Heisenberg Group Hn Research Articles

Second-Order Regularity for Degenerate Parabolic Quasi-Linear Equations in the Heisenberg Group

In the Heisenberg group Hn, we obtain the local second-order HWloc2,2-regularity for the weak solution u to a class of degenerate parabolic quasi-linear equations ∂tu=∑i=12nXiAi(Xu) modeled on the parabolic p-Laplacian equation. Specifically, when 2≤p≤4, we demonstrate the integrability of (∂tu)2, namely, ∂tu∈Lloc2; when 2≤p<3, we demonstrate the HWloc2,2-regularity of u, namely, XXu∈Lloc2. For the HWloc2,2-regularity, when p≥2, the range of p is optimal compared to the Euclidean case.

A Stone–von Neumann equivalence of categories for smooth representations of the Heisenberg group

The classical Stone–von Neumann theorem relates the irreducible unitary representations of the Heisenberg group Hn to non-trivial unitary characters of its center Z, and plays a crucial role in the construction of the oscillator representation for the metaplectic group. In this paper we extend these ideas to non-unitary and non-irreducible representations, thereby obtaining an equivalence of categories between certain representations of Z and those of Hn. Our main result is a smooth equivalence, which involves the fundamental ideas of du Cloux on differentiable representations and smooth imprimitivity systems for Nash groups. We show how to extend the oscillator representation to the smooth setting and give an application to degenerate Whittaker models for representations of reductive groups. We also include an algebraic equivalence, which can be regarded as a generalization of Kashiwara’s lemma from the theory of D-modules.

The prescribed mean curvature equation for t-graphs in the sub-Finsler Heisenberg group [formula omitted]

We study the sub-Finsler prescribed mean curvature equation, associated to a strictly convex body K0⊆R2n, for t-graphs on a bounded domain Ω in the Heisenberg group Hn. 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.

Horizontal p-Kirchhoff equation on the Heisenberg group

We study the existence and multiplicity of solutions both for the isotropic and the anisotropic horizontal p-Kirchhoff equations on the Heisenberg group Hn, 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‾⁎:=QnQ∑i=1n1pi−n can be the effective critical exponent in the setting of anisotropic horizontal Sobolev space.

Generalized Schrödinger operators on the Heisenberg group and Hardy spaces

Let L=−ΔHn+μ be a generalized Schrödinger operator on the Heisenberg group Hn, where ΔHn 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 HL1(Hn) introduced in terms of the maximal function associated with the heat semigroup e−tL; in particular, we obtain an atomic decomposition of HL1(Hn), and prove the Riesz transform characterization of HL1(Hn). The dual space of HL1(Hn) is also studied.

A Detailed Study of Kirchhoff-type Critical Elliptic Equations and p-Sub-Laplacian Operators within the Heisenberg Group Hn Framework

A Detailed Study of Kirchhoff-type Critical Elliptic Equations and p-Sub-Laplacian Operators within the Heisenberg Group Hn Framework

Weighted estimates for bilinear fractional integral operator on the Heisenberg group

In this article, we consider an analogue of Kenig and Stein's bilinear fractional integral operator on the Heisenberg group Hn. We completely characterize exponents α,β and γ such that the operator is bounded from Lp(Hn,|x|αp)×Lq(Hn,|x|βq) to Lr(Hn,|x|−γr). Also, analogous sharp results are obtained on the Euclidean space.

A characterization of gauge balls in ℍ n by horizontal curvature

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

Cohomology of annuli, duality and L∞-differential forms on Heisenberg groups

In the last few years the authors proved Poincaré and Sobolev type inequalities in Heisenberg groups Hn for differential forms in the Rumin's complex. The need to substitute the usual de Rham complex of differential forms for Euclidean spaces with the Rumin's complex is due to the different stratification of the Lie algebra of Heisenberg groups. The crucial feature of Rumin's complex is that dc is a differential operator of order 1 or 2 according to the degree of the form.Roughly speaking, Poincaré and Sobolev type inequalities are quantitative formulations of the well known topological problem whether a closed form is exact. More precisely, for suitable p and q, we mean that every exact differential form ω in Lp admits a primitive ϕ in Lq such that ‖ϕ‖Lq≤C‖ω‖Lp. The cases of the norm Lp, p≥1 and q<∞ have been already studied in a series of papers by the authors. In the present paper we deal with the limiting case where q=∞: it is remarkable that, unlike in the scalar case, when the degree of the forms ω is at least 2, we can take q=∞ in the left-hand side of the inequality. The corresponding inequality in the Euclidean setting RN (p=N and q=∞) was proven by Bourgain & Brezis.

Parabolic Anderson model on Heisenberg groups: The Itô setting

In this note we focus our attention on a stochastic heat equation defined on the Heisenberg group Hn of order n. This equation is written as ∂tu=12Δu+uW˙α, where Δ is the hypoelliptic Laplacian on Hn and {W˙α;α>0} is a family of Gaussian space-time noises which are white in time and have a covariance structure generated by (−Δ)−α in space. Our aim is threefold: (i) Give a proper description of the noise Wα; (ii) Prove that one can solve the stochastic heat equation in the Itô sense as soon as α>n2; (iii) Give some basic moment estimates for the solution u(t,x).

Gradient estimates for Schrödinger operators with characterizations of 𝐵𝑀𝑂_{ℒ} on Heisenberg groups

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

Analogues of theorems of Chernoff and Ingham on the Heisenberg group

We prove an analogue of Chernoff’s theorem for the Laplacian Δℍ on the Heisenberg group ℍn. As an application, we prove Ingham type theorems for the group Fourier transform on ℍn and also for the spectral projections associated to the sublaplacian.

Regularity for Quasi-Linear p-Laplacian Type Non-Homogeneous Equations in the Heisenberg Group

When 2−1/Q&lt;p≤2, we establish the Cloc0,1 and Cloc1,α-regularities of weak solutions to quasi-linear p-Laplacian type non-homogeneous equations in the Heisenberg group Hn, where Q=2n+2 is the homogeneous dimension of Hn.

Interval Oscillation Theorems for the Weighted p -Sub-Laplacian Equation in the Heisenberg Group

In this paper, we derive a Riccati-type inequality in the Heisenberg group H n . Based on it, some oscillation criteria are established for the weighted p -sub-Laplacian equations in H n . Our results generalize the oscillation theorems for p -sub-Laplacian equations in R n to ones in H n .

Sampling in spaces of entire functions of exponential type in [formula omitted

In this paper we consider the question of sampling for spaces of entire functions of exponential type in several variables. The novelty resides in the growth condition we impose on the entire functions, that is, that their restriction to a hypersurface is square integrable with respect to a natural measure. The hypersurface we consider is the boundary bU of the Siegel upper half-space U and it is fundamental that bU can be identified with the Heisenberg group Hn. We consider entire functions in Cn+1 of exponential type with respect to the hypersurface bU whose restriction to bU are square integrable with respect to the Haar measure on Hn. For these functions we prove a version of the Whittaker–Kotelnikov–Shannon Theorem. Instrumental in our work are spaces of entire functions in Cn+1 of exponential type with respect to the hypersurface bU whose restrictions to bU belong to some homogeneous Sobolev space on Hn. For these spaces, using the group Fourier transform on Hn, we prove a Paley–Wiener type theorem and a Plancherel–Pólya type inequality.

Nodal solutions for Q-Laplacian problem with exponential nonlinearities on the Heisenberg group

This paper deals with the existence of nodal solutions (sign-changing) to the following elliptic equation involving Q-Laplacian:{−ΔQu=λf(ξ,u)inΩ,u=0on∂Ω, where ΔQ(⋅):=divHn(|∇Hn(⋅)|HnQ−2∇Hn(⋅)) is the Q-Laplacian operator, Ω is an open, smooth bounded domain in the Heisenberg group Hn, Q=2n+2 is the homogeneous dimension of Hn, λ is a positive parameter, and nonlinear term f(ξ,u) behaves like exp⁡(β|s|QQ−1) as s→∞. Under 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 solutions for the above problem in subcritical and critical exponential growth cases.

Spherical means on the Heisenberg group: Stability of a maximal function estimate

Consider the surface measure μ on a sphere in a nonvertical hyperplane on the Heisenberg group ℍn, n ≥ 2, and the convolution f * μ. Form the associated maximal function Mf = supt>0 ∣f * μt∣ generated by the automorphic dilations. We use decoupling inequalities due to Wolff and Bourgain—Demeter to prove Lp-boundedness of M in an optimal range.

Homogeneous Fourier and Weyl multipliers on Sobolev spaces related to the Heisenberg group

Inspired by the work of A. Bonami and S. Poornima that a non-constant function which is homogeneous of degree 0 cannot be a Fourier multiplier on homogeneous Sobolev spaces, we establish analogous results for Fourier multipliers on the Heisenberg group Hn and Weyl multipliers on Cn acting on Sobolev Spaces.

Existence for singular critical exponential (𝑝, 𝑄) equations in the Heisenberg group

AbstractThe paper deals with the existence of nontrivial solutions for(p,Q)(p,Q)equations in the Heisenberg groupHn\mathbb{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)(p,Q)growth of the elliptic operator and the fact that the equation is studied in the entire Heisenberg group.

Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group

Abstract This paper is devoted to the weak and strong estimates for the linear and multilinear fractional Hausdorff operators on the Heisenberg group H n {{\mathbb{H}}}^{n} . A sharp strong estimate for T Φ m {T}_{\Phi }^{m} is obtained. As an application, we derive the sharp constant for the product Hardy operator on H n {{\mathbb{H}}}^{n} . Some weak-type ( p , q ) \left(p,q) ( 1 ≤ p ≤ ∞ ) \left(1\le p\le \infty ) estimates for T Φ , 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 T Φ , 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.