On the relative fundamental solutions for a second order differential operator on the Heisenberg group

T. Godoy; L. Saal

Displaying similar documents to “On the relative fundamental solutions for a second order differential operator on the Heisenberg group”

Subalgebra of $L_{1} (G)$ associated with Laplacian on a Lie group

A. Hulanicki (1974)

Colloquium Mathematicae

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Littlewood-Paley characterization of Hölder-Zygmund spaces on stratified Lie groups

Guorong Hu (2019)

Czechoslovak Mathematical Journal

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We give a characterization of the Hölder-Zygmund spaces $𝒞^{σ} (G)$ ( $0 < σ < \infty$ ) on a stratified Lie group $G$ in terms of Littlewood-Paley type decompositions, in analogy to the well-known characterization of the Euclidean case. Such decompositions are defined via the spectral measure of a sub-Laplacian on $G$ , in place of the Fourier transform in the classical setting. Our approach mainly relies on almost orthogonality estimates and can be used to study other function spaces such as Besov and Triebel-Lizorkin...

Transferring $L^{p}$ eigenfunction bounds from $S^{2 n + 1}$ to hⁿ

Valentina Casarino, Paolo Ciatti (2009)

Studia Mathematica

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By using the notion of contraction of Lie groups, we transfer $L^{p} - L ²$ estimates for joint spectral projectors from the unit complex sphere $S^{2 n + 1}$ in $ℂ^{n + 1}$ to the reduced Heisenberg group hⁿ. In particular, we deduce some estimates recently obtained by H. Koch and F. Ricci on hⁿ. As a consequence, we prove, in the spirit of Sogge’s work, a discrete restriction theorem for the sub-Laplacian L on hⁿ.

Leibniz's rule on two-step nilpotent Lie groups

Krystian Bekała (2016)

Colloquium Mathematicae

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Let be a nilpotent Lie algebra which is also regarded as a homogeneous Lie group with the Campbell-Hausdorff multiplication. This allows us to define a generalized multiplication $f g = {(f^{\lor} * g^{\lor})}^{\land}$ of two functions in the Schwartz class (*), where $^{\lor}$ and $^{\land}$ are the Abelian Fourier transforms on the Lie algebra and on the dual * and ∗ is the convolution on the group . In the operator analysis on nilpotent Lie groups an important notion is the one of symbolic calculus which can be viewed as a higher order...

Growth and smooth spectral synthesis in the Fourier algebras of Lie groups

Jean Ludwig, Lyudmila Turowska (2006)

Studia Mathematica

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Let G be a Lie group and A(G) the Fourier algebra of G. We describe sufficient conditions for complex-valued functions to operate on elements u ∈ A(G) of certain differentiability classes in terms of the dimension of the group G. Furthermore, generalizing a result of Kirsch and Müller [Ark. Mat. 18 (1980), 145-155] we prove that closed subsets E of a smooth m-dimensional submanifold of a Lie group G having a certain cone property are sets of smooth spectral synthesis. For such sets we...

Hall algebra of morphism category

QingHua Chen, Liwang Zhang (2024)

Czechoslovak Mathematical Journal

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This paper investigates a universal PBW-basis and a minimal set of generators for the Hall algebra $ℋ (C_{2} (𝒫))$ , where $C_{2} (𝒫)$ is the category of morphisms between projective objects in a finitary hereditary exact category $𝒜$ . When $𝒜$ is the representation category of a Dynkin quiver, we develop multiplication formulas for the degenerate Hall Lie algebra $ℒ$ , which is spanned by isoclasses of indecomposable objects in $C_{2} (𝒫)$ . As applications, we demonstrate that $ℒ$ contains a Lie subalgebra isomorphic to the central...

Local superderivations on Lie superalgebra $𝔮 (n)$

Haixian Chen, Ying Wang (2018)

Czechoslovak Mathematical Journal

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Let $𝔮 (n)$ be a simple strange Lie superalgebra over the complex field $ℂ$ . In a paper by A. Ayupov, K. Kudaybergenov (2016), the authors studied the local derivations on semi-simple Lie algebras over $ℂ$ and showed the difference between the properties of local derivations on semi-simple and nilpotent Lie algebras. We know that Lie superalgebras are a generalization of Lie algebras and the properties of some Lie superalgebras are similar to those of semi-simple Lie algebras, but $𝔭 (n)$ is an exception....

$𝔤$ -quasi-Frobenius Lie algebras

David N. Pham (2016)

Archivum Mathematicum

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A Lie version of Turaev’s $\bar{G}$ -Frobenius algebras from 2-dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a $𝔤$ -quasi-Frobenius Lie algebra for $𝔤$ a finite dimensional Lie algebra. The latter consists of a quasi-Frobenius Lie algebra $(𝔮, β)$ together with a left $𝔤$ -module structure which acts on $𝔮$ via derivations and for which $β$ is $𝔤$ -invariant. Geometrically, $𝔤$ -quasi-Frobenius Lie algebras are the Lie algebra structures associated to...

The groups of automorphisms of the Witt $W_{n}$ and Virasoro Lie algebras

Vladimir V. Bavula (2016)

Czechoslovak Mathematical Journal

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Let $L_{n} = K [x_{1}^{\pm 1}, ..., x_{n}^{\pm 1}]$ be a Laurent polynomial algebra over a field $K$ of characteristic zero, $W_{n} : = {Der}_{K} (L_{n})$ the Lie algebra of $K$ -derivations of the algebra $L_{n}$ , the so-called Witt Lie algebra, and let $Vir$ be the Virasoro Lie algebra which is a $1$ -dimensional central extension of the Witt Lie algebra. The Lie algebras $W_{n}$ and $Vir$ are infinite dimensional Lie algebras. We prove that the following isomorphisms of the groups of Lie algebra automorphisms hold: ${Aut}_{Lie} (Vir) ≃ {Aut}_{Lie} (W_{1}) ≃ {\pm 1} ⋉ K^{*}$ , and give a short proof that ${Aut}_{Lie} (W_{n}) ≃ {Aut}_{K - alg} (L_{n}) ≃ {GL}_{n} (ℤ) ⋉ K^{* n}$ .

The variety of dual mock-Lie algebras

Luisa M. Camacho, Ivan Kaygorodov, Viktor Lopatkin, Mohamed A. Salim (2020)

Communications in Mathematics

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We classify all complex $7$ - and $8$ -dimensional dual mock-Lie algebras by the algebraic and geometric way. Also, we find all non-trivial complex $9$ -dimensional dual mock-Lie algebras.

Semibounded Unitary Representations of Double Extensions of Hilbert–Loop Groups

K. H. Neeb (2014)

Annales de l’institut Fourier

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A unitary representation $π$ of a, possibly infinite dimensional, Lie group $G$ is called semibounded if the corresponding operators $i d π (x)$ from the derived representation are uniformly bounded from above on some non-empty open subset of the Lie algebra $𝔤$ of $G$ . We classify all irreducible semibounded representations of the groups ${\hat{ℒ}}_{φ} (K)$ which are double extensions of the twisted loop group $ℒ_{φ} (K)$ , where $K$ is a simple Hilbert–Lie group (in the sense that the scalar product on its Lie algebra is invariant)...

When unit groups of continuous inverse algebras are regular Lie groups

Helge Glöckner, Karl-Hermann Neeb (2012)

Studia Mathematica

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It is a basic fact in infinite-dimensional Lie theory that the unit group $A^{\times}$ of a continuous inverse algebra A is a Lie group. We describe criteria ensuring that the Lie group $A^{\times}$ is regular in Milnor’s sense. Notably, $A^{\times}$ is regular if A is Mackey-complete and locally m-convex.

A.e. convergence of anisotropic partial Fourier integrals on Euclidean spaces and Heisenberg groups

D. Müller, E. Prestini (2010)

Colloquium Mathematicae

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We define partial spectral integrals $S_{R}$ on the Heisenberg group by means of localizations to isotropic or anisotropic dilates of suitable star-shaped subsets V containing the joint spectrum of the partial sub-Laplacians and the central derivative. Under the assumption that an L²-function f lies in the logarithmic Sobolev space given by $l o g (2 + L_{α}) f \in L ²$ , where $L_{α}$ is a suitable “generalized” sub-Laplacian associated to the dilation structure, we show that $S_{R} f (x)$ converges a.e. to f(x) as R → ∞.

$\mathcal{L}$ -homomorphisms of Lie algebras

Aleksander A. Lashkhi (1981)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Si studiano gli omomorfismi reticolari ( $\mathcal{L}$ -omomorfismi) di algebre di Lie sopra anelli commutativi con unità. Le algebre di Lie sopra un campo e le $p$ -algebre di Lie non ammettono $\mathcal{L}$ -omomorfismi propri. Si assegnano condizioni necessarie e sufficienti affinchè un'algebra di Lie periodica o mista possieda un « $\mathcal{L}$ -omomorfismo su una catena di lunghezza $n$ .

$\mathcal{L}$ -homomorphisms of Lie algebras

Aleksander A. Lashkhi (1981)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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$L^{p}$ spectral multipliers on the free group $N_{3, 2}$

Alessio Martini, Detlef Müller (2013)

Studia Mathematica

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Let L be a homogeneous sublaplacian on the 6-dimensional free 2-step nilpotent Lie group $N_{3, 2}$ on three generators. We prove a theorem of Mikhlin-Hörmander type for the functional calculus of L, where the order of differentiability s > 6/2 is required on the multiplier.

SCAP-subalgebras of Lie algebras

Sara Chehrazi, Ali Reza Salemkar (2016)

Czechoslovak Mathematical Journal

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A subalgebra $H$ of a finite dimensional Lie algebra $L$ is said to be a $SCAP$ -subalgebra if there is a chief series $0 = L_{0} \subset L_{1} \subset ... \subset L_{t} = L$ of $L$ such that for every $i = 1, 2, ..., t$ , we have $H + L_{i} = H + L_{i - 1}$ or $H \cap L_{i} = H \cap L_{i - 1}$ . This is analogous to the concept of $SCAP$ -subgroup, which has been studied by a number of authors. In this article, we investigate the connection between the structure of a Lie algebra and its $SCAP$ -subalgebras and give some sufficient conditions for a Lie algebra to be solvable or supersolvable.

The diffeomorphism group of a non-compact orbifold

A. Schmeding

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We endow the diffeomorphism group $D i f f_{O r b} (Q,)$ of a paracompact (reduced) orbifold with the structure of an infinite-dimensional Lie group modeled on the space of compactly supported sections of the tangent orbibundle. For a second countable orbifold, we prove that $D i f f_{O r b} (Q,)$ is C⁰-regular, and thus regular in the sense of Milnor. Furthermore, an explicit characterization of the Lie algebra associated to $D i f f_{O r b} (Q,)$ is given.

On the spectral radius in $L_{1} (G)$

E. Porada (1971)

Colloquium Mathematicae

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Tempered reductive homogeneous spaces

Yves Benoist, Toshiyuki Kobayashi (2015)

Journal of the European Mathematical Society

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Let $G$ be a semisimple algebraic Lie group and $H$ a reductive subgroup. We find geometrically the best even integer $p$ for which the representation of $G$ in $L^{2} (G / H)$ is almost $L^{p}$ . As an application, we give a criterion which detects whether this representation is tempered.

A complete analogue of Hardy's theorem on semisimple Lie groups

Rudra P. Sarkar (2002)

Colloquium Mathematicae

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A result by G. H. Hardy ([11]) says that if f and its Fourier transform f̂ are $O (| x |^{m} e^{- α x ²})$ and $O (| x | ⁿ e^{- x ² / (4 α)})$ respectively for some m,n ≥ 0 and α > 0, then f and f̂ are $P (x) e^{- α x ²}$ and $P^{'} (x) e^{- x ² / (4 α)}$ respectively for some polynomials P and P’. If in particular f is as above, but f̂ is $o (e^{- x ² / (4 α)})$ , then f = 0. In this article we will prove a complete analogue of this result for connected noncompact semisimple Lie groups with finite center. Our proof can be carried over to the real reductive groups of the Harish-Chandra class.