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In this note we show that a connected, simply connected nilpotent Lie group with an integrable left-invariant complex structure on a generating and suitably complemented subbundle of the tangent bundle admits a Cauchy-Riemann (CR) embedding in complex space defined by polynomials. We also show that a similar conclusion holds on suitable quotients of nilpotent Lie groups. Our results extend the CR embeddings constructed by Naruki [Publ. Res. Inst. Math. Sci. 6 (1970), pp. 113–187] in 1970. In particular, our generalisation to quotients allows us to see a class of Levi degenerate CR manifolds as quotients of nilpotent Lie groups. [ABSTRACT FROM AUTHOR]

We prove an L p -spectral multiplier theorem for sub-Laplacians on Heisenberg type groups under the sharp regularity condition s > d 1 / p - 1 / 2 , where d is the topological dimension of the underlying group. Our approach relies on restriction type estimates where the multiplier is additionally truncated along the spectrum of the Laplacian on the center of the group. [ABSTRACT FROM AUTHOR]

In this paper, we study the Orlicz estimates for the parabolic Schrödinger operator $ L = {\partial _t} - {\Delta _X} + V, $ where the nonnegative potential $ V $ belongs to a reverse Hölder class on nilpotent Lie groups $ {\Bbb G} $ and $ {\Delta _X} $ is the sub-Laplace operator on $ {\Bbb G} $. Under appropriate growth conditions of the Young function, we obtain the regularity estimates of the operator $ L $ in the Orlicz space by using the domain decomposition method. Our results generalize some existing ones of the $ L^{p} $ estimates.

We study the families of measures on Carnot groups that have vanishing pp-module, which we call Mp{M}_{p}-exceptional families. We found necessary and sufficient Conditions for the family of intrinsic Lipschitz surfaces passing through a common point to be Mp{M}_{p}-exceptional for p≥1p\ge 1. We describe a wide class of Mp{M}_{p}-exceptional intrinsic Lipschitz surfaces for p∈(0,∞)p\in \left(0,\infty ).

In this paper, we study the Orlicz estimates for the parabolic Schrödinger operator where the nonnegative potential belongs to a reverse Hölder class on nilpotent Lie groups and is the sub-Laplace operator on. Under appropriate growth conditions of the Young function, we obtain the regularity estimates of the operator in the Orlicz space by using the domain decomposition method. Our results generalize some existing ones of the estimates. In this paper, we study the Orlicz estimates for the parabolic Schrödinger operator where the nonnegative potential belongs to a reverse Hölder class on nilpotent Lie groups and is the sub-Laplace operator on. Under appropriate growth conditions of the Young function, we obtain the regularity estimates of the operator in the Orlicz space by using the domain decomposition method. Our results generalize some existing ones of the estimates. [ABSTRACT FROM AUTHOR]

We study the phase retrieval property for orbits of general irreducible representations of nilpotent groups, for the classes of simply connected Lie groups, and for finite groups. We prove by induction that in the Lie group case, all irreducible representations do phase retrieval. For the finite group case, we mostly focus on p-groups. Here our main result states that every irreducible representation of an arbitrary p-group with exponent p and size ≤ p 2 + p / 2 does phase retrieval. Despite the fundamental differences between the two settings, our inductive proof methods are very similar. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the nonnegative solutions of the differential inequality u p ≤ ℒ s ⁢ u on the Grushin space 픾 α n for (p , s , α) ∈ (1 , ∞) × (0 , 1) × (0 , ∞) , where the ℒ s are the fractional powers of the Grushin operator ℒ . We show that any nonnegative solution of the fractional order differential inequality displayed above is zero if and only if p ≤ Q Q - 2 ⁢ s , where Q is the homogeneous dimension of 픾 α n . Moreover, we also consider the similar problems of nonnegative weak solutions of the fractional sub-Laplacian differential inequality on nilpotent Lie groups. [ABSTRACT FROM AUTHOR]

The goal of this paper is to investigate which one of the non-isometric left-invariant Lorentz metrics g on four-dimensional nilpotent Lie groups H 3 × R and G 4 satisfies the Ricci soliton equation 2 Ric [ g ] + L X g + α g = 0 , here X is a vector field and α is a constant. Among the left-invariant Lorentzian metrics on H 3 × R , g λ - is a shrinking while g λ + and g μ are expanding and also g 0 1 , g 0 2 , g 0 3 have Ricci solitons. We exhibit among the non-isometric left-invariant Lorentz metric on the group G 4 only g 1 λ , g 2 λ have Lorentz Ricci solitons and g 2 λ is a shrinking. [ABSTRACT FROM AUTHOR]

In this paper, we study the conjecture of Bansal, Kumar and Sharma, which is an analog of Hardy’s theorem for Gabor transform in the setup of connected nilpotent Lie groups. To approach this conjecture, we use the orbit method and the Plancherel theory. When the Lie group G is simply connected, we show that the conjecture is true. [ABSTRACT FROM AUTHOR]

In this paper, we consider N, a simply connected two-step nilpotent Lie group with N, its corresponding (two-step nilpotent) Lie algebra, and we study Newton's method for solving the equation f (x) = 0, where f: N !N is a mapping. Under certain generalized Lipschitz condition, we obtain the convergence radius of Newton's method and the estimation of the uniqueness ball of the zero point of f. Some applications to special cases including Kantorovich's condition and γ-condition are provided. The determination of an approximate zero point of an analytic mapping is also presented. [ABSTRACT FROM AUTHOR]

In this paper, we define and prove an analog of the Heisenberg uncertainty inequality for Gabor transform in the setup of connected, simply connected nilpotent Lie groups. When G is connected nilpotent and has a non-compact center, a proof of such an analog is given for functions in the Schwartz space of G. The representation theory and a localized Plancherel formula are fundamental tools in the proof of our results. [ABSTRACT FROM AUTHOR]

We investigate to what extent a nilpotent Lie group is determined by its C ∗ -algebra. We prove that, within the class of exponential Lie groups, direct products of Heisenberg groups with abelian Lie groups are uniquely determined even by their unitary dual, while nilpotent Lie groups of dimension ≤ 5 are uniquely determined by the Morita equivalence class of their C ∗ -algebras. We also find that this last property is shared by the filiform Lie groups and by the 6 -dimensional free two-step nilpotent Lie group. [ABSTRACT FROM AUTHOR]

We study a problem of isometric compact 2-step nilmanifolds $M/\Gamma$ using some information on their geodesic flows, where $M$ is a simply connected 2-step nilpotent Lie group with a left invariant metric and $\Gamma$ is a cocompact discrete subgroup of isometries of $M$. Among various works concerning this problem, we consider the algebraic aspect of it. In fact, isometry groups of simply connected Riemannian manifolds can be characterized in a purely algebraic way, namely by normalizers. So, suitable factorization of normalizers and expression of a vector bundle as an associated fiber bundle to a principal bundle, lead us to a general framework, namely groupoids. In this way, drawing upon advanced ingredients of Lie groupoids, normal subgroupoid systems and other notions, not only an answer in some sense to our rigidity problem has been given, but also the dependence between normalizers, automorphisms and especially almost inner automorphisms, has been clarified.

We survey the work of Elias M. Stein in the field of analysis on the Heisenberg group and other nilpotent Lie groups, together with its applications to complex analysis in several variables and partial differential equations. [ABSTRACT FROM AUTHOR]

Let N be a nilpotent Lie group and K a compact subgroup of the automorphism group Aut(N) of N. It is well-known that if (K ⋉ N , K) is a Gelfand pair then N is at most 2-step nilpotent Lie group. The notion of Gelfand pair was generalized when K is a non-compact group. In this work, we give an example of a 3-step nilpotent Lie group and a non-compact subgroup K of Aut(N) such that (K ⋉ N , N) is a generalized Gelfand pair. [ABSTRACT FROM AUTHOR]

We give a direct proof for the positivity of Kirillov's character on the convolution algebra of smooth, compactly supported functions on a connected, simply connected nilpotent Lie group G. Then we use this positivity result to construct a representation of G × G and establish a G × G-equivariant isometric isomorphism between our representation and the Hilbert–Schmidt operators on the underlying representation of G. In fact, we provide a framework in which we establish the positivity of Kirillov's character for coadjoint orbits of groups such as SL (2 , ℝ) under additional hypotheses that are automatically satisfied in the nilpotent case. These hypotheses include the existence of a real polarization and the Pukanzsky condition. [ABSTRACT FROM AUTHOR]

We study local control of a mechanism with the growth vector (4,7). We study controllability and extremal trajectories of the nilpotent approximation as an example of the control theory on a Lie group. We provide solutions to the system and show examples of local extremal trajectories. [ABSTRACT FROM AUTHOR]

We prove that there do not exist quasi-isometric embeddings of connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics into a metric measure space satisfying the curvature-dimension condition RCD(0, N) with N ∈ ℝ and N > 1. In fact, we can prove that a sub-Riemannian manifold whose generic degree of nonholonomy is not smaller than 2 cannot be bi-Lipschitzly embedded in any Banach space with the Radon-Nikodym property. We also get that every regular sub-Riemannian manifold do not satisfy the curvature-dimension condition CD(K, N), where K, N ∈ ℝ and N > 1. Along the way to the proofs, we show that the minimal weak upper gradient and the horizontal gradient coincide on the Carnot-Carath´eodory spaces which may have independent interests. [ABSTRACT FROM AUTHOR]

Let N be a connected, simply connected two-step nilpotent Lie group and K a compact subgroup of Aut(N), the group of automorphisms of N. Let δ denote a unitary equivalence class of irreducible unitary representa- tions of K. In this paper, we give an explicit formula for spherical functions of type δ for the semidirect K ∝ N. [ABSTRACT FROM AUTHOR]

We propose a general scheme for the search of a fundamental solution to the hypoelliptic diffusion equation in a "sufficiently good" sub-Riemannian manifold and the small-time asymptotics for the solution, which includes the generalized Fourier transform and the orbit method closely related to it, as well as an application of the perturbative method to the nilpotent approximation, and Trotter's formula. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish a local uncertainty inequality for arbitrary connected, simply connected nilpotent Lie groups. This allows us to prove a couple of global uncertainty inequalities. In the nilpotent case, this type of result is only obtained for the Heisenberg group. [ABSTRACT FROM AUTHOR]

We establish analogues of Heisenberg uncertainty inequality for some classes of Lie groups, such as connected and simply connected nilpotent Lie groups, diamond Lie groups and Heisenberg motion groups. [ABSTRACT FROM AUTHOR]

We show that the distribution of symmetry of a naturally reductive nilpotent Lie group coincides with the invariant distribution induced by the set of fixed vectors of the isotropy. This extends a known result on compact naturally reductive spaces. We also address the study of the quotient by the foliation of symmetry. [ABSTRACT FROM AUTHOR]

Hardy's uncertainty principle for the Gabor transform is proved for locally compact abelian groups having noncompact identity component and groups of the form $\mathbb{R}^{n}\times K$ , where $K$ is a compact group having irreducible representations of bounded dimension. We also show that Hardy's theorem fails for a connected nilpotent Lie group $G$ which admits a square integrable irreducible representation. Further, a similar conclusion is made for groups of the form $G\times D$ , where $D$ is a discrete group. [ABSTRACT FROM AUTHOR]

We establish a Fourier inversion theorem for general connected, simply connected nilpotent Lie groups G=exp(g) by showing that operator fields defined on suitable sub-manifolds of g∗ are images of Schwartz functions under the Fourier transform. As an application of this result, we provide a complete characterisation of a large class of invariant prime closed two-sided ideals of L1(G) as kernels of sets of irreducible representations of G. [ABSTRACT FROM AUTHOR]

Using the extension method, we study the one-parameter symmetry groups of the heat equation ∂tp = Δp, where Δ = X 1 2 + X 2 2 is the sub-Laplacian constructed by a Goursat distribution span({X1, X2}) in ℝn, where the vector fields X1 and X2 satisfy the commutation relations [X1, Xj] = Xj+1 (where Xn+1 = 0) and [Xj, Xk] = 0 for j ≥ 1 and k ≥ 1. We show that there are no such groups for n ≥ 4 (with exception of the linear transformations of solutions which are admitted by every linear equation). [ABSTRACT FROM AUTHOR]

Abstract Let G be a connected, simply connected nilpotent group and π be an irreducible unitary representation of G that is square-integrable modulo its center Z (G) on L 2 (R d). We prove that under reasonably weak conditions on G and π there exist a discrete subset Γ of G / Z (G) and some (relatively) compact set F ⊆ R d such that { | F | − 1 / 2 π (γ) 1 F | γ ∈ Γ } forms an orthonormal basis of L 2 (R d). This construction generalizes the well-known example of Gabor orthonormal bases in time-frequency analysis. The main theorem covers graded Lie groups with one-dimensional center. In the presence of a rational structure, the set Γ can be chosen to be a uniform subgroup of G / Z. [ABSTRACT FROM AUTHOR]

We establish a general CCR (liminarity) property for uniformly bounded irreducible representations of nilpotent Lie groups on reflexive Banach spaces, extending the well-known property of unitary irreducible representations of these groups on Hilbert spaces. We also prove that this conclusion fails for many representations on non-reflexive Banach spaces. Our approach to these results blends the method of transference from abstract harmonic analysis and a systematic use of spaces of smooth vectors with respect to Lie group representations. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to formulate and prove an analogue of the classical Heisenberg–Pauli–Weyl uncertainty inequality for connected nilpotent Lie groups with noncompact center. Representation theory and a localized Plancherel formula play an important role in the proof. [ABSTRACT FROM AUTHOR]

Several classes of locally compact groups have been shown to possess a qualitative uncertainty principle for the Gabor transform. These include Moore groups, the Heisenberg group ℍ n {\mathbb{H}_{n}} , the group ℍ n × D {\mathbb{H}_{n}\times D} (where D is a discrete group) and other low-dimensional nilpotent Lie groups. [ABSTRACT FROM AUTHOR]

Left-invariant control affine systems on simply connected nilpotent Lie groups of dimension ≤4 are considered. First, we classify these control systems under two natural equivalence relations. Second, the associated cost-extended systems (with invariant quadratic Lagrangian) are classified. The latter classification is reinterpreted in terms of invariant affine distributions equipped with quadratic cost; also, we obtain a classification of the invariant sub-Riemannian (and Riemannian) structures. A few remarks conclude the paper. [ABSTRACT FROM AUTHOR]

Let $${\mathcal {N}}_m$$ be the group of $$m\times m$$ upper triangular real matrices with all the diagonal entries 1. Then it is an $$(m-1)$$ -step nilpotent Lie group, diffeomorphic to $${\mathbb {R}}^{\frac{1}{2} m(m-1)}$$ . It contains all the integer matrices as a lattice $$\Gamma _m$$ . The automorphism group of $${\mathcal {N}}_m \ (m\ge 4)$$ turns out to be extremely small. In fact, $$\mathrm {Aut}({\mathcal {N}})=\mathcal {I} \rtimes \mathrm {Out}({\mathcal {N}})$$ , where $$\mathcal {I}$$ is a connected, simply connected nilpotent Lie group, and $$\mathrm {Out}({\mathcal {N}})={{\tilde{K}}}={(\mathbb {R}^*)^{m-1}\rtimes \mathbb {Z}_2}$$ . With a nice left-invariant Riemannian metric on $${\mathcal {N}}$$ , the isometry group is $$\mathrm {Isom}({\mathcal {N}})= {\mathcal {N}} \rtimes K$$ , where $$K={(\mathbb {Z}_2)^{m-1}\rtimes \mathbb {Z}_2}\subset {{\tilde{K}}}$$ is a maximal compact subgroup of $$\mathrm {Aut}({\mathcal {N}})$$ . We prove that, for odd $$m\ge 4$$ , there is no infra-nilmanifold which is essentially covered by the nilmanifold $$\Gamma _m\backslash {\mathcal {N}}_m$$ . For $$m=2n\ge 4$$ (even), there is a unique infra-nilmanifold which is essentially (and doubly) covered by the nilmanifold $$\Gamma _m\backslash {\mathcal {N}}_m$$ . [ABSTRACT FROM AUTHOR]

We give relations between main operators of quantum mechanics on one of most general classes of nilpotent Lie groups. Namely, we show relations between momentum and position operators as well as Euler and Coulomb potential operators on homogeneous groups. Homogeneous group analogues of some well-known inequalities such as Hardy's inequality, Heisenberg-Kennard type and Heisenberg-Pauli-Weyl type uncertainty inequalities, as well as Caffarelli-Kohn-Nirenberg inequality are derived, with best constants. The obtained relations yield new results already in the setting of both isotropic and anisotropic Rn, and of the Heisenberg group. The proof demonstrates that the method of establishing equalities in sharper versions of such inequalities works well in both isotropic and anisotropic settings. [ABSTRACT FROM AUTHOR]

We determine the space of primary ideals in the group algebra $$L^{1}(G) $$ of a connected nilpotent Lie group by identifying for every $$\pi \in \widehat{G} $$ the family $$\mathcal I^\pi $$ of primary ideals with hull $$\{\pi \} $$ with the family of invariant subspaces of a certain finite dimensional sub-space $$\mathcal P_Q^\pi $$ of the space of polynomials $$\mathcal P(G) $$ on G. [ABSTRACT FROM AUTHOR]

Nilpotent Lie groups with stepwise square integrable representations were recently investigated by J.A. Wolf. We give an alternative approach to these representations by relating them to the stratifications of the duals of nilpotent Lie algebras, thus proving that they correspond to a subset with relative Hausdorff topology, dense interior, and total Plancherel measure in the unitary dual of the Lie group under consideration. [ABSTRACT FROM AUTHOR]

A locally compact group G is said to be approximable by discrete subgroups if there exists a sequence of discrete subgroups of G such that for any nonempty open set O of G, there exists an integer k such that , for every n ≥ k. It has been proved that the identity component of a Lie group approximable by discrete subgroups is nilpotent. In this paper we shall extend this result to pro-Lie groups. In particular, every connected locally compact group approximable by discrete subgroups is nilpotent. An application of discrete approximation is given. [ABSTRACT FROM AUTHOR]

The Kodaira-Thurston manifold is a quotient of a nilpotent Lie group by a cocompact lattice. We compute the family Gromov-Witten invariants which count pseudoholomorphic tori in the Kodaira-Thurston manifold. For a fixed symplectic form the Gromov-Witten invariant is trivial so we consider the twistor family of left-invariant symplectic forms which are orthogonal for some fixed metric on the Lie algebra. This family defines a loop in the space of symplectic forms. This is the first example of a genus one family Gromov-Witten computation for a non-Kähler manifold. [ABSTRACT FROM AUTHOR]

We investigate locally compact topological groups for which a generalized analog of the Heisenberg uncertainty inequality hold. In particular, it is shown that this inequality holds for $\mathbb{R}^{n} \times K$ (where K is a separable unimodular locally compact group of type I), Euclidean motion group and several general classes of nilpotent Lie groups which include thread-like nilpotent Lie groups, 2-NPC nilpotent Lie groups and several low-dimensional nilpotent Lie groups. [ABSTRACT FROM AUTHOR]

By employing a new reduction procedure we derive explicit expressions for the fundamental solutions of a family P k , λ of degenerate second order differential operators on R N + ℓ . Here λ is a complex parameter located in the strip | Re ( λ ) | < N + k − 1 . As is pointed out in [2] P k , 0 has a geometric background and arises as a Grushin-type operator induced by a sub-Riemannian structure on a k + 1 -step nilpotent Lie group. Our method leads to new formulas for the inverse of the Kohn-Laplacian Δ λ which has been widely studied before in the framework of pseudo-convex domains and CR geometry. As an application we show that in all cases the fundamental solutions have a meromorphic extension in the parameter λ to C ∖ Q . All poles are simple and Q ⊂ R is an explicitly given discrete set. We recover the invertibility of Δ 1 modulo the classical Szegö projection. This phenomenon had been observed before in [11] . [ABSTRACT FROM AUTHOR]