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In this paper we show that, for a sub-Laplacian Δ on a 3-dimensional manifold M, no point interaction centered at a point q 0 ∈ M exists. When M is complete w.r.t. the associated sub-Riemannian structure, this means that Δ acting on C 0 ∞ ( M ∖ { q 0 } ) is essentially self-adjoint in L 2 ( M ) . A particular example is the standard sub-Laplacian on the Heisenberg group. This is in stark contrast with what happens in a Riemannian manifold N, whose associated Laplace-Beltrami operator acting on C 0 ∞ ( N ∖ { q 0 } ) is never essentially self-adjoint in L 2 ( N ) , if dim ⁡ N ≤ 3 . We then apply this result to the Schrodinger evolution of a thin molecule, i.e., with a vanishing moment of inertia, rotating around its center of mass.

In the paper, we investigate the linear wave equation with viscoelastic damping on the Heisenberg group H n . Basing on the group Fourier transform on H n and on the properties of the Hermite functions, we derive some L 2 ( H n ) − L 2 ( H n ) estimates with additional L 1 ( H n ) regularity on initial data for the solution and its higher-order horizontal gradients.

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.

Under the framework of the complex column-vector loop algebra Cp, we propose a scheme for generating nonisospectral integrable hierarchies of evolution equations which generalizes the applicable scope of the Tu scheme. As applications of the scheme, we work out a nonisospectral integrable Schrodinger hierarchy and its expanding integrable model. The latter can be reduced to some nonisospectral generalized integrable Schrodinger systems, including the derivative nonlinear Schrodinger equation once obtained by Kaup and Newell. Specially, we obtain the famous Fokker-Plank equation and its generalized form, which has extensive applications in the stochastic dynamic systems. Finally, we investigate the Lie group symmetries, fundamental solutions and group-invariant solutions as well as the representation of the tensor of the Heisenberg group H3 and the matrix linear group SL(2,R) for the generalized Fokker-Plank equation (GFPE).

We show that if a distribution does not depend on the vertical coordinates, then the Schouten curvature tensor coincides with the Riemannian curvature. The Schouten curvature tensor and the nonholonomicity tensor are used to write the Jacobi equation for the distribution. This leads to a study of second-order optimality conditions for horizontal geodesics in sub-Riemannian geometry. We study conjugate points for horizontal geodesics on the Heisenberg group as an example.

In this paper we investigate the property of engulfing for H-convex functions defined on the Heisenberg group $${\mathbb {H}^n}$$ H n . Starting from the horizontal sections introduced by Capogna and Maldonado (Proc Am Math Soc 134:3191–3199, 2006) , we consider a new notion of section, called $${\mathbb {H}^n}$$ H n -section, as well as a new condition of engulfing associated to the $${\mathbb {H}^n}$$ H n -sections, for an H-convex function defined in $$\mathbb {H}^n.$$ H n . These sections, that arise as suitable unions of horizontal sections, are dimensionally larger; as a matter of fact, the $${\mathbb {H}^n}$$ H n -sections, with their engulfing property, will lead to the definition of a quasi-distance in $${\mathbb {H}^n}$$ H n in a way similar to Aimar et al. in the Euclidean case (J Fourier Anal Appl 4:377–381, 1998). A key role is played by the property of round H-sections for an H-convex function, and by its connection with the engulfing properties.

We give the direct method of moving planes for solutions to the conformally invariant fractional power subLaplace equation on the Heisenberg group. The method is based on four maximum principles derived here. Then symmetry and nonexistence of positive cylindrical solutions are proved.

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 .

Two spheres with centers p and q and signed radii r and s are said to be in contact if |p–q|2=(r–s)2. Using Lie’s line-sphere correspondence, we show that if F is a field in which –1 is not a square, then there is an isomorphism between the set of spheres in F3 and the set of lines in a suitably constructed Heisenberg group that is embedded in (F[i])3; under this isomorphism, contact between spheres translates to incidences between lines.In the past decade there has been significant progress in understanding the incidence geometry of lines in three space. The contact-incidence isomorphism allows us to translate statements about the incidence geometry of lines into statements about the contact geometry of spheres. This leads to new bounds for Erdős’ repeated distances problem in F3, and improved bounds for the number of point-sphere incidences in three dimensions. These new bounds are sharp for certain ranges of parameters.

We study the regularity of minima of scalar variational integrals of $p$-growth, $1, Comment: arXiv admin note: text overlap with arXiv:1711.03284

In 1936 W. Burau discovered an interesting family of n×n matrices that give a linear representation of Artin’s classical braid group Bn, n=1,2,…. A natural question followed very quickly: is the so-called Burau representation faithful? Over the years it was proved to be faithful for n≤3, nonfaithful for n≥5, but the case of n=4 remains open to this day, in spite of many papers on the topic. This paper introduces braid groups, describes the problem in ways that make it accessible to readers with a minimal background, reviews the literature, and makes a contribution that reinforces conjectures that the Burau representation of B4 is faithful.

In this paper, we prove the uniqueness and simplicity of the principal frequency (or the first eigenvalue) of the Dirichlet p-sub-Laplacian for general vector fields. As a byproduct, we establish the Caccioppoli inequalities and also discuss the particular cases on the Grushin plane and on the Heisenberg group.

Embedded symmetry within the Heisenberg group is used to couple geometric insight and analytic calculation to obtain a new sharp Stein–Weiss inequality with mixed homogeneity on the line of duality. SL(2,R) invariance and Riesz potentials define a natural bridge for encoded information that connects distinct geometric structures. The intrinsic character of the Heisenberg group makes it the natural playing field on which to explore the laws of symmetry and the interplay between analysis and geometry on a manifold.

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., Comment: From the previous version, we changed the title, added a section about the action of the group on the boundary, and made general edits. 29 pages, 11 figures

A variant of Gromov's Hölder-equivalence problem, motivated by a pinching problem in Riemannian geometry, is discussed. A partial result is given. The main tool is a general coarea inequality satisfied by packing energies of maps.; On introduit une variante, invariante par homéomorphisme quasi-symétrique, du problème d'équivalence höldérienne de Gromov. On obtient un résultat partiel, qui a une conséquence en géométrie riemannienne. Il repose sur une forme générale de l'inégalité de la coaire pour les p-énergies des fonctions.

In this paper, we define and study Gevrey spaces associated with a Hormander family of (globally defined) vector fields and its corresponding sub‐Laplacian. We show some natural relations between the various Gevrey spaces in this setting on general manifolds, and more particular properties on Lie groups with polynomial growth of the volume. In the case of the Heisenberg group and of SU(2), we show that all our descriptions coincide.

In this paper, we apply De Giorgi-Moser iteration to establish the Hölder regularity of quasiminimizers to generalized Orlicz functional on the Heisenberg group by using the Riesz potential, maximal function, Calderón-Zygmund decomposition, and covering Lemma on the context of the Heisenberg Group. The functional includes the p -Laplace functional on the Heisenberg group which has been studied and the variable exponential functional and the double phase growth functional on the Heisenberg group that have not been studied.

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 inℍn{{\mathbb{H}}}^{n}.

The aim of this paper is to show that a supercritical Neumann problem has at least one positive radial solution via recent variational principle and defining an energy functional with a hidden symmetry associated with the problem. Also, we study the problem in the setting of the Heisenberg group $${\mathbb {H}}^n$$ , which is a special case of locally compact groups.

Any free nilpotent Lie algebra is determined by its rank and step. We consider free nilpotent Lie algebras of steps 3 and 4 and corresponding connected and simply connected Lie groups. We construct Casimir functions of such groups, i.e., invariants of the coadjoint representation. For free 3-step nilpotent Lie groups, we get a full description of coadjoint orbits. It turns out that general coadjoint orbits are affine subspaces, and special coadjoint orbits are affine subspaces or direct products of nonsingular quadrics. The knowledge of Casimir functions is useful for investigation of integration properties of dynamical systems and optimal control problems on Carnot groups. In particular, for some wide class of time-optimal problems on 3-step free Carnot groups, we conclude that extremal controls corresponding to two-dimensional coadjoint orbits have the same behavior as in time-optimal problems on the Heisenberg group or on the Engel group.

The paper deals with the existence of nontrivial solutions for ( p , Q ) (p,Q) equations in the Heisenberg group H n \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.

We study projectional properties of Poisson cut-out sets $E$ in non-Euclidean spaces. In the first Heisenbeg group, endowed with the Kor\'anyi metric, we show that the Hausdorff dimension of the vertical projection $\pi(E)$ (projection along the center of the Heisenberg group) almost surely equals $\min\{2,dim_H(E)\}$ and that $\pi(E)$ has non-empty interior if $dim_H(E)>2$. As a corollary, this allows us to determine the Hausdorff dimension of $E$ with respect to the Euclidean metric in terms of its Heisenberg Hausdorff dimension $dim_H (E)$. We also study projections in the one-point compactification of the Heisenberg group, that is, the $3$-sphere $S^3$ endowed with the visual metric $d$ obtained by identifying $S^3$ with the boundary of the complex hyperbolic plane. In $S^3$, we prove a projection result that holds simultaneously for all radial projections (projections along so called "chains"). This shows that the Poisson cut-outs in $S^3$ satisfy a strong version of the Marstrand's projection theorem, without any exceptional directions., Comment: 36 pages

This note concerns low-dimensional intrinsic Lipschitz graphs, in the sense of Franchi, Serapioni, and Serra Cassano, in the Heisenberg group $\mathbb{H}^n$, $n\in \mathbb{N}$. For $1\leq k\leq n$, we show that every intrinsic $L$-Lipschitz graph over a subset of a $k$-dimensional horizontal subgroup $\mathbb{V}$ of $\mathbb{H}^n$ can be extended to an intrinsic $L'$-Lipschitz graph over the entire subgroup $\mathbb{V}$, where $L'$ depends only on $L$, $k$, and $n$. We further prove that $1$-dimensional intrinsic $1$-Lipschitz graphs in $\mathbb{H}^n$, $n\in \mathbb{N}$, admit corona decompositions by intrinsic Lipschitz graphs with smaller Lipschitz constants. This complements results that were known previously only in the first Heisenberg group $\mathbb{H}^1$. The main difference to this case arises from the fact that for $1\leq k, Comment: 30 pages; v2: minor revision, results unchanged

The existence of at least one positive radial solution of the generalized Kohn–Laplacian problem (1) −ΔHnu+R(ξ)u=∑i=1kai(|ξ|Hn)|u|pi−2u −∑j=1mbj(|ξ|Hn)|u|qj−2u ξ∈Ω,u>0ξ∈Ω,∂u∂n=0ξ∈∂Ω, is proved, w...

We use the representation theory of nilpotent Lie groups of step two to find the fundamental solution for the Paneitz operator on the anisotropic quaternionic Heisenberg group. For the isotropic ca...

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 ) ...

It is well known that the cohomology of any non-trivial 1-dimensional local system on a nilmanifold vanishes (this result, due to J. Dixmier, was also announced and proved in some particular case by Alaniya). A complex nilmanifold is a quotient of a nilpotent Lie group equipped with a left-invariant complex structure by an action of a discrete, co-compact subgroup. We prove a Dolbeault version of Dixmier’s and Alaniya’s theorem, showing that the Dolbeault cohomology $$ {H}^{0,p}\left(\mathfrak{g},L\right) $$ of a nilpotent Lie algebra with coefficients in any non-trivial 1-dimensional local system vanishes. Note that the Dolbeault cohomology of the corresponding local system on the manifold is not necessarily zero. This implies that the twisted version of Console–Fino theorem is false (Console–Fino proved that the Dolbeault cohomology of a complex nilmanifold is equal to the Dolbeault cohomology of its Lie algebra, when the complex structure is rational). As an application, we give a new proof of a theorem due to H. Sawai, who obtained an explicit description of LCK nilmanifolds. An LCK structure on a manifold M is a Kahler structure on its cover $$ \tilde{M} $$ such that the deck transform map acts on $$ \tilde{M} $$ by homotheties. We show that any complex nilmanifold admitting an LCK structure is Vaisman, and is obtained as a compact quotient of the product of a Heisenberg group and the real line.

This paper aims to define and study currents and slices of currents in the Heisenberg group $${\mathbb {H}}^n$$ H n . Currents, depending on their integration properties and on those of their boundaries, can be classified into subspaces and, assuming their support to be compact, we can work with currents of finite mass, define the notion of slices of Heisenberg currents and show some important properties for them. While some such properties are similarly true in Riemannian settings, others carry deep consequences because they do not include the slices of the middle dimension n, which opens new challenges and scenarios for the possibility of developing a compactness theorem. Furthermore, this suggests that the study of currents on the first Heisenberg group $${\mathbb {H}}^1$$ H 1 diverges from the other cases, because that is the only situation in which the dimension of the slice of a hypersurface, $$2n-1$$ 2 n - 1 , coincides with the middle dimension n, which triggers a change in the associated differential operator in the Rumin complex.

We consider nonlinear sub-elliptic systems with VMO-coefficients for the case 1 < p < 2 under controllable growth conditions, as well as natural growth conditions, respectively, in the Heisenberg group. On the basis of a generalization of the technique of 𝓐-harmonic approximation introduced by Duzaar-Grotowski-Kronz, and an appropriate Sobolev-Poincaré type inequality established in the Heisenberg group, we prove partial Hölder continuity results for vector-valued solutions of discontinuous sub-elliptic problems. The primary model covered by our analysis is the non-degenerate sub-elliptic p-Laplacian system with VMO-coefficients, involving sub-quadratic growth terms.

In this note, we show that sub-Riemannian manifolds can contain branching normal minimizing geodesics. This phenomenon occurs if and only if a normal geodesic has a discontinuity in its rank at a non-zero time, which in particular for a strictly normal geodesic means that it contains a non-trivial abnormal subsegment. The simplest example is obtained by gluing the three-dimensional Martinet flat structure with the Heisenberg group in a suitable way. We then use this example to construct more general types of branching., 11 pages, 1 figure. Final version, to appear on Geometric And Functional Analysis (GAFA)

We study locally compact metric spaces that enjoy various forms of homogeneity with respect to Mobius self-homeomorphisms. We investigate connections between such homogeneity and the combination of isometric homogeneity with invertibility. In particular, we provide a new characterization of snowflakes of boundaries of rank-one symmetric spaces of non-compact type among locally compact and connected metric spaces. Furthermore, we investigate the metric implications of homogeneity with respect to uniformly strongly quasi-Mobius self-homeomorphisms, connecting such homogeneity with the combination of uniform bi-Lipschitz homogeneity and quasi-invertibility. In this context we characterize spaces containing a cut point and provide several metric properties of spaces containing no cut points. These results are motivated by a desire to characterize the snowflakes of boundaries of rank-one symmetric spaces up to bi-Lipschitz equivalence.

In this paper, we consider the generalized Weyl–Heisenberg group $${\mathbb {H}}(G_\tau )$$ associated with the semi direct product group $$G_\tau =H \times _\tau K$$ in which H is a locally compact group, K is a locally compact abelian group and $$\tau :H \rightarrow \hbox {Aut}(K)$$ is a continuous homomorphism. We give a square integrable representation on $${\mathbb {H}}(G_\tau )$$ and then as a result, we obtain admissible wavelets on this group. Moreover, we define the localization operator on $${\mathbb {H}}(G_\tau )$$ and we show that it is a bounded and compact operator. Furthermore, we investigate the boundedness and compactness of localization operators with two admissible wavelets on $$L^p({\hat{K}})$$ , ( $${\hat{K}}$$ the dual group of K), for the generalized Weyl–Heisenberg group. Finally, some example are given.

In this paper, we study on normal complex contact metric manifold admitting a semi symmetric metric connection. We obtain curvature properties of a normal complex contact metric manifold admitting a semi symmetric metric connection. We also prove that this type of manifold is not conformal flat, concircular flat, and conharmonic flat. Finally, we examine complex Heisenberg group with the semi symmetric metric connection.

We give an integral representation of the fundamental solution of the heat and resolvent equation associated to the Casimir–Laplace operator for the quaternionic Heisenberg group. As an application we obtain the corresponding wave kernel and Green function.

Let $${\mathcal {L}}=-\varDelta _{{\mathbb {H}}^n}+V$$ be a Schrodinger operator on the Heisenberg group $${\mathbb {H}}^n$$ , where $$\varDelta _{{\mathbb {H}}^n}$$ is the sublaplacian on $${\mathbb {H}}^n$$ and the nonnegative potential V belongs to the reverse Holder class $$RH_q$$ with $$q\ge Q/2$$ . Here $$Q=2n+2$$ is the homogeneous dimension of $${\mathbb {H}}^n$$ . In this paper the author first introduces a class of Morrey spaces associated with the Schrodinger operator $${\mathcal {L}}$$ on $${\mathbb {H}}^n$$ . Then by using some pointwise estimates of the kernels related to the nonnegative potential V, the author establishes the boundedness properties of the Littlewood–Paley function $${\mathfrak {g}}_{{\mathcal {L}}}$$ and the Lusin area integral $${\mathcal {S}}_{{\mathcal {L}}}$$ (with respect to the heat semigroup $$\{e^{-s{\mathcal {L}}}\}_{s>0}$$ ) acting on the Morrey spaces. It can be shown that the same conclusions also hold for the operators $${\mathfrak {g}}_{\sqrt{{\mathcal {L}}}}$$ and $${\mathcal {S}}_{\sqrt{{\mathcal {L}}}}$$ with respect to the Poisson semigroup $$\{e^{-s\sqrt{{\mathcal {L}}}}\}_{s>0}$$ .

In this work, we study non-characteristic domains of Heisenberg groups. We prove that bounded domains which are diffeomorphic to the solid torus having the center of the group as rotation axis, are non-characteristic. Then we state the following conjecture: the bounded non-characteristic domains of the Heisenberg group of dimension 1 are diffeomorphic to a solid torus having the center of the group as revolution axis.

A Semmes surface in the Heisenberg group is a closed set $S$ that is upper Ahlfors-regular with codimension one and satisfies the following condition, referred to as Condition B. Every ball $B(x,r)$ with $x \in S$ and $0 < r < \operatorname{diam} S$ contains two balls with radii comparable to $r$ which are contained in different connected components of the complement of $S$. Analogous sets in Euclidean spaces were introduced by Semmes in the late $80$'s. We prove that Semmes surfaces in the Heisenberg group are lower Ahlfors-regular with codimension one and have big pieces of intrinsic Lipschitz graphs. In particular, our result applies to the boundary of chord-arc domains and of reduced isoperimetric sets., 39 pages, 4 figures

A theorem of Dorronsoro from the 1980s quantifies the fact that real-valued Sobolev functions on Euclidean spaces can be approximated by affine functions almost everywhere, and at all sufficiently small scales. We prove a variant of Dorronsoro's theorem in Heisenberg groups: functions in horizontal Sobolev spaces can be approximated by affine functions which are independent of the last variable. As an application, we deduce new proofs for certain vertical vs. horizontal Poincare inequalities for real-valued functions on the Heisenberg group, originally due to Austin-Naor-Tessera and Lafforgue-Naor.

We show that, under a natural scaling, the small-time behavior of the logarithmic derivatives of the subelliptic heat kernel on $SU(2)$ converges to their analogues on the Heisenberg group at time 1. Realizing $SU(2)$ as $\mathbb{S}^3$, we then generalize these results to higher-order odd-dimensional spheres equipped with their natural subRiemannian structure, where the limiting spaces are now the higher-dimensional Heisenberg groups., 32 pages, corrected errors in preliminary estimates in Section 2.1

The almost sure value of the Hausdorff dimension of limsup sets generated by randomly distributed rectangles in the Heisenberg group is computed in terms of directed singular value functions., Comment: 21 pages

We discuss the existence and uniqueness of solution for the second boundary value problem of potential theory often referred to as the Neumann problem, on a gauge ball for the canonical sub-Laplacian in H-type groups. In this way we extend the classical results of the problem as well as its generalization to the Heisenberg group.

The existence of at least one positive radial solution of the Neumann problem $$ -\Delta _{\mathbb{H}^{n}} u+R(\xi ) u=a \bigl( \vert \xi \vert _{\mathbb{H}^{n}} \bigr) \vert u \vert ^{p-2} u - b\bigl( \vert \xi \vert _{\mathbb{H}^{n}}\bigr) \vert u \vert ^{q-2}u, $$−ΔHnu+R(ξ)u=a(|ξ|Hn)|u|p−2u−b(|ξ|Hn)|u|q−2u, is proved on the Heisenberg group $\mathbb{H}^{n}$Hn, via the variational principle, where $a(|\xi |_{\mathbb{H}^{n}})$a(|ξ|Hn), $b(|\xi |_{\mathbb{H}^{n}})$b(|ξ|Hn) are nonnegative radial functions and $R(\xi )$R(ξ) satisfies suitable conditions.

In this paper, we introduce the definitions of the Fourier transform and spherical Fourier transform on quaternionic Heisenberg group, then we present some of their properties, in particular, we wi...

In this note, we prove the existence of a weak solution for nonlinear p-sub-Laplacian equations with the Dirichlet boundary condition on the Heisenberg group. In the proof, we use a version of the ...

The aim of this paper is to study Sasakian immersions of (non-compact) complete regular Sasakian manifolds into the Heisenberg group and into $$ {\mathbb{B}}^N\times {\mathbb{R}}$$ equipped with their standard Sasakian structures. We obtain a complete classification of such manifolds in the η-Einstein case.

We prove that the HRT (Heil, Ramanathan, and Topiwala) Conjecture is equivalent to the conjecture that co-central translates of square-integrable functions on the Heisenberg group are linearly independent.

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.

In this paper we study Hardy inequalities in the Heisenberg group $${\mathbb {H}}^n$$ , with respect to the Carnot–Caratheodory distance $$\delta $$ from the origin. We firstly show that, letting Q be the homogenous dimension, the optimal constant in the (unweighted) Hardy inequality is strictly smaller than $$n^2 = (Q-2)^2/4$$ . Then, we prove that, independently of n, the Heisenberg group does not support a radial Hardy inequality, i.e., a Hardy inequality where the gradient term is replaced by its projection along $$\nabla \!_{\mathbb {H}}\delta $$ . This is in stark contrast with the Euclidean case, where the radial Hardy inequality is equivalent to the standard one, and has the same constant. Motivated by these results, we consider Hardy inequalities for non-radial directions, i.e., directions tangent to the Carnot–Caratheodory balls. In particular, we show that the associated constant is bounded on homogeneous cones $$C_\Sigma $$ with base $$\Sigma \subset {\mathbb {S}}^{2n}$$ , even when $$\Sigma $$ degenerates to a point. This is a genuinely sub-Riemannian behavior, as such constant is well known to explode for homogeneous cones in the Euclidean space.

In this paper, we study questions related to the theory of stochastic processes on Lie nilpotent groups. In particular, we consider the stochastic process on the Heisenberg group H3(ℝ) whose trajectories satisfy the horizontal conditions in the stochastic sense relative to the standard contact structure on H3 (ℝ). It is shown that this process is a homogeneous Markov process relative to the Heisenberg group operation. There was found a representation in the form of a Wiener integral for a one-parameter linear semigroup of operators for which the Heisenberg sublaplacian generated by basis vector fields of the corresponding Lie algebra L(H3) is producing. The main method of solving the problem in this paper is using the path integrals technique, which indicates the common direction of further development of the results.

In the setting of Heisenberg group, we characterize those functionsΦ, for which the fractional Hausdorff operatorsTΦ,βand Hausdorff operatorsTΦ,T˜Φare bounded onLpspaces with power weights,BMOspace, and Hardy spaces, respectively. Meanwhile, the corresponding operator norms ofTΦandT˜Φare worked out.