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In this paper, we prove new Liouville-type properties of fully nonlinear subelliptic equations modelled on the Heisenberg group. A key tool that can be of independent interest is a strong maximum principle for solutions of such differential equations. [ABSTRACT FROM AUTHOR]

In this article, we study the recent development of the qualitative uncertainty principle on certain Lie groups. In particular, we consider that if the Weyl transform on certain step-two nilpotent Lie groups is of finite rank, then the function has to be zero almost everywhere as long as the nonvanishing set for the function has finite measure. Further, we consider that if the Weyl transform of each Fourier–Wigner piece of a suitable function on the Heisenberg motion group is of finite rank, then the function has to be zero almost everywhere whenever the nonvanishing set for each Fourier–Wigner piece has finite measure. [ABSTRACT FROM AUTHOR]

We construct some linear nonhomogeneous differential operator on the Heisenberg group whose kernel is interconnected with the Lie algebra of the group of conformal mappings. In more detail, the kernel of coincides with first two coordinate functions of mappings of the Lie algebra of conformal mappings. We derive the integral representation formula and give a coercive estimate for . [ABSTRACT FROM AUTHOR]

For any parameter vector (r,t,ω) in ℍ := ℝ × ℝn × ℝn, the representation matrix S(r,t,ω) of the Heisenberg group (ℍ,∗) is extended to a new form SB,W,A(r,t,ω) with nonsingular matrices B,W,A which are related to the directional translation 풯Bt : f → f(⋅− Bt), directional modulation ℳWω : f(x) →eix′Wω and dilation 풟A : f →|det(A)|− 1 2f(A−1⋅) if B′W = I n, the identity matrix of order n. In the case B′W≠I n, an intrinsic group structure under the matrix SB,W,A(r,t,ω) is investigated and a unitary operator representation is offered. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear discontinuous sub-elliptic system with drift on the Heisenberg group, where the coefficients in system are discontinuous and satisfy the vanishing mean oscillation condition and growth conditions with the growth index $ 2 2 < p < ∞ , and the non-homogeneous terms satisfy the controllable growth conditions and the natural growth conditions, respectively. The partial Hölder regularity to weak solutions and the partial Morrey regularity to horizontal gradients of weak solutions are proved by the $ \mathcal {A} $ A -harmonic approximation. One of the difficulties in this paper is how to reasonably define the weak solution and avoid the requirement of integrability of Tu. In fact, we used $ T={X_i}{X_{n+i}}-{X_{n+i}}{X_i} $ T = X i X n + i − X n + i X i . [ABSTRACT FROM AUTHOR]

We prove the convergence of a wide class of continued fractions, including generalized continued fractions over quaternions and octonions. Fractional points in these systems are not bounded away from the unit sphere, so that the iteration map is not uniformly expanding. We bypass this problem by analyzing digit sequences for points that converge to the unit sphere under iteration, expanding on previous methods of Dani–Nogueira. [ABSTRACT FROM AUTHOR]

In the Heisenberg group ℍ1 with a sub-Finsler structure, an (X,Y )-Lipschitz surface which is complete, oriented, connected and stable must be a vertical plane. In particular, the result holds for entire intrinsic graphs of Euclidean Lipschitz functions. [ABSTRACT FROM AUTHOR]

In this work, we study the following Schrödinger-Poisson system { − Δ H u + μ ϕ u = λ u − γ , in Ω , − Δ H ϕ = u 2 , in Ω , u > 0 , 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, μ = ± 1 , 0 < γ < 1 , and λ > 0 are some real parameters. For the above system, we prove the existence and uniqueness of positive solution for μ = 1 and each λ > 0 . Multiple solutions of the system are also considered for μ = − 1 and λ > 0 small enough using the critical point theory for nonsmooth functional. [ABSTRACT FROM AUTHOR]

A class of critical Kirchhoff-Poisson systems on Heisenberg group was studied. Due to the existence of critical and non-local terms, the space embedding was not compact. Under the assumptions that the nonlinear terms were appropriate, the space compactness was overcame by the variational method and established the existence of at least one solution to this system. Furthermore, using deformation lemma and topological degree theory, it was proved that the solution was a sign-changing solution. [ABSTRACT FROM AUTHOR]

Let L = - Δ ℍ n + V {L=-{\Delta}_{\mathbb{H}^{n}}+V} be a Schrödinger operator on Heisenberg groups ℍ n {\mathbb{H}^{n}} , where Δ ℍ n {{\Delta}_{\mathbb{H}^{n}}} is the sub-Laplacian, the nonnegative potential V belongs to the reverse Hölder class B 풬 / 2 {B_{\mathcal{Q}/2}} . Here 풬 {\mathcal{Q}} is the homogeneous dimension of ℍ n {\mathbb{H}^{n}} . In this article, we introduce the fractional heat semigroups { e - t ⁢ L α } t > 0 {\{e^{-tL^{\alpha}}\}_{t>0}} , α > 0 {\alpha>0} , associated with L. By the fundamental solution of the heat equation, we estimate the gradient and the time-fractional derivatives of the fractional heat kernel K α , t L ⁢ ( ⋅ , ⋅ ) {K_{\alpha,t}^{L}(\,\cdot\,,\cdot\,)} , respectively. As an application, we characterize the space BMO L γ ⁢ ( ℍ n ) {\mathrm{BMO}_{L}^{\gamma}(\mathbb{H}^{n})} via { e - t ⁢ L α } t > 0 {\{e^{-tL^{\alpha}}\}_{t>0}} . [ABSTRACT FROM AUTHOR]

Let n ∈ N and H n be the Heisenberg group of dimension 2 n + 1 . Let Ω be a bounded open subset of H n and p ∈ (1 , Q) , where Q is the homogeneous dimension of H n . Within an appropriate framework, we prove an interior pointwise gradient estimate for a weak solution to the problem - div H (A (x , X u)) = g - div H f in Ω , u = 0 on ∂ Ω , where X represents the horizontal gradient on the Heisenberg group. [ABSTRACT FROM AUTHOR]

In this paper, the following conformable fractional sub-elliptic equation is investigated: \[ \begin{cases} -\Delta_H^{\alpha} u = f(\xi,u,\nabla_H^\alpha u), & {\rm in} \ \Omega,\\ u > 0, & {\rm in} \ \Omega,\\ u=0, & {\rm on} \ \partial \Omega. \end{cases} \] { − Δ H α u = f (ξ , u , ∇ H α u) , in Ω , u > 0 , in Ω , u = 0 , on ∂ Ω. Ω represents the bounded domain in space $ \mathbb {H}^n $ H n , $ \alpha \in (0,1) $ α ∈ (0 , 1) , and f>0 is a bounded function. This paper is organized into three sections. In the first section, we present some of the features of the Heisenberg Laplacian. To prove the solution to the above equation, we use the Arzelà–Ascoli theorem and Bolzano–Weierstrass theorem in the second section. We then prove the uniqueness and cylindrical symmetry of the solution in the third section. [ABSTRACT FROM AUTHOR]

This paper is concerned with a PDE-based approach to the horizontally quasiconvex (h-quasiconvex, for short) envelope of a given continuous function in the Heisenberg group. We provide a characterization for upper semicontinuous, h-quasiconvex functions in terms of the viscosity subsolution to a first-order nonlocal Hamilton-Jacobi equation. We also construct the corresponding envelope of a continuous function by iterating the nonlocal operator. One important step in our arguments is to prove the uniqueness and existence of viscosity solutions to the Dirichlet boundary problem for the nonlocal Hamilton-Jacobi equation. Applications of our approach to the h-convex hull of a given set in the Heisenberg group are discussed as well. [ABSTRACT FROM AUTHOR]

(1) Suppose μ is a smooth measure on a smooth hypersurface of positive Gaussian curvature in R 2 n . If n ≥ 2 , then W (μ) , the Weyl transform of μ is a compact operator, and if p > n ≥ 6 , then W (μ) belongs to the p-Schatten class. (2) There exist Schatten class operators with linearly dependent quantum translates. [ABSTRACT FROM AUTHOR]

In this paper, we study the structure of truncated connections on the multidimensional Heisenberg group endowed with a left-invariant sub-Riemannian structure. We find that sub-Riemannian geodesic lines are parabolas whose orthogonal projections onto the corresponding contact planes are straight lines. In addition to such parabolas, some straight lines lying in contact planes are also geodesics. [ABSTRACT FROM AUTHOR]

Here, we consider an anisotropic equation \[ -\Delta _{{\mathbb {H}^n},\overrightarrow {p}}u +a(q)|u|^{p^–2}u=\lambda w(q)|u|^{m-2}u-h(q)|u|^{l-2}u, \] in the Heisenberg group {\mathbb {H}^n}, where the operator \Delta _{{\mathbb {H}^n},\overrightarrow {p}} is the horizontal anisotropic p-Laplacian on the Heisenberg group and is defined in the sequel. By the variational methods, we prove the existence of the entire weak solutions. [ABSTRACT FROM AUTHOR]

Let H denote the 3-dimensional Heisenberg Lie group. The main purpose of this paper is to classify all possible linear control systems on the homogeneous spaces of H through its closed subgroups. We expose a detailed study on the control behavior (i.e., controllability property and control sets) of some particular dynamics evolving on non simply connected homogeneous (state) spaces of dimension two and three. [ABSTRACT FROM AUTHOR]

In this paper, we study rough Hausdorff operators on variable exponent Lebesgue spaces in the setting of the Heisenberg group. We prove the boundedness of rough Hausdorff operators by giving some sufficient conditions. [ABSTRACT FROM AUTHOR]

A flag domain in ℝ 3 is a subset of ℝ 3 of the form { (x , y , t) : y < A ⁢ (x) } , where A : ℝ → ℝ is a Lipschitz function. We solve the Dirichlet and Neumann problems for the sub-elliptic Kohn–Laplacian △ ♭ = X 2 + Y 2 in flag domains Ω ⊂ ℝ 3 , with L 2 -boundary values. We also obtain improved regularity for solutions to the Dirichlet problem if the boundary values have first order L 2 -Sobolev regularity. Our solutions are obtained as sub-elliptic single and double layer potentials, which are best viewed as integral operators on the first Heisenberg group. We develop the theory of these operators on flag domains, and their boundaries. [ABSTRACT FROM AUTHOR]

In this paper, we define the notion of Bott connection in the Heisenberg group (ℍ 3 , g) and derive the expression of the Bott connection by using the Levi-Civita connection. Moreover, we derive the expressions of killing vector fields by using the killing equation and obtain some explicit formulas for killing magnetic curves associated to the Bott connection. Furthermore, we give some examples of killing magnetic curves. [ABSTRACT FROM AUTHOR]

The left-invariant sub-Lorentzian problem on the Heisenberg group is considered. An optimal synthesis is constructed, the sub-Lorentzian distance and spheres are described. [ABSTRACT FROM AUTHOR]

We establish the second variation of sub-Riemannian surface measure for minimal non-horizontal submanifolds of a sub-Riemannian stratified Lie group. We obtain some applications for codimension one. Furthermore, we present a new proof of the fact that the hyperbolic paraboloid is stable in the Heisenberg group. [ABSTRACT FROM AUTHOR]

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 Z 2 -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. [ABSTRACT FROM AUTHOR]

In this article, we reproduce results of classical regularity theory of quasilinear elliptic equations in the divergence form, in the setting of Heisenberg Group. The considered cases encompass a very wide class of equations with isotropic growth conditions that are generalizations of the p -Laplacian and include equations with polynomial or exponential type growth. Some more general conditions have also been explored. [ABSTRACT FROM AUTHOR]

We establish an area formula for the spherical measure of intrinsically regular submanifolds of low codimension in Heisenberg groups. The spherical measure is constructed by an arbitrary homogeneous distance. Among the arguments of the proof, we point out the differentiability properties of intrinsic graphs and a chain rule for intrinsically differentiable functions. [ABSTRACT FROM AUTHOR]

We prove local Lipschitz regularity for weak solutions to a class of degenerate parabolic PDEs modeled on the parabolic p-Laplacian ∂tu = ... Xi(|∇0u|p-2Xiu) in a cylinder Ω × &#8477+, where Ω is domain in the Heisenberg group ℍn, and 2 ≤ p ≤ 4. The result continues to hold in the more general setting of contact subRiemannian manifolds. [ABSTRACT FROM AUTHOR]

Consider the equation { - Δ H 3 ⁢ u = f ⁢ (ξ , u , ∇ H ⁡ u , ∇ H 2 ⁡ u , ∇ H 3 ⁡ u , ∇ H 4 ⁡ u , ∇ H 5 ⁡ u) in ⁢ Ω , u > 0 in ⁢ Ω , u = ∂ ∂ ⁡ ν ⁢ (∇ H 2 ⁡ u) = ∂ ∂ ⁡ ν ⁢ (∇ H 3 ⁡ u) = 0 on ⁢ ∂ ⁡ Ω , where Ω is a domain of the finite-dimensional space ℍ n and f is a positive and bounded function. We prove the existence of a solution for the above equation. In addition, we prove the uniqueness and the cylindrical symmetry of the solution. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Fisher–KPP equation on the Heisenberg group. We discuss the existence of global solutions, asymptotic behavior of global solutions and blow‐up solutions. Moreover, we extend the obtained results to the time‐fractional Fisher–KPP equation on the Heisenberg group. [ABSTRACT FROM AUTHOR]

Let \mathcal {L}=-\Delta _{\mathbb {H}^n}+V be a Schrödinger operator with the nonnegative potential V belonging to the reverse Hölder class B_{Q}, where Q is the homogeneous dimension of the Heisenberg group \mathbb {H}^n. In this paper, we obtain pointwise bounds for the spatial derivatives of the heat and Poisson kernels related to \mathcal {L}. As an application, we characterize the space BMO_{\mathcal {L}}(\mathbb {H}^n), associated to the Schrödinger operator \mathcal {L}, in terms of two Carleson type measures involving the spatial derivatives of the heat kernel of the semigroup \{e^{-s\mathcal {L}}\}_{s>0} and the Poisson kernel of the semigroup \{e^{-s\sqrt {\mathcal {L}}}\}_{s>0}, respectively. At last, we pose a conjecture about the converse characterization of BMO_{\mathcal {L}}(\mathbb {H}^n). [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and nonexistence of positive solutions for the nonlinear integral equations with weights related to the sharp Hardy–Littlewood–Sobolev (hereinafter referred to as HLS) inequality on bounded domains of the Heisenberg group H n : f q − 1 (ξ) = ∫ Ω G (ξ) f (η) G (η) | η − 1 ξ | Q − α d η + λ ∫ Ω f (η) | η − 1 ξ | Q − α − β d η , ξ ∈ Ω ¯ , where q>1, 0 < α < Q , 0 < β < Q − α , Q = 2n + 2 is the homogeneous dimension of H n , λ ∈ R , Ω ⊂ H n is a smooth bounded domain and G (ξ) is nonnegative continuous in Ω ¯ . [ABSTRACT FROM AUTHOR]

We introduce an extended class of cross-Toeplitz operators which act between Fock–Segal–Bargmann spaces with different weights. It is natural to consider these operators in the framework of representation theory of the Heisenberg group. Our main technique is representation of cross-Toeplitz by two-sided relative convolutions from the Heisenberg group. In turn, two-sided convolutions are reduced to usual (one-sided) convolutions on the Heisenberg group of the doubled dimensionality. This allows us to utilise the powerful group-representation technique of coherent states, co- and contra-variant transforms, twisted convolutions, symplectic Fourier transform, etc. We discuss connections of (cross-)Toeplitz operators with pseudo-differential operators, localisation operators in time–frequency analysis, and characterisation of kernels in terms of ladder operators. The paper is written in a detailed and reasonably self-contained manner to be suitable as an introduction into group-theoretical methods in phase space and time–frequency operator theory. [ABSTRACT FROM AUTHOR]

Let 1 < p ≤ 4 when n = 1 , and 1 < p < 3 + 1 n - 1 when n ≥ 2 . We obtain the second-order horizontal Sobolev HW loc 2 , 2 -regularity of p-harmonic functions in the Heisenberg group ℍ n . This improves the known range of p obtained by Domokos and Manfredi in 2005. [ABSTRACT FROM AUTHOR]

In this paper, we make an effort to establish Lieb's and Lions' type theorems on Heisenberg group and then apply them to study the existence of solution for variational problem on Heisenberg group. [ABSTRACT FROM AUTHOR]

Spectral analysis and synthesis studies translation invariant function spaces, so-called varieties over topological groups. The basic building blocks are the finite dimensional varieties. In the commutative case finite dimensional varieties are spanned by exponential polynomials. In non-commutative situations no relevant results exist. In this paper we consider finite dimensional left translation invariant linear spaces of continuous complex valued functions over the Heisenberg group. Using basic knowledge about Lie algebra we describe all left varieties of this type. In particular, it turns out that those function spaces are spanned by exponential polynomials as well. [ABSTRACT FROM AUTHOR]

In this paper, we present necessary conditions for the existence of weak solution of the parabolic-type equations and systems on the Heisenberg group. The main technique for proving the results relies on the method of test functions. [ABSTRACT FROM AUTHOR]

In this paper, we consider a critical (p, q) equation on the Heisenberg group of the following form: - Δ H , p u - Δ H , q u + V (ξ) (| u | p - 2 u + | u | q - 2 u) = μ ∫ H n F (ξ , u) | η - 1 ξ | λ d ξ f (η , u) + | u | q ∗ - 2 u , where the operator - Δ H , ℘ φ = div H (| D H φ | H ℘ - 2 D H φ) , with ℘ ∈ { p , q } , is the proverbial horizontal ℘ -Laplacian on the Heisenberg group, 1 < p < (2 Q - λ) 2 Q q < q < Q , q ∗ = q Q / (Q - q) is the critical exponent, and Q = 2 n + 2 is the homogeneous dimension of H n , μ and λ are some real parameters. Under the appropriate assumptions of potential functions V and f, the existence of entire solutions to the above equation on the Heisenberg group is obtained by using the mountain pass theorem and the concentration compactness principle. The results presented here extend or complete recent papers and are new to critical equations involving (p, q)-Laplacian operators and convolution terms on Heisenberg group. [ABSTRACT FROM AUTHOR]

Consider the sub-Riemannian Heisenberg group H. In this paper, we answer the following question: given a compact set K⊆R and a continuous map f:K→H, when is there a horizontal Cm curve F:R→H such that F/K=f? Whitney originally answered this question for real valued mappings, and Fefferman provided a complete answer for real valued functions defined on subsets of Rn. We also prove a finiteness principle for Cm,ω√ horizontal curves in the Heisenberg group in the sense of Brudnyi and Shvartsman. [ABSTRACT FROM AUTHOR]

Here, the existence and multiplicity of weak solutions to a generalized (p (⋅) , q (⋅)) -Laplace equation involving Leray–Lions type operators with Hardy potential are studied under Dirichlet boundary conditions on the Heisenberg groups. [ABSTRACT FROM AUTHOR]

We study X -valued CR functions of Theodoresco class B 1 on the Heisenberg group H 1 = R × C equipped with the non-embeddable CR structure discovered by Nirenberg [On a question of Hans Lewy. Russian Math Surveys. 1974;29:251–262] where X is an arbitrary complex Fréchet space. If L ¯ φ ≡ L ¯ + φ ∂ / ∂ x is Nirenberg's perturbation of Lewy's operator L ¯ = ∂ / ∂ w ¯ − i w ∂ / ∂ x , we show that for every open neighbourhood U ⊂ H 1 of the origin, no two solutions f a : U → X , a ∈ { 1 , 2 } , of Theodoresco class B 1 , to the tangential Cauchy–Riemann equations L ¯ φ f = 0 are functionally independent. [ABSTRACT FROM AUTHOR]

In this article, the degenerated horizontal (p, q)-Laplacian with weights − Δ H , p a (u) − Δ H , q b (u) = f (ξ , u , D H u) and the competing horizontal (p, q)-Laplacian with weights − Δ H , p a (u) + Δ H , q b (u) = f (ξ , u , D H u) including the Dirchlet boundary condition are studied. The existence and approximation results for these problems are studied where 1 < q < p < ∞ , f : Ω × R × H n → R is a Carathéodory function, Ω is a bounded smooth domain in the Heisenberg group H n and Δ H , p stands for the horizontal p-Laplacian on H n. The proofs are based on weighted Heisenberg Sobolev spaces, Nemytskij operators, Browder-Minty Theorem, and finite dimensional approximation. [ABSTRACT FROM AUTHOR]

The spectral synthesis problem for the phase space associated with the reduced Heisenberg group is studied. The paper deals with the case of subspaces in invariant under the twisted shifts and the action of the unitary group . It is shown that any such subspace is generated by the root vectors of a special Hermite operator contained in this subspace. As a corollary, we obtain the spectral synthesis theorem for subspaces in invariant under the unilateral shifts and the action of the unitary group . [ABSTRACT FROM AUTHOR]

Let L = - Δ H n + V be a Schrödinger operator on Heisenberg groups H n , where Δ H n is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class B q , q ≥ Q / 2 and Q = 2 n + 2 is the homogeneous dimension of H n . We establish a T1 criterion for the boundedness of γ -Schrödinger–Calderón–Zygmund operators on Campanato type spaces BMO L α (H n) . As an application, by the aid of regularity estimate for fractional heat semigroup { e - t L β } t > 0 , we prove the BMO L α -boundedness of operators generated by fractional heat semigroups including the maximal operators, the square functions, the Laplace transform type multipliers and the fractional integral associated with Schrödinger operator L via T1 theorem, respectively. [ABSTRACT FROM AUTHOR]

We consider an asymmetric left-invariant norm ∥ ⋅ ∥ K in the first Heisenberg group ℍ 1 induced by a convex body K ⊂ ℝ 2 containing the origin in its interior. Associated to ∥ ⋅ ∥ K there is a perimeter functional, that coincides with the classical sub-Riemannian perimeter in case K is the closed unit disk centered at the origin of ℝ 2 . Under the assumption that K has C 2 boundary with strictly positive geodesic curvature we compute the first variation formula of perimeter for sets with C 2 boundary. The localization of the variational formula in the non-singular part of the boundary, composed of the points where the tangent plane is not horizontal, allows us to define a mean curvature function H K out of the singular set. In the case of non-vanishing mean curvature, the condition that H K be constant implies that the non-singular portion of the boundary is foliated by horizontal liftings of translations of ∂ ⁡ K dilated by a factor of 1 H K . Based on this we can define a sphere 핊 K with constant mean curvature 1 by considering the union of all horizontal liftings of ∂ ⁡ K starting from (0 , 0 , 0) until they meet again in a point of the vertical axis. We give some geometric properties of this sphere and, moreover, we prove that, up to non-homogeneous dilations and left-translations, they are the only solutions of the sub-Finsler isoperimetric problem in a restricted class of sets. [ABSTRACT FROM AUTHOR]

We provide an elementary proof of the support of the law of a hypoelliptic Brownian motion on the Heisenberg group H. We consider a control norm associated to left-invariant vector fields on H, and describe the support in terms of the space of finite energy horizontal curves. [ABSTRACT FROM AUTHOR]

In this paper, we determine, in the Heisenberg group, the parametric Killing magnetic flux surfaces and their corresponding Killing scalar flux functions. An examples of each are given with a graphic representation in Euclidean space. [ABSTRACT FROM AUTHOR]

When 2 − 1 / Q < p ≤ 2 , we establish the C loc 0 , 1 and C loc 1 , α -regularities of weak solutions to quasi-linear p-Laplacian type non-homogeneous equations in the Heisenberg group H n , where Q = 2 n + 2 is the homogeneous dimension of H n . [ABSTRACT FROM AUTHOR]

This paper deals with the existence of solutions for the noncooperative Schrödinger–Kirchhoff system involving the p-Laplacian operator and critical nonlinearities on the Heisenberg group. Under some suitable conditions, together with the limit index theory and the concentration–compactness principle, we obtain the existence and multiplicity of solutions for this system. To our best knowledge, the existence results for the noncooperative system with p-Laplacian and critical nonlinearities are new on the Heisenberg group. [ABSTRACT FROM AUTHOR]

We obtain a Liouville-type theorem for cylindrical viscosity solutions of fully nonlinear CR invariant equations on the Heisenberg group. As a by-product, we also prove a comparison principle with finite singularities for viscosity solutions to more general fully nonlinear operators on the Heisenberg group. [ABSTRACT FROM AUTHOR]