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In this paper, we study the existence of at least one bounded weak solution for Kohn–Spencer Laplacian with a weight depending on the solution and convection term of the form \[ -div_{\mathbb{H}^n}(\nu(\xi,u) |D_{\mathbb{H}^n}u|^{p-2}_{\mathbb{H}^n}D_{\mathbb{H}^n}u)=f(\xi,u,D_{\mathbb{H}^n} u) \] − di v H n (ν (ξ , u) | D H n u | H n p − 2 D H n u) = f (ξ , u , D H n u) in a bounded domain $ \Omega \subset \mathbb {H}^n $ Ω ⊂ H n . We show the set of solutions is uniformly bounded by a special Moser's iteration. [ABSTRACT FROM AUTHOR]

In this paper, we focus on studying the Cauchy problem for semilinear damped wave equations involving the sub-Laplacian L on the Heisenberg group H n with power type nonlinearity | u | p and initial data taken from Sobolev spaces of negative order homogeneous Sobolev space H ˙ L - γ (H n) , γ > 0 , on H n . In particular, in the framework of Sobolev spaces of negative order, we prove that the critical exponent is the exponent p crit (Q , γ) = 1 + 4 Q + 2 γ , for γ ∈ (0 , Q 2) , where Q : = 2 n + 2 is the homogeneous dimension of H n . More precisely, we establish A global-in-time existence of small data Sobolev solutions of lower regularity for p > p crit (Q , γ) in the energy evolution space; A finite time blow-up of weak solutions for 1 < p < p crit (Q , γ) under certain conditions on the initial data by using the test function method. Furthermore, to precisely characterize the blow-up time, we derive sharp upper bound and lower bound estimates for the lifespan in the subcritical case. [ABSTRACT FROM AUTHOR]

We study the biholomorphic action of the Heisenberg group H n on the Siegel domain D n + 1 ( n ≥ 1 ). Such H n -action allows us to obtain decompositions of both D n + 1 and the weighted Bergman spaces A λ 2 (D n + 1) ( λ > - 1 ). Through the use of symplectic geometry we construct a natural set of coordinates for D n + 1 adapted to H n . This yields a useful decomposition of the domain D n + 1 . The latter is then used to compute a decomposition of the Bergman spaces A λ 2 (D n + 1) ( λ > - 1 ) as direct integrals of Fock spaces. This effectively shows the existence of an interplay between Bergman spaces and Fock spaces through the Heisenberg group H n . As an application, we consider T (λ) (L ∞ (D n + 1) H n) the C ∗ -algebra acting on the weighted Bergman space A λ 2 (D n + 1) ( λ > - 1 ) generated by Toeplitz operators whose symbols belong to L ∞ (D n + 1) H n (essentially bounded and H n -invariant). We prove that T (λ) (L ∞ (D n + 1) H n) is commutative and isomorphic to VSO (R +) (very slowly oscillating functions on R + ), for every λ > - 1 and n ≥ 1 . [ABSTRACT FROM AUTHOR]

We look at a momentum map associated with the Heisenberg group. We show that the cocycle associated with its momentum mapping is the value of a modulus of an associated coadjoint orbit. [ABSTRACT FROM AUTHOR]

In the setting of horizontal curves in the Heisenberg group, we prove a C m , ω finiteness principle, a C m , ω Lusin approximation result, a C ∞ Whitney extension result, and a C ∞ Lusin approximation result. Combined with previous work, this completes the study of Whitney extension and Lusin approximation for horizontal curves of class C m , C m , ω , and C ∞ in the Heisenberg group. [ABSTRACT FROM AUTHOR]

We consider the logarithmic Sobolev inequality on the Heisenberg group. One can derive the logarithmic Sobolev inequality from the Sobolev inequality, and we consider an application to the uncertainty inequality on the Heisenberg group. Moreover, one can also obtain a dissipative estimate of a solution of the heat equation on the Heisenberg group from the logarithmic Sobolev inequality. [ABSTRACT FROM AUTHOR]

In this paper, we prove the fractional Hardy inequality on polarisable metric measure spaces. The integral Hardy inequality for 1 < p ≤ q < ∞ 1

In this paper, we aim at identifying the level sets of the gauge norm in the Heisenberg group ℍ n via the prescription of their (non-constant) horizontal mean curvature. We establish a uniqueness result in ℍ 1 under an assumption on the location of the singular set, and in ℍ n for n ≥ 2 in the proper class of horizontally umbilical hypersurfaces. [ABSTRACT FROM AUTHOR]

We provide a semi-constructive criterion for ellipticity of the differential operator on the Heisenberg group H 1. [ABSTRACT FROM AUTHOR]

For any parameter vector (r , t , ω) in ℍ : = ℝ × ℝ n × ℝ n , the representation matrix S (r , t , ω) of the Heisenberg group (ℍ , ∗) is extended to a new form S B , W , A (r , t , ω) with nonsingular matrices B , W , A which are related to the directional translation B t : f → f (⋅ − B t) , directional modulation ℳ W ω : f (x) → e i x ′ W ω and dilation A : f → | det (A) | − 1 2 f (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 S B , W , A (r , t , ω) is investigated and a unitary operator representation is offered. [ABSTRACT FROM AUTHOR]

This article is devoted to the study of a critical Choquard-Kirchhoff pp-sub-Laplacian equation on the entire Heisenberg group Hn{{\mathbb{H}}}^{n}, where the Kirchhoff function KK can be zero at zero, i.e., the equation can be degenerate, and involving a nonlinearity, which is critical in the sense of the Hardy-Littlewood-Sobolev inequality. We first establish the concentration-compactness principle for the pp-sub-Laplacian Choquard equation on the Heisenberg group, and we then prove existence results.

Abstract We consider the logarithmic Sobolev inequality on the Heisenberg group. One can derive the logarithmic Sobolev inequality from the Sobolev inequality, and we consider an application to the uncertainty inequality on the Heisenberg group. Moreover, one can also obtain a dissipative estimate of a solution of the heat equation on the Heisenberg group from the logarithmic Sobolev inequality.

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 paper, we introduce the notion of the semi-symmetric metric connection in the Heisenberg group. Moreover, by using the method of Riemannian approximations, we define the notions of intrinsic curvature for regular curves, the intrinsic geodesic curvature of regular curves on a surface, and the intrinsic Gaussian curvature of the surface away from characteristic points in the Heisenberg group with the semi-symmetric metric connection. Finally, we derive the expressions of those curvatures and prove the Gauss–Bonnet theorem related to the semi-symmetric metric connection in the Heisenberg group. [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 consider the extended metaplectic representation of the semi-direct product of the Heisenberg group and the symplectic group (the Jacobi group). We give explicit formulas for the Berezin symbols and for the complex Weyl symbols of the corresponding representation operators. Then we deduce formulas for the symbols of the representation operators in the classical Weyl calculus. As an application, we find the classical Weyl symbol of the exponential of an operator whose Weyl symbol is a polynomial on ℝ2n of degree ≤2, recovering a result of L. Hörmander. [ABSTRACT FROM AUTHOR]

We study the nonholonomic motion of a point particle on the Heisenberg group around the fixed “sun” placed at the origin whose potential is given by the fundamental solution of the sub-Laplacian. In contrast with several recent papers that approach this problem as a variational one (hence a control problem) we study the equations of dynamical motion which are non-variational in nonholonomic mechanics. We find three independent first integrals of the system and show that its bounded trajectories are wound up around certain surfaces of the fourth order. We also describe some particular cases of trajectories. [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]

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]

In this paper, we investigate the existence and uniqueness of mild solutions to the fractional Navier–Stokes equations related to time derivative of order α ∈ (0 , 1) . And the mild solution is associated with the sublaplacian provided by the left invariant vector fields on the Heisenberg group. We demonstrate that when the nonlinear external force term matches the applicable conditions, the global mild solution can be obtained by using improved Ascoli–Arzela theorem and Schaefer's fixed point theorem. [ABSTRACT FROM AUTHOR]

We consider a parabolic nonlinear equation on the Heisenberg group. Applying the Gidas–Spruck type estimates, we prove that under suitable conditions, the equation does not have positive solutions. As an application of the nonexistence result, we provide optimal universal estimates for positive solutions.

Abstract 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 , $$ \textstyle\begin{cases} -\Delta _{H}u+\mu \phi u=\lambda u^{-\gamma}, &\text{in } \Omega , \\ -\Delta _{H}\phi =u^{2}, &\text{in } \Omega , \\ u>0, &\text{in } \Omega , \\ u=\phi =0, &\text{on } \partial \Omega , \end{cases} $$ where Δ H $\Delta _{H}$ is the Kohn-Laplacian on the first Heisenberg group H 1 $\mathbb{H}^{1}$ , and Ω ⊂ H 1 $\Omega \subset \mathbb{H}^{1}$ is a smooth bounded domain, μ = ± 1 $\mu =\pm 1$ , 0 < γ < 1 $0 0 $\lambda >0$ are some real parameters. For the above system, we prove the existence and uniqueness of positive solution for μ = 1 $\mu =1$ and each λ > 0 $\lambda >0$ . Multiple solutions of the system are also considered for μ = − 1 $\mu =-1$ and λ > 0 $\lambda >0$ small enough using the critical point theory for nonsmooth functional.

We deal with a wide class of nonlinear nonlocal equations led by integro-differential operators of order (s,p), with summability exponent p in (1,∞) and differentiability order s in (0,1), whose prototype is the fractional subLaplacian in the Heisenberg group. We present very recent boundedness and regularity estimates (up to the boundary) for the involved weak solutions, and we introduce the nonlocal counterpart of the Perron Method in the Heisenberg group, by recalling some results on the fractional obstacle problem. Throughout the paper we also list various related open problems.

This research focuses on introducing a novel algorithm for RGB image encryption. The proposed approach is a hybrid fusion of mathematical principles, namely group theory and chaos theory. It leverages the favorable characteristics of the Heisenberg subgroup of G L 2 n + 1 (Z) and a 4D chaotic laser system derived from the Lorenz-Haken equations. The algorithm aims to provide robust encryption by truncating four sequences from the 4D chaotic map. The initial Heisenberg matrix is generated using two input parameters, while the second matrix is constructed using two additional parameters.To enhance encryption efficiency, four square boxes of dimensions 16 × 16 are designed specifically for the 4D laser system. Each box contains cells that represent the rank of values relative to similar values in the same stream. Furthermore, the source image is divided into three monochromatic components. The foreground of the Heisenberg matrix is multiplied by each channel extracted from the source image. The encryption process proceeds with digram substitution, wherein two-pixel couples are replaced using the values stored in the four box squares. The first value is selected from box 1, followed by the second value from box 4. The remaining values needed to complete the rectangle are retrieved from boxes 2 and 3. For the subsequent digram, a circular rotation of the boxes is performed. The first value is obtained from box 2, the second value from box 3, and so forth, ensuring a continuous circular pattern. Finally, a second multiplication is applied, involving the second plane of the Heisenberg matrix and the result obtained in the previous stage. Experimental results demonstrate the algorithm's effectiveness, exhibiting resistance to differential attacks, statistical attacks, and superior performance compared to existing encryption techniques. [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]

A geodetically convex set in the Heisenberg group H n , n ≥ 1 is defined to be a set with the property that a geodesic joining any two points in the set lies completely in it. Here we classify the geodetically convex sets to be either an empty set, a singleton set, an arc of a geodesic or the whole space H n . We also show that a geodetically convex function on H n is a constant function. These results generalize the known results of H 1 to higher dimensional Heisenberg group. [ABSTRACT FROM AUTHOR]

This paper is primarily devoted to a class of interpolation inequalities of Hardy and Rellich types on the Heisenberg group H n . Consequently, several weighted Hardy type, Heisenberg–Pauli–Weyl uncertainty principle and Hardy–Rellich type inqualities are established on H n . Moreover, new weighted Sobolev type embeddings are derived. Finally, an integral inequality for vector fields in a domain of the Heisenberg group is obtained, leading to several specific weighted Hardy type inequalities by making careful choices of vector fields. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem for the degenerate parabolic equations on the Heisenberg groups with power law non-linearities. We obtain Fujita-type critical exponents, which depend on the homogeneous dimension of the Heisenberg groups. The analysis includes the case of porous medium equations. Our proof approach is based on methods of nonlinear capacity estimates specifically adapted to the nature of the Heisenberg groups. We also use the Kaplan eigenfunctions method in combination with the Hopf-type lemma on the Heisenberg groups. [ABSTRACT FROM AUTHOR]

In this paper, using variational methods and critical point theory we establish the existence of multiple solutions for a class of elliptic equations on Carnot groups depending on one real positive parameter and involving a subcritical nonlinearity. Some recent results are extended and improved. [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]

In this paper, we study the rough fractional Hausdorff operator on variable exponent Morrey–Herz spaces in the setting of the Heisenberg group. We define Morrey–Herz spaces with three variable exponents and then give sufficient and necessary conditions for the boundedness of the rough fractional Hausdorff operator. [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]

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]

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]

In this paper, we study the existence and multiplicity of solutions for the following Kirchhoff-Choquard type equation involving the fractional p-Laplacian on the Heisenberg group: ... where (-Δ)ps is the fractional p-Laplacian on the Heisenberg group ℍN, M is the Kirchhoff function, V (ξ) is the potential function, 0 < s < 1,1 < p < N/s, μ > 0, f (ξ, u) is the nonlinear function, 0 < λ < Q, Q = 2N + 2, and Qλ* = 2Q-λ/Q-2 is the Sobolev critical exponent. Using the Krasnoselskii genus theorem, the existence of infinitely many solutions is obtained if μ is sufficiently large. In addition, using the fractional version of the concentrated compactness principle, we prove that problem has m pairs of solutions if μ is sufficiently small. As far as we know, the results of our study are new even in the Euclidean case. [ABSTRACT FROM AUTHOR]

We look at a momentum map associated with the Heisenberg group. We show that the cocycle associated with its momentum mapping is the value of a modulus of an associated coadjoint orbit.

In this paper, we introduce the notion of the semi-symmetric metric connection in the Heisenberg group. Moreover, by using the method of Riemannian approximations, we define the notions of intrinsic curvature for regular curves, the intrinsic geodesic curvature of regular curves on a surface, and the intrinsic Gaussian curvature of the surface away from characteristic points in the Heisenberg group with the semi-symmetric metric connection. Finally, we derive the expressions of those curvatures and prove the Gauss–Bonnet theorem related to the semi-symmetric metric connection in the Heisenberg group.

We study the sub-Finsler prescribed mean curvature equation, associated to a strictly convex body K 0 ⊆ R 2 n , for t -graphs on a bounded domain Ω in the Heisenberg group H n. When the prescribed datum H is constant and strictly smaller than the Finsler mean curvature of ∂Ω, we prove the existence of a Lipschitz solution to the Dirichlet problem for the sub-Finsler CMC equation by means of a Finsler approximation scheme. [ABSTRACT FROM AUTHOR]

We prove L p estimates for a class of Littlewood-Paley functions on homogeneous groups including the Heisenberg groups under a sharp integrability condition on the kernels. The results will be shown by methods without use of Fourier transform estimates. [ABSTRACT FROM AUTHOR]

We study the existence and multiplicity of solutions both for the isotropic and the anisotropic horizontal p -Kirchhoff equations on the Heisenberg group H n , via Krasnoselskii's genus. Finally, an interesting question for the anisotropic embedding in the subelliptic literature is presented and it is conjectured that the exponent p ‾ ⁎ : = Q n Q ∑ i = 1 n 1 p i − n can be the effective critical exponent in the setting of anisotropic horizontal Sobolev space. [ABSTRACT FROM AUTHOR]