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Topological states of matter can be classified only in terms of global topological invariants. These global topological invariants are encoded in terms of global observable topological phase factors in the state vectors of electrons. In condensed matter, the energy spectrum of the Hamiltonian operator has a band structure, meaning that it is piecewise continuous. The energy in each continuous piece depends on the quasi-momentum which varies in the Brillouin zone. Thus, the Brillouin zone of quasi-momentum variables constitutes the base localization space of the energy eigenstates of electrons. This is a continuous topological parameter space bearing the homotopy of a torus. Since the base localization space has the homotopy of a torus, if we vary the quasi-momentum in a direction, when the edge of the zone is reached, we obtain a closed path. Then, if we lift this loop from the base space to the sections of the sheaf-theoretic fibration induced by the localization of the energy eigenfunctions, we obtain a global topological phase factor which encodes the topological structure of the Brillouin zone. Because it is homotopically equivalent to a torus, the global phase factor turns out to be quantized, taking integer values. The experimental significance of this model stems from the recent discovery that there are observable global topological phase factors in fairly ordinary materials. In this communication, we show that it is the unitary representation theory of the discrete Heisenberg group in terms of commutative modular symplectic variables, giving rise to a joint commutative representation space endowed with an integral and Z2-invariant symplectic form that articulates the specific form of the topological conditions characterizing both the quantum Hall effect and the spin quantum Hall effect under a unified sheaf-theoretic cohomological framework.

The concentration-compactness principle for the Trudinger–Moser-type inequality in the Euclidean space was established crucially relying on the Pólya–Szegő inequality which allows to adapt the symmetrization argument. As far as we know, the first concentration-compactness principle of Trudinger–Moser type in non-Euclidean settings, such as the Heisenberg (and more general stratified) groups where the Pólya–Szegő inequality fails, was found in [J. Li, G. Lu and M. Zhu, Concentration-compactness principle for Trudinger–Moser inequalities on Heisenberg groups and existence of ground state solutions, Calc. Var. Partial Differential Equations 57 2018, 3, Paper No. 84] by developing a nonsmooth truncation argument. In this paper, we establish the concentration-compactness principle of Trudinger–Moser type on any compact Riemannian manifolds as well as on the entire complete noncompact Riemannian manifolds with Ricci curvature lower bound. Our method is a symmetrization-free argument on Riemannian manifolds where the Pólya–Szegő inequality fails. This method also allows us to give a completely symmetrization-free argument on the entire Heisenberg (or stratified) groups which refines and improves a proof in the paper of Li, Lu and Zhu. Our results also show that the bounds for the suprema in the concentration-compactness principle on compact manifolds are continuous and monotone increasing with respect to the volume of the manifold.

We extend to the parabolic setting some of the ideas originated with Xiao Zhong's proof in [31] of the Hölder regularity of $p-$harmonic functions in the Heisenberg group $\mathbb{H}^n$. Given a number $p\ge 2$, in this paper we establish the $C^{\infty}$ smoothness of weak solutions of a class of quasilinear PDE in $\mathbb{H}^n$ modeled on the equation $$∂_t u= \sum_{i=1}^{2n} X_i \bigg((1+|\nabla_0 u|^2)^{\frac{p-2}{2}} X_i u\bigg).$$

In this paper, we are concerned with the following a new critical nonlocal Schrödinger-Poisson system on the Heisenberg group:

In this paper, sufficient conditions are obtained for the nonexistence of solutions to the Cauchy problems for ultra-parabolic equations involving fractional Laplacian operator. The nonexistence result is also extended to the corresponding system. The results to such problems are derived in the Heisenberg group. The proofs of our nonexistence theorems are based on the nonlinear capacity method. This method is based on the choice of a suitable test function in the weak formulation of the considered problems.

We define Kakeya sets in the Heisenberg group and show that the Heisenberg Hausdorff dimension of Kakeya sets in the first Heisenberg group is at least 3. This lower bound is sharp since, under our definition, the \{xoy\}-plane is a Kakeya set with Heisenberg Hausdorff dimension 3. [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]

The paper deals with the existence of nontrivial solutions for (p , Q) equations in the Heisenberg group H n with critical exponential growth at infinity and a singular behavior at the origin. The main features and novelty of the paper are the above generality on the right-hand side of the equation, the (p , Q) growth of the elliptic operator and the fact that the equation is studied in the entire Heisenberg group. [ABSTRACT FROM AUTHOR]

The aim of this article is to derive some Lewy-Stampacchia estimates and the existence of solutions for equations driven by a nonlocal integro-differential operator on the Heisenberg group. [ABSTRACT FROM AUTHOR]

We formulate and prove an analogue of Beurling's theorem for the Fourier transform on the Heisenberg group. As a consequence, we deduce Hardy and Cowling-Price theorems. [ABSTRACT FROM AUTHOR]

An Euclidean submanifold is said to be of finite Chen-type if its coordinate functions are a finite sum of eigenfunctions of its Laplacian [6]. In this article, we study helicoidal invariant surfaces in the 3-dimensional Heisenberg group H3, which has finite type immersion. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of least-energy nodal (sign-changing) solutions for a class of critical Schrödinger-Poisson system on the Heisenberg group given by - Δ H u + μ ϕ | u | q - 2 u = λ f (ξ , u) + | u | 2 u , in Ω , - Δ H ϕ = | u | q , in Ω , u = ϕ = 0 , on ∂ Ω , where Δ H is the Kohn-Laplacian on the first Heisenberg group H 1 , and Ω ⊂ H 1 is a smooth bounded domain, λ > 0 and μ ∈ R are some real parameters. Under the suitable conditions on f, together with the constraint variational method and the quantitative deformation lemma, we obtain the existence, energy estimates and the convergence property of the least energy sign-changing solution. However, there are several difficulties arising in the framework of Heisenberg groups, also due to the presence of the non-local coefficient as well as critical nonlinearities. [ABSTRACT FROM AUTHOR]

This note concerns Loomis–Whitney inequalities in Heisenberg groups H n : | K | ≲ ∏ j = 1 2 n | π j (K) | n + 1 n (2 n + 1) , K ⊂ H n. Here π j , j = 1 , ... , 2 n , are the vertical Heisenberg projections to the hyperplanes { x j = 0 } , respectively, and | · | refers to a natural Haar measure on either H n , or one of the hyperplanes. The Loomis–Whitney inequality in the first Heisenberg group H 1 is a direct consequence of known L p improving properties of the standard Radon transform in R 2 . 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 H 1 . The same approach, combined with multilinear interpolation, also yields the following strong type bound: ∫ H n ∏ j = 1 2 n f j (π j (p)) d p ≲ ∏ j = 1 2 n ‖ f j ‖ n (2 n + 1) n + 1 for all nonnegative measurable functions f 1 , ... , f 2 n on R 2 n . These inequalities and their geometric corollaries are thus ultimately based on planar geometry. Among the applications of Loomis–Whitney inequalities in H n , we mention the following sharper version of the classical geometric Sobolev inequality in H n : ‖ u ‖ 2 n + 2 2 n + 1 ≲ ∏ j = 1 2 n ‖ X j u ‖ 1 2 n , u ∈ B V (H n) , where X j , j = 1 , ... , 2 n , are the standard horizontal vector fields in H n . Finally, we also establish an extension of the Loomis–Whitney inequality in H n , where the Heisenberg vertical coordinate projections π 1 , ... , π 2 n are replaced by more general families of mappings that allow us to apply the same inductive approach based on the L 3 / 2 - L 3 boundedness of an operator in the plane. [ABSTRACT FROM AUTHOR]

Let L = - Δ H n + V be the Schrödinger operator on the Heisenberg group H n , where Δ H n is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class B q for q ≥ Q / 2 . Suppose that b ∈ B M O ρ θ (H n) , which is larger than B M O (H n) . We prove that the operator T β 1 , β 2 = (- Δ H n + V) - β 1 V β 2 is bounded from the Herz space K ˙ p 1 α , p (H n) into K ˙ p 2 α , p (H n) . By a maximal estimate, we obtain the boundedness of the commutators [ b , T β 1 , β 2 ] and [ b , T β ] from K ˙ p 1 α , p (H n) into K ˙ p 2 α , p (H n) , where T β = (- Δ H n + V) - β . [ABSTRACT FROM AUTHOR]

This paper is the first of a three part series in which we explore geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of Cn × Cm. In this paper, we present a streamlined calculation for a general integral formula for the complex Green operator N and the projection onto the nullspace of □b. The main application of our formulas is the critical case of codimension two quadrics in C4 where we discuss the known solvability and hypoellipticity criteria of Peloso and Ricci [J. Funct. Anal. 203 (2003), pp. 321–355] We also provide examples to show that our formulas yield explicit calculations in some well-known cases: the Heisenberg group and a Cartesian product of Heisenberg groups. [ABSTRACT FROM AUTHOR]

A small ball problem and Chung's law of iterated logarithm for a hypoelliptic Brownian motion in Heisenberg group are proven. In addition, bounds on the limit in Chung's law are established. [ABSTRACT FROM AUTHOR]

Here, the existence of at least one positive radial solution of the Nonlinear nonhomogeneous Neumann problem − Δ H n u + R (ξ) u = a (| ξ | H n ) | u | q − 2 u 1 + | u | q − p − b (| ξ | H n ) | u | q − 2 u , ξ ∈ Ω is proved on the Heisenberg group H n , via variational principle, where Ω is the Korányi ball, a and b are nonnegative radial functions and R (ξ) holds in some suitable conditions. [ABSTRACT FROM AUTHOR]

The quantum transition probability assignment is an equiareal transformation from the annulus of symplectic spinorial amplitudes to the disk of complex state vectors, which makes it equivalent to the equiareal projection of Archimedes. The latter corresponds to a symplectic synchronization method, which applies to the quantum phase space in view of Weyl’s quantization approach involving an Abelian group of unitary ray rotations. We show that Archimedes’ method of synchronization, in terms of a measure-preserving transformation to an equiareal disk, imposes the integrality of the quantum of action, and requires the extension of the classical moment map from the real line to the circle. Additionally, the same synchronization method is encoded in the structure of the Heisenberg group, viewed as a principal bundle with a connection, whose curvature and anholonomy is expressed in terms of area bounding loops in relation to the underlying Abelian shadow on a symplectic plane. In this manner, we show that the geometric phase pertains to the minimal synchronized area π ħ of the 2-d symplectic Abelian shadow of the symplectic ball, modulo Z . The integrality condition naturally leads to the consideration of modular commutative observables pertaining to the role of the discrete Heisenberg group. We prove that the structural transition from non-commutativity to modular commutativity in accordance to Weyl’s group-theoretic commutation relations takes place via universal factorization through the discrete Heisenberg group. In this way, we derive a homology-theoretic formulation of the synchronization method in terms of the area-bounding cells of the modular lattice R 2 Z 2 in relation to any Abelian symplectic shadow. Thus, we finally obtain the physical interpretation of the analytic representation of quantum states as theta functions corresponding to the sections of a complex line bundle with an integral symplectic structure. [ABSTRACT FROM AUTHOR]

Using a generalized Bochner type theorem for Olshanski spherical pairs, we prove a Lévy-Khinchin formula for the infinite dimensional Heisenberg group H ∞ = M (∞ , C) × R relatively to the action of the product group K ∞ = U (∞) × U (∞) . [ABSTRACT FROM AUTHOR]

In this paper, Green’s first and second identities are extended to (p, q)-sub-Laplacians on the Heisenberg group with some of their applications. Particularly, we apply Green’s identities to a nonlinear boundary value problem that involves the (p, q)-sub-Laplacian in order to show uniqueness of a positive weak solution. [ABSTRACT FROM AUTHOR]

This paper is devoted to the weak and strong estimates for the linear and multilinear fractional Hausdorff operators on the Heisenberg group Hn{{\mathbb{H}}}^{n}. A sharp strong estimate for TΦm{T}_{\Phi }^{m} is obtained. As an application, we derive the sharp constant for the product Hardy operator on Hn{{\mathbb{H}}}^{n}. Some weak-type (p,q)\left(p,q) (1≤p≤∞)\left(1\le p\le \infty ) estimates for TΦ,β{T}_{\Phi ,\beta } are also obtained. As applications, we calculate some sharp weak constants for the fractional Hausdorff operator on the Heisenberg group. Besides, we give an explicit weak estimate for TΦ,β→m{T}_{\Phi ,\overrightarrow{\beta }}^{m} under some mild assumptions on Φ\Phi . We extend the results of Guo et al. [Hausdorff operators on the Heisenberg group, Acta Math. Sin. (Engl. Ser.) 31 (2015), no. 11, 1703–1714] to the fractional setting.

Abstract The aim of this paper is to study the oscillation of solutions of the nonlinear degenerate elliptic equation in the Heisenberg group H n $H^{n}$ . We first derive a critical inequality in H n $H^{n}$ . Based on it, we establish a Picone-type differential inequality and a Sturm-type comparison principle. Then we obtain an oscillation theorem. Our result generalizes the related conclusions for the nonlinear elliptic equations in R n $R^{n}$ .

In this article, we derive a generalised nonlinear Picone's identity for p sub-Laplacian on the Heisenberg group. As an application of Picone's identity, we prove a Hardy type inequality and Picone's inequality. We also establish some qualitative results involving the system of nonlinear equations involving p-sub-Laplacian.

This article deals with existence and multiplicity of normalized solutions for critical Schrödinger-Poisson system involving the p -subLaplacian in the Heisenberg group. Under appropriate assumptions, together with the truncation technique, the concentration-compactness principle, the genus theory, we prove that the existence and multiplicity of the normalized solutions in the L p -subcritical case. As far as we know, this study seems to be the first contribution regarding existence of normalized solutions for the critical p -subLaplacian Schrödinger-Poisson system in the Heisenberg group. Moreover, the results of the paper are completely new even in the Euclidean case. [ABSTRACT FROM AUTHOR]

This paper gives the first example of a unipotent group that is not virtually abelian and preserves a strictly convex domain.

We discuss the notion of domain variation solution for some degenerate elliptic two-phase free boundary problems as well as the viscosity definition of the problem when the operator is degenerate.

We examine the geometry of the level sets of particular horizontally p-harmonic functions in the Heisenberg group. We find sharp, natural geometric conditions ensuring that the level sets of the p-capacitary potential of a bounded annulus in the Heisenberg group are strictly starshaped.

We give a complete analytic and geometric description of the horofunction boundary for polygonal sub-Finsler metrics, that is, those that arise as asymptotic cones of word metrics, on the Heisenberg group. We develop theory for the more general case of horofunction boundaries in homogeneous groups by connecting horofunctions to Pansu derivatives of the distance function.

Let L = − Δ H n + μ be a generalized Schrödinger operator on the Heisenberg group H n , where Δ H n is the sub-Laplacian, and μ is a nonnegative Radon measure satisfying certain conditions. In this paper, we first establish some estimates of the fundamental solution and the heat kernel of L. Applying these estimates, we then study the Hardy spaces H L 1 (H n) introduced in terms of the maximal function associated with the heat semigroup e − t L ; in particular, we obtain an atomic decomposition of H L 1 (H n) , and prove the Riesz transform characterization of H L 1 (H n). The dual space of H L 1 (H n) is also studied. [ABSTRACT FROM AUTHOR]

We complete the study started in the paper [P. Pucci, L. Temperini, On the concentration-compactness principle for Folland-Stein spaces and for fractional horizontal Sobolev spaces, Math. Eng. 5 (2023), Paper no. 007], giving some applications of its abstract results to get existence of solutions of certain critical equations in the entire Heinseberg group. In particular, different conditions for existence are given for critical horizontal p-Laplacian equations. [ABSTRACT FROM AUTHOR]

Let { T t } t > 0 be the semigroup of linear operators generated by a Schrödinger operator L = - Δ H n + V on Heisenberg group H n , where Δ H n is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class B Q / 2 and Q is the homogeneous dimension of H n . Let ∫ 0 ∞ λ d E L (λ) be the spectral resolution of L , we prove that if a function F satisfies a Mihlin condition with exponent α > Q / 2 then the operator F (L) = ∫ 0 ∞ F (λ) d E L (λ) is bounded on Hardy space H L 1 (H n) . [ABSTRACT FROM AUTHOR]

In this paper we study evolutive first order Mean Field Games in the Heisenberg group; each agent can move in the whole space but it has to follow "horizontal" trajectories which are given in terms of the vector fields generating the group and the kinetic part of the cost depends only on the horizontal velocity. The Hamiltonian is not coercive in the gradient term and the coefficients of the first order term in the continuity equation may have a quadratic growth at infinity. The main results of this paper are two: the former is to establish the existence of a weak solution to the Mean Field Game systems while the latter is to represent this solution following the Lagrangian formulation of the Mean Field Games. We also provide some generalizations to Heisenberg-type structures. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of nonlocal Schrödinger‐Poisson type system with sublinear term. With the aid of the Ekeland's variational principle and the mountain pass theorem, the existence of negative energy solution, positive energy solution, and positive ground state solution is obtained, respectively. Moreover, we also obtain the multiplicity of solutions by using the Clark theorem. The novelty of this paper is that the equation has not only nonlocal term but also nonlocal coefficient. [ABSTRACT FROM AUTHOR]

The existence of at least one positive radial solution of the generalized Kohn–Laplacian problem (1) − Δ H n u + R (ξ) u = ∑ i = 1 k a i (| ξ | H n ) | u | p i − 2 u − ∑ j = 1 m b j (| ξ | H n ) | u | q j − 2 u ξ ∈ Ω , u > 0 ξ ∈ Ω , ∂ u ∂ n = 0 ξ ∈ ∂ Ω , is proved, where Δ H n is the Kohn–Laplacian (Heisenberg–Laplacian) operator, Ω is a Korányi ball, a i , 1 ≤ i ≤ k and b j , 1 ≤ j ≤ m are nonnegative radial functions and R (ξ) satisfies some suitable conditions. [ABSTRACT FROM AUTHOR]

We study contact 3-manifolds using the techniques of sub-Riemannian geometry and geometric measure theory, in particular establishing properties of their Lipschitz homotopy groups. We prove a biLipschitz version of the Theorem of Darboux: a contact (2 n + 1) -manifold endowed with a sub-Riemannian structure is locally biLipschitz equivalent to the Heisenberg group H n with its Carnot-Carathéodory metric. Then each contact (2 n + 1) -manifold endowed with a sub-Riemannian structure is purely k -unrectifiable for k > n. We then extend results of Dejarnette et al. [5] and Wenger and Young [20] on the Lipschitz homotopy groups of H 1 to an arbitrary contact 3-manifold endowed with a Carnot-Carathéodory metric, namely that for any contact 3-manifold the first Lipschitz homotopy group is uncountably generated and all higher Lipschitz homotopy groups are trivial. Therefore, in the sense of Lipschitz homotopy groups, a contact 3-manifold is a K (π, 1) -space with an uncountably generated first homotopy group. Along the way, we prove that each open distributional embedding between purely 2-unrectifiable sub-Riemannian manifolds induces an injective map on the associated first Lipschitz homotopy groups. Therefore, each open subset of a contact 3-manifold determines an uncountable subgroup of the first Lipschitz homotopy group of the contact 3-manifold. [ABSTRACT FROM AUTHOR]

Given a surface S in a 3D contact sub-Riemannian manifold M, we investigate the metric structure induced on S by M, in the sense of length spaces. First, we define a coefficient K̂ at characteristic points that determines locally the characteristic foliation of S. Next, we identify some global conditions for the induced distance to be finite. In particular, we prove that the induced distance is finite for surfaces with the topology of a sphere embedded in a tight coorientable distribution, with isolated characteristic points. [ABSTRACT FROM AUTHOR]

Let µA be the singular measure on the Heisenberg group Hn supported on the graph of the quadratic function '(y) = ytAy, where A is a 2n × 2n real symmetric matrix. If det(2A ± J) 6= 0, we prove that the operator of convolution by µA on the right is bounded from L(2n+2)/(2n+1)(Hn) to L2n+2(Hn). We also study the type set of the measures d (y, s) = (y)|y|-dµA(y, s), for 0 6 < 2n, where is a cut-off function around the origin on R2n. Moreover, for = 0 we characterize the type set of 0. [ABSTRACT FROM AUTHOR]

The paper deals with the existence of solutions for (p,Q)(p,Q) coupled elliptic systems in the Heisenberg group, with critical exponential growth at infinity and singular behavior at the origin. We derive existence of nonnegative solutions with both components nontrivial and different, that is solving an actual system, which does not reduce into an equation. The main features and novelties of the paper are the presence of a general coupled critical exponential term of the Trudinger-Moser type and the fact that the system is set in ℍn{{\mathbb{H}}}^{n}.

In this paper we revisit the proof of the Alt-Caffarelli-Friedman monotonicity formula. Then, in the framework of the Heisenberg group, we discuss the existence of an analogous monotonicity formula introducing a necessary condition for its existence, recently proved in [18].