligned} \sup _{\Vert \nabla _{{\mathbb {H}}}u\Vert _{Q}^{Q}+\Vert u\Vert _{Q}^{Q}\le 1} \int _{{\mathbb {H}}^{n}}\Phi _Q\left( \alpha _{Q}\left( 1+\alpha \Vert u\Vert _{Q}^{Q} \right) ^{\frac{1}{Q-1}}|u|^{\frac{Q}{Q-1}}\right) d\xi

Under the framework of the complex column-vector loop algebra Cp, we propose a scheme for generating nonisospectral integrable hierarchies of evolution equations which generalizes the applicable scope of the Tu scheme. As applications of the scheme, we work out a nonisospectral integrable Schrodinger hierarchy and its expanding integrable model. The latter can be reduced to some nonisospectral generalized integrable Schrodinger systems, including the derivative nonlinear Schrodinger equation once obtained by Kaup and Newell. Specially, we obtain the famous Fokker-Plank equation and its generalized form, which has extensive applications in the stochastic dynamic systems. Finally, we investigate the Lie group symmetries, fundamental solutions and group-invariant solutions as well as the representation of the tensor of the Heisenberg group H3 and the matrix linear group SL(2,R) for the generalized Fokker-Plank equation (GFPE).

We show that if a distribution does not depend on the vertical coordinates, then the Schouten curvature tensor coincides with the Riemannian curvature. The Schouten curvature tensor and the nonholonomicity tensor are used to write the Jacobi equation for the distribution. This leads to a study of second-order optimality conditions for horizontal geodesics in sub-Riemannian geometry. We study conjugate points for horizontal geodesics on the Heisenberg group as an example.

In this paper we investigate the property of engulfing for H-convex functions defined on the Heisenberg group $${\mathbb {H}^n}$$ 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 $${\mathbb {H}^n}$$ H n -section, as well as a new condition of engulfing associated to the $${\mathbb {H}^n}$$ H n -sections, for an H-convex function defined in $$\mathbb {H}^n.$$ H n . These sections, that arise as suitable unions of horizontal sections, are dimensionally larger; as a matter of fact, the $${\mathbb {H}^n}$$ H n -sections, with their engulfing property, will lead to the definition of a quasi-distance in $${\mathbb {H}^n}$$ 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.

We give the direct method of moving planes for solutions to the conformally invariant fractional power subLaplace equation on the Heisenberg group. The method is based on four maximum principles derived here. Then symmetry and nonexistence of positive cylindrical solutions are proved.

In this paper, we recall some properties of the Fourier transform on the quaternionic Heisenberg group, then we prove some qualitative uncertainty principles associated with this transform.

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.

The aim of this paper is to show that a supercritical Neumann problem has at least one positive radial solution via recent variational principle and defining an energy functional with a hidden symmetry associated with the problem. Also, we study the problem in the setting of the Heisenberg group $${\mathbb {H}}^n$$ , which is a special case of locally compact groups.

Any free nilpotent Lie algebra is determined by its rank and step. We consider free nilpotent Lie algebras of steps 3 and 4 and corresponding connected and simply connected Lie groups. We construct Casimir functions of such groups, i.e., invariants of the coadjoint representation. For free 3-step nilpotent Lie groups, we get a full description of coadjoint orbits. It turns out that general coadjoint orbits are affine subspaces, and special coadjoint orbits are affine subspaces or direct products of nonsingular quadrics. The knowledge of Casimir functions is useful for investigation of integration properties of dynamical systems and optimal control problems on Carnot groups. In particular, for some wide class of time-optimal problems on 3-step free Carnot groups, we conclude that extremal controls corresponding to two-dimensional coadjoint orbits have the same behavior as in time-optimal problems on the Heisenberg group or on the Engel group.

Many fundamental results of pluripotential theory on the quaternionic space $${\mathbb {H}}^n$$ are extended to the Heisenberg group. We introduce notions of a plurisubharmonic function, the quaternionic Monge–Ampere operator, differential operators $$d_0$$ and $$d_1$$ and a closed positive current on the Heisenberg group. The quaternionic Monge–Ampere operator is the coefficient of $$ (d_0d_1u)^n$$ . We establish the Chern–Levine–Nirenberg type estimate, the existence of quaternionic Monge–Ampere measure for a continuous quaternionic plurisubharmonic function and the minimum principle for the quaternionic Monge–Ampere operator. Unlike the tangential Cauchy–Riemann operator $$ {\overline{\partial }}_b $$ on the Heisenberg group which behaves badly as $$ \partial _b{\overline{\partial }}_b\ne -{\overline{\partial }}_b\partial _b $$ , the quaternionic counterpart $$d_0$$ and $$d_1$$ satisfy $$ d_0d_1=-d_1d_0 $$ . This is the main reason that we have a good theory for the quaternionic Monge–Ampere operator than $$ (\partial _b{\overline{\partial }}_b)^n$$ .

It is well known that the cohomology of any non-trivial 1-dimensional local system on a nilmanifold vanishes (this result, due to J. Dixmier, was also announced and proved in some particular case by Alaniya). A complex nilmanifold is a quotient of a nilpotent Lie group equipped with a left-invariant complex structure by an action of a discrete, co-compact subgroup. We prove a Dolbeault version of Dixmier’s and Alaniya’s theorem, showing that the Dolbeault cohomology $$ {H}^{0,p}\left(\mathfrak{g},L\right) $$ of a nilpotent Lie algebra with coefficients in any non-trivial 1-dimensional local system vanishes. Note that the Dolbeault cohomology of the corresponding local system on the manifold is not necessarily zero. This implies that the twisted version of Console–Fino theorem is false (Console–Fino proved that the Dolbeault cohomology of a complex nilmanifold is equal to the Dolbeault cohomology of its Lie algebra, when the complex structure is rational). As an application, we give a new proof of a theorem due to H. Sawai, who obtained an explicit description of LCK nilmanifolds. An LCK structure on a manifold M is a Kahler structure on its cover $$ \tilde{M} $$ such that the deck transform map acts on $$ \tilde{M} $$ by homotheties. We show that any complex nilmanifold admitting an LCK structure is Vaisman, and is obtained as a compact quotient of the product of a Heisenberg group and the real line.

This paper aims to define and study currents and slices of currents in the Heisenberg group $${\mathbb {H}}^n$$ H n . Currents, depending on their integration properties and on those of their boundaries, can be classified into subspaces and, assuming their support to be compact, we can work with currents of finite mass, define the notion of slices of Heisenberg currents and show some important properties for them. While some such properties are similarly true in Riemannian settings, others carry deep consequences because they do not include the slices of the middle dimension n, which opens new challenges and scenarios for the possibility of developing a compactness theorem. Furthermore, this suggests that the study of currents on the first Heisenberg group $${\mathbb {H}}^1$$ H 1 diverges from the other cases, because that is the only situation in which the dimension of the slice of a hypersurface, $$2n-1$$ 2 n - 1 , coincides with the middle dimension n, which triggers a change in the associated differential operator in the Rumin complex.

In this note, we show that sub-Riemannian manifolds can contain branching normal minimizing geodesics. This phenomenon occurs if and only if a normal geodesic has a discontinuity in its rank at a non-zero time, which in particular for a strictly normal geodesic means that it contains a non-trivial abnormal subsegment. The simplest example is obtained by gluing the three-dimensional Martinet flat structure with the Heisenberg group in a suitable way. We then use this example to construct more general types of branching., 11 pages, 1 figure. Final version, to appear on Geometric And Functional Analysis (GAFA)

In this paper, we consider the generalized Weyl–Heisenberg group $${\mathbb {H}}(G_\tau )$$ associated with the semi direct product group $$G_\tau =H \times _\tau K$$ in which H is a locally compact group, K is a locally compact abelian group and $$\tau :H \rightarrow \hbox {Aut}(K)$$ is a continuous homomorphism. We give a square integrable representation on $${\mathbb {H}}(G_\tau )$$ and then as a result, we obtain admissible wavelets on this group. Moreover, we define the localization operator on $${\mathbb {H}}(G_\tau )$$ and we show that it is a bounded and compact operator. Furthermore, we investigate the boundedness and compactness of localization operators with two admissible wavelets on $$L^p({\hat{K}})$$ , ( $${\hat{K}}$$ the dual group of K), for the generalized Weyl–Heisenberg group. Finally, some example are given.

We give an integral representation of the fundamental solution of the heat and resolvent equation associated to the Casimir–Laplace operator for the quaternionic Heisenberg group. As an application we obtain the corresponding wave kernel and Green function.

Let $${\mathcal {L}}=-\varDelta _{{\mathbb {H}}^n}+V$$ be a Schrodinger operator on the Heisenberg group $${\mathbb {H}}^n$$ , where $$\varDelta _{{\mathbb {H}}^n}$$ is the sublaplacian on $${\mathbb {H}}^n$$ and the nonnegative potential V belongs to the reverse Holder class $$RH_q$$ with $$q\ge Q/2$$ . Here $$Q=2n+2$$ is the homogeneous dimension of $${\mathbb {H}}^n$$ . In this paper the author first introduces a class of Morrey spaces associated with the Schrodinger operator $${\mathcal {L}}$$ on $${\mathbb {H}}^n$$ . Then by using some pointwise estimates of the kernels related to the nonnegative potential V, the author establishes the boundedness properties of the Littlewood–Paley function $${\mathfrak {g}}_{{\mathcal {L}}}$$ and the Lusin area integral $${\mathcal {S}}_{{\mathcal {L}}}$$ (with respect to the heat semigroup $$\{e^{-s{\mathcal {L}}}\}_{s>0}$$ ) acting on the Morrey spaces. It can be shown that the same conclusions also hold for the operators $${\mathfrak {g}}_{\sqrt{{\mathcal {L}}}}$$ and $${\mathcal {S}}_{\sqrt{{\mathcal {L}}}}$$ with respect to the Poisson semigroup $$\{e^{-s\sqrt{{\mathcal {L}}}}\}_{s>0}$$ .

In this work, we study non-characteristic domains of Heisenberg groups. We prove that bounded domains which are diffeomorphic to the solid torus having the center of the group as rotation axis, are non-characteristic. Then we state the following conjecture: the bounded non-characteristic domains of the Heisenberg group of dimension 1 are diffeomorphic to a solid torus having the center of the group as revolution axis.

We show that, under a natural scaling, the small-time behavior of the logarithmic derivatives of the subelliptic heat kernel on $SU(2)$ converges to their analogues on the Heisenberg group at time 1. Realizing $SU(2)$ as $\mathbb{S}^3$, we then generalize these results to higher-order odd-dimensional spheres equipped with their natural subRiemannian structure, where the limiting spaces are now the higher-dimensional Heisenberg groups., 32 pages, corrected errors in preliminary estimates in Section 2.1

The aim of this paper is to study Sasakian immersions of (non-compact) complete regular Sasakian manifolds into the Heisenberg group and into $$ {\mathbb{B}}^N\times {\mathbb{R}}$$ equipped with their standard Sasakian structures. We obtain a complete classification of such manifolds in the η-Einstein case.

In this paper we study Hardy inequalities in the Heisenberg group $${\mathbb {H}}^n$$ , with respect to the Carnot–Caratheodory distance $$\delta $$ 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 $$\nabla \!_{\mathbb {H}}\delta $$ . 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–Caratheodory balls. In particular, we show that the associated constant is bounded on homogeneous cones $$C_\Sigma $$ with base $$\Sigma \subset {\mathbb {S}}^{2n}$$ , even when $$\Sigma $$ 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.

In this paper, we study questions related to the theory of stochastic processes on Lie nilpotent groups. In particular, we consider the stochastic process on the Heisenberg group H3(ℝ) whose trajectories satisfy the horizontal conditions in the stochastic sense relative to the standard contact structure on H3 (ℝ). It is shown that this process is a homogeneous Markov process relative to the Heisenberg group operation. There was found a representation in the form of a Wiener integral for a one-parameter linear semigroup of operators for which the Heisenberg sublaplacian generated by basis vector fields of the corresponding Lie algebra L(H3) is producing. The main method of solving the problem in this paper is using the path integrals technique, which indicates the common direction of further development of the results.

This paper is concerned with the following Schrodinger-Poisson type system: $$\left\{\begin{array}{ll}{-\Delta_{H} u+\mu \phi u=\lambda|u|^{q-2} u+|u|^{2} u,} \;\;\;\;\;\;\;\;\;\;\;\;\;\; {\text { in } \Omega} ,\\{-\Delta_{H} \phi=u^{2},} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; {\text { in } \Omega}, \\{\phi=u=0,}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ {\text { on } \partial \Omega},\end{array}\right.$$ where Δh is the Kohn-Laplacian on the first Heisenberg group ℍ1 and Ω ⊂ ℍ1 is a smooth bounded domain, 1 0 some real parameters. By the Green’s representation formula, the concentration compactness and the critical point theory, we prove that the above system has at least two positive solutions for μ < S × meas(Ω)−½ and λ small enough, where S is the best Sobolev constant. Moreover, we show also that there is a positive ground state solution for the above system. Our result is new even in the Euclidean case.

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.

Let M be a left module over a ring R with identity and let $$\beta $$ be a skew-symmetric R-bilinear form on M. The generalized Heisenberg group consists of the set $$M\times M\times R = \{(x, y, t):x, y\in M, t\in R\}$$ with group law $$\begin{aligned} (x_1, y_1, t_1)(x_2, y_2, t_2) = (x_1+x_2, y_1+y_2, t_1+\beta (x_1, y_2)+t_2). \end{aligned}$$ Under the assumption of 2 being a unit in R, we prove that the generalized Heisenberg group decomposes into a product of its subset and subgroup, similar to the well-known polar decomposition in linear algebra. This leads to a parametrization of the generalized Heisenberg group that resembles a parametrization of the Lorentz transformation group by relative velocities and space rotations.

The paper deals with the k-plane transform $${\mathcal {R}}_{k}$$ on the Heisenberg group. We study the properties of the transform $${\mathcal {R}}_{k}$$ and obtain three types of inversion formulas for $${\mathcal {R}}_{k}$$. The first inversion is deduced with the help of the group Fourier transform, together with the partial Riesz potential and Heisenberg sublaplacian. By this formula, another two formulas are established in terms with the adjoint of $${\mathcal {R}}_{k}$$ and the wavelet, respectively.

The Nappi-Witten model is a Wess-Zumino-Witten model in which the target space is the nonreductive Heisenberg group $H_4$. We consider the representation theory underlying this conformal field theory. Specifically, we study the category of weight modules, with finite-dimensional weight spaces, over the associated affine vertex operator algebra $\mathsf{H}_4$. In particular, we classify the irreducible $\mathsf{H}_4$-modules in this category and compute their characters. We moreover observe that this category is nonsemisimple, suggesting that the Nappi-Witten model is a logarithmic conformal field theory., 21 pages; introduction expanded, references added, minor notation changes

In this paper, we introduce a curve shortening flow in a 3-dimensional pseudohermitian manifold with vanishing torsion. The flow preserves the Legendrian condition and decreases the length of curves. The stationary solution of our flow is a Legendrian geodesic. We classify the singularity and prove some convergence results. Moreover, we study the flow in Heisenberg group especially with Type I singularity.

In this paper, we discuss some sufficient conditions for the boundedness of rough Hausdorff operators on the central Morrey and Herz spaces with the Muckenhoupt weights on the Heisenberg group. The boundedness for the commutators of rough Hausdorff operators with symbols in weighted CMO space on such spaces is given as well.

This paper is a development of the results and techniques of the two papers (Carvalho-Neto and Planas in J Differ Equ 259:2948–2980, 2015; Oka in J Math Anal Appl 473:382–407, 2019) for the aim of addressing the existence and uniqueness of local and global mild solutions, on the Heisenberg group $$\mathbb {H}^d$$ , of the time-fractional Navier–Stokes system with time derivative of order $$\alpha \in (0, 1)$$ . The proof relies on Schaefer’s fixed point theorem.

The k-Cauchy–Fueter complex in quaternionic analysis was generalized to the Heisenberg group in a previous paper (Ren et al., in Adv Appl Clifford Algebras 30(2): Paper No. 20, 2020). But it is not known whether this differential complex is exact or not. In this paper, we apply the Penrose transform (the twistor method) to a double fibration of homogeneous spaces of $$\mathrm{SO}(2N,{\mathbb {C}})$$ to prove its exactness on the Heisenberg group.

The Kepler–Heisenberg problem is that of determining the motion of a planet around a sun in the Heisenberg group, thought of as a three-dimensional sub-Riemannian manifold. The sub-Riemannian Hamiltonian provides the kinetic energy, and the gravitational potential is given by the fundamental solution to the sub-Laplacian. The dynamics are at least partially integrable, possessing two first integrals as well as a dilational momentum which is conserved by orbits with zero energy. The system is known to admit closed orbits of any rational rotation number, which all lie within the fundamental zero-energy integrable subsystem. Here, we demonstrate that, under mild conditions, zero-energy orbits are self-similar. Consequently, we find that these zero-energy orbits stratify into three families: future collision, past collision, and quasi-periodicity without collision. If a collision occurs, it occurs in finite time.

We use the Weyl correspondence approach to investigate the spectral operators of the Laguerre and the Kautz systems. We show that the basic functions in the Laguerre and the Kautz systems are the eigenvectors of modified Sturm-Liouville operators. The results are further generalized to multiple parameters of one complex variable in both the unit disc and the upper half-plane contexts.

The study of the degenerate part of the absolute of the discrete Heisenberg group required solving a problem on the number of geodesics in this group and in its semigroup. Analytically, this problem reduces to the study of the asymptotic behavior of Gaussian q-binomial coefficients, and the required property is the almost multiplicativity of group characters. The problem has a natural formulation in terms of an (apparently, new) asymptotic property of Young diagrams.

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 BMO estimates for the logarithm of the Jacobian. 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 of quasiconformal densities and metrics in domains in $${\mathbb {H}}^{1}$$. The theorems are discussed for the sub-Riemannian and the Koranyi distances. This extends results due to Astala–Gehring, Astala–Koskela, Koskela and Bonk–Koskela–Rohde.

In this paper, we prove the strong unique continuation property for the following degenerate elliptic equation 0.1$$\begin{aligned} \Delta _zu +|z|^2\partial _t^2u = Vu,\quad (z,t) \in {\mathbb {R}}^N \times {\mathbb {R}} \end{aligned}$$where the potential V satisfies either of the following growth assumptions 0.2$$\begin{aligned}&\left| V(z,t) \right| \le \frac{f(\rho (z,t))}{\rho (z,t)^2},\ \text {where} \rho \text { is as in }(2.1)\text { and }\nonumber \\&\quad f\text { satisfies the Dini integrability condition as in }(1.3). \end{aligned}$$or when $$\begin{aligned}&\left| V(z,t) \right| \le C \frac{\psi (z,t)^{\epsilon }}{\rho (z,t)^2},\ \text {for some }\epsilon >0\text { with }\psi \text { as in }(2.6)\text { and } N\text { even.} \end{aligned}$$This extends some of the previous results obtained in [18] for this subfamily of Baouendi–Grushin operators. As corollaries, we obtain new unique continuation properties for solutions u to $$\begin{aligned} \Delta _{{\mathbb {H}}} u = Vu \end{aligned}$$with certain symmetries as expressed in (1.6) where $$\Delta _{{\mathbb {H}}}$$ corresponds to the sub-Laplacian on the Heisenberg group $${\mathbb {H}}^n$$.

We propose analogues of Green’s and Picone’s identities for the p-sub-Laplacian on stratified Lie groups. In particular, these imply a generalised Diaz-Saa inequality. Using these we derive a comparison principle and uniqueness of positive solutions to nonlinear hypoelliptic equations on general stratified Lie groups extending to this setting previously known results on Euclidean and Heisenberg groups.

In this paper we establish an upper estimate and a 3G-theorem for the Green function of the Khon Laplacian $$\Delta _{{\mathbb {H}}}$$ on a domain D of the Heisenberg group $${{\mathbb {H}}^n}$$ . We also establish a generalized triangle property which allows us to introduce a new Kato class for the ball.

In this paper, we study some global properties of curves in the Heisenberg group $$H_{1}$$. In particular, we obtain Fenchel-type theorem and Fary–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 $$S^{3}$$.

We prove an analogue of Pfannschmidt’s uncertainty principle for the group Fourier transform on the Heisenberg group. Gutzmer’s formula for the Hermite expansions plays an important role in the proof.

In this paper, we establish some necessary and sufficient conditions for the boundedness of generalized multilinear Hausdorff operators on weighted central Morrey, Herz, and Morrey–Herz spaces on the Heisenberg group. Moreover, we also give some sufficient conditions for the boundedness of generalized multilinear Hausdorff operators on such spaces with the Muckenhoupt weights. Our results generalize some previous known results.

In this paper we consider the wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups with time-dependent H\"older propagation speeds. The examples are the time-dependent wave equation for the sub-Laplacian on the Heisenberg group or on general stratified Lie groups, or $p$-evolution equations for higher order operators, already in all these cases our results being new. We establish sharp well-posedness results in the spirit of the classical result by Colombini, de Giorgi and Spagnolo. In particular, we describe an interesting loss of regularity phenomenon depending on the step of the group and on the order of the considered operator., Comment: 22 pages

In this paper, a shift-invariant system of the form $$\{L_{a\nu }g_s:\nu \in \varGamma ,s\in {{\mathbb {Z}}}\}$$ for $$a>0$$ is studied on the Heisenberg group $${{\mathbb {H}}^n}$$ , where $$L_x$$ denotes the left translation operator on $${{\mathbb {H}}^n}$$ and $$\varGamma $$ is a lattice in $${{\mathbb {H}}^n}$$ . The characterizations for the shift-invariant system to be a Bessel sequence and a frame sequence are given in terms of certain operators arising from the fiber map corresponding to this system. For a shift invariant system to be a frame sequence (Riesz sequence) the characterization is given in terms of dual Gramian (Gramian). Moreover, the problem of characterizing a pair of shift-invariant systems to be dual frames is also discussed.

We study stable surfaces, i.e., second order minima of the area for variations of fixed volume, in sub-Riemannian space forms of dimension $3$. We prove a stability inequality and provide sufficient conditions ensuring instability of volume-preserving area-stationary $C^2$ surfaces with a non-empty singular set of curves. Combined with previous results, this allows to describe any complete, orientable, embedded and stable $C^2$ surface $\Sigma$ in the Heisenberg group $\mathbb{H}^1$ and the sub-Riemannian sphere $\mathbb{S}^3$ of constant curvature $1$. In $\mathbb{H}^1$ we conclude that $\Sigma$ is a Euclidean plane, a Pansu sphere or congruent to the hyperbolic paraboloid $t=xy$. In $\mathbb{S}^3$ we deduce that $\Sigma$ is one of the Pansu spherical surfaces discovered in [28]. As a consequence, such spheres are the unique $C^2$ solutions to the sub-Riemannian isoperimetric problem in $\mathbb{S}^3$., Comment: 31 pages, no figures

In this paper, we compute sub-Riemannian limits of Gaussian curvature for a Euclidean $$C^2$$ -smooth surface in the BCV spaces and the twisted Heisenberg group away from characteristic points and signed geodesic curvature for Euclidean $$C^2$$ -smooth curves on surfaces. We get Gauss–Bonnet theorems in the BCV spaces and the twisted Heisenberg group.