The Heat Kernel and Green Functions of Sub-Laplacians on the Quaternion Heisenberg Group

We introduce the quaternion Heisenberg group and show that it is a special case of the model step two nilpotent Lie group studied by Beals, Gaveau and Greiner. Using the heat kernel, we give formulas for Green functions of sub-Laplacians on the quaternion Heisenberg group. This research has been supported by the Natural Sciences and Engineering Research Council of Canada.     

Hardy-Hausdorff spaces on the Heisenberg group

In this paper, by using the tent spaces on the Siegel upper half space, which are defined in terms of Choquet integrals with respect to Hausdorff capacity on the Heisenberg group, the Hardy-Hausdorff spaces on the Heisenberg group are introduced. Then, by applying the properties of the tent spaces on the Siegel upper half space and the Sobolev type spaces on the Heisenberg group, the atomic decomposition of the Hardy-Hausdorff spaces is obtained. Finally, we prove that the predual spaces of Q spaces on the Heisenberg group are the Hardy-Hausdorff spaces.     

Asymptotics for Certain Harmonic Functions and the Martin Compactification on the Quaternionic Heisenberg Group

In this paper, we make the asymptotic estimates of the heat kernel for the quaternionic Heisenberg group in various cases. We also use these results to deduce the asymptotic estimates of certain harmonic functions on the quaternionic Heisenberg group. Moreover a Martin compactification of the quaternionic Heisenberg group is constructed, and we prove that the Martin boundary of this group is homeomorphic to the unit ball in the quaternionic field.     

BMO Spaces and John-Nirenberg Estimates for the Heisenberg Group Targets

In this paper, the BMO spaces for the Heisenberg group targets are studied. Some properties of the BMO spaces and the John-Nirenberg estimates are obtained.     

Geodesics on $${\mathbb{H}}$$ -type Quaternion Groups with Sub-Lorentzian Metric and Their Physical Interpretation

We study the existence and cardinality of normal geodesics of different causal types on \mathbb H(eisenberg){\mathbb {H}(eisenberg)} -type quaternion group equipped with the sub-Lorentzian metric. We present explicit formulas for geodesics and describe reachable sets by geodesics of different causal character. We compare results with the sub-Riemannian quaternion group and with the sub-Lorentzian Heisenberg group, showing that there are similarities and distinctions. We show that the geodesics on \mathbbH{\mathbb{H}} -type quaternion groups with the sub-Lorentzian metric satisfy the equations describing the motion of a relativistic particle in a constant homogeneous electromagnetic field.     

Higher‐order Sobolev‐type embeddings on Carnot–Carathéodory spaces

A sufficient condition for higher‐order Sobolev‐type embeddings on bounded domains of Carnot–Carathéodory spaces is established for the class of rearrangement‐invariant function spaces. The condition takes form of a one‐dimensional inequality for suitable integral operators depending on the isoperimetric function relative to the Carnot–Carathéodory structure of the relevant sets. General results are then applied to particular Sobolev spaces built upon Lebesgue, Lorentz and Orlicz spaces on John domains in the Heisenberg group. In the case of the Heisenberg group, the condition is shown to be necessary as well.     

Joint eigenfunctions of invariant differential operators on the quaternion Heisenberg group

Let L be the sublaplacian on the quaternion Heisenberg group N and T the Dirac type operator with respect to central variables of N.In this article,we characterize the H c-valued joint eigenfunctions of L and T having eigenvalues from the quaternionic Heisenberg fan.     

Two-dimensional quaternion wavelet transform

In this paper we introduce the continuous quaternion wavelet transform (CQWT). We express the admissibility condition in terms of the (right-sided) quaternion Fourier transform. We show that its fundamental properties, such as inner product, norm relation, and inversion formula, can be established whenever the quaternion wavelets satisfy a particular admissibility condition. We present several examples of the CQWT. As an application we derive a Heisenberg type uncertainty principle for these extended wavelets.     

Remainder terms for several inequalities on some groups of Heisenberg-type

We give estimates of the remainder terms for several conformally-invariant Sobolev-type inequalities on the Heisenberg group. By considering the variations of associated functionals, we give a stability for two dual inequalities: The fractional Sobolev(FS) and Hardy-Littlewood-Sobolev(HLS) inequalities, in terms of distance to the submanifold of extremizers. Then we compare their remainder terms to improve the inequalities in another way. We also compare, in the limit case, the remainder terms of Beckner-Onofri(BO) inequality and its dual logarithmic Hardy-Littlewood-Sobolev(Log-HLS) inequality. Besides, we also list without proof some results for other groups of Iwasawa-type. Our results generalize earlier works on Euclidean spaces of Chen et al.(2013) and Dolbeault and Jankowiak(2014) onto some groups of Heisenberg-type. We worked for "almost"all fractions especially for comparing results, and the stability of HLS is also absolutely new, even for Euclidean case.     

Espaces de Besov et estimations de Strichartz généralisées sur le groupe de Heisenberg

In this paper, we prove dispersive and Strichartz inequalities on the Heisenberg group. The proof involves the analysis of Besov-type spaces on the Heisenberg group.

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Le travail du troisième auteur est partiellement finacé par la NNSF de Chine.     

Variable Hardy Spaces on the Heisenberg Group

We consider Hardy spaces with variable exponents defined by grand maximal function on the Heisenberg group. Then we introduce some equivalent characterizations of variable Hardy spaces. By using atomic decomposition and molecular decomposition we get the boundedness of singular integral operators on variable Hardy spaces. We investigate the Littlewood-Paley characterization by virtue of the boundedness of singular integral operators.     

海森堡群上半空间内最佳Hardy不等式

证明了一类海森堡群上半空间内与次拉普拉斯算子相关的最佳Hardy不等式.作为应用,我们得到了相应的最佳Rellich型不等式.     

Trudinger‐Moser inequalities on the entire Heisenberg group

Continuing our previous work (Cohn, Lam, Lu, Yang, Nonlinear Analysis, 2011), we obtain a class of Trudinger‐Moser inequalities on the entire Heisenberg group, which indicate what the best constants are. All the existing proofs of similar inequalities on unbounded domain of the Euclidean space or the Heisenberg group are based on rearrangement argument. In this note, we propose a new approach to solve this problem. Specifically we get the global Trudinger‐Moser inequality by gluing local estimates with the help of cut‐off functions. Our method still works for similar problems when the Heisenberg group is replaced by the Euclidean space or complete noncompact Riemannian manifolds.     

PARTIAL SCHAUDER ESTIMATES FOR A SUB-ELLIPTIC EQUATION

In this paper,we establish the partial Schauder estimates for the Kohn Laplace equation in the Heisenberg group based on the mean value theorem,the Taylor formula and a priori estimates for the derivatives of the Newton potential.     

On Weighted α-Besov Spaces and α-Bloch Spaces of Quaternion-Valued Functions

In this paper, we give the definitions of weighted α-Besov-type spaces and α-Bloch spaces of quaternion-valued functions, then we obtain characterizations of these quaternion α-Bloch spaces by quaternion α-Besov-type spaces. Relations between Q _p norms and weighted α-Besov norms are also considered. The role of ρ?^α sequences in securing non-Bloch functions is highlighted in quaternion sense.     

Different Faces of the Shearlet Group

Recently, shearlet groups have received much attention in connection with shearlet transforms applied for orientation sensitive image analysis and restoration. The square integrable representations of the shearlet groups provide not only the basis for the shearlet transforms but also for a very natural definition of scales of smoothness spaces, called shearlet coorbit spaces. The aim of this paper is twofold: first we discover isomorphisms between shearlet groups and other well-known groups, namely extended Heisenberg groups and subgroups of the symplectic group. Interestingly, the connected shearlet group with positive dilations has an isomorphic copy in the symplectic group, while this is not true for the full shearlet group with all nonzero dilations. Indeed we prove the general result that there exist, up to adjoint action of the symplectic group, only one embedding of the extended Heisenberg algebra into the Lie algebra of the symplectic group. Having understood the various group isomorphisms it is natural to ask for the relations between coorbit spaces of isomorphic groups with equivalent representations. These connections are examined in the second part of the paper. We describe how isomorphic groups with equivalent representations lead to isomorphic coorbit spaces. In particular we apply this result to square integrable representations of the connected shearlet groups and metaplectic representations of subgroups of the symplectic group. This implies the definition of metaplectic coorbit spaces. Besides the usual full and connected shearlet groups we also deal with Toeplitz shearlet groups.     

Locally conformally homogeneous pseudo-Riemannian spaces

Locally homogeneous Riemannian spaces were studied in [1–4]. Locally conformally homogeneous Riemannian spaces were considered in [10]. Moreover, the theorem claiming that every such space is either conformally flat or conformally equivalent to a locally homogeneous Riemannian space was proved.In this article, we study locally conformally homogeneous pseudo-Riemannian spaces and prove a theorem on their structure. Using three-dimensional Lie groups and the six-dimensional Heisenberg group [11], we construct some examples showing the difference between the Riemannian and pseudo-Riemannian cases for such spaces.     

A pseudo-differential calculus on the Heisenberg group

In this note we present a symbolic pseudo-differential calculus on the Heisenberg group. We particularise to this group our general construction [4], [2] and [3] of pseudo-differential calculi on graded groups. The relation between the Weyl quantisation and the representations of the Heisenberg group enables us to consider here scalar-valued symbols. We find that the conditions defining the symbol classes are similar but different to the ones in [1]. Applications are given to Schwartz hypoellipticity and to subelliptic estimates on the Heisenberg group.     

Weighted Estimates for Commutators of Hausdorff Operators on the Heisenberg Group

The aim of this paper is to give some sufficient conditions for the boundedness of commutators of Hausdorff operators with symbols in weighted central BMO type spaces on the Herz spaces, central Morrey spaces and Morrey-Herz spaces associated with both power weights and Muckenhoupt weights on the Heisenberg group.

     

Hausdorff dimension distribution of quasiconformal mappings on the Heisenberg group

We construct quasiconformal mappings on the Heisenberg group which change the Hausdorff dimension of Cantor-type sets in an arbitrary fashion. On the other hand, we give examples of subsets of the Heisenberg group whose Hausdorff dimension cannot be lowered by any quasiconformal mapping. For a general set of a certain Hausdorff dimension we obtain estimates of the Hausdorff dimension of the image set in terms of the magnitude of the quasiconformal distortion.