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Let G be a semisimple Lie group with Iwasawa decomposition G = KAN. Let X = G/K be the associated symmetric space and assume that X is of rank one. Let M be the centraliser of A in K and consider an orthonormal basis $Y_{δ, j} : δ \in K ̂ ₀, 1 \leq j \leq d_{δ}$ of L²(K/M) consisting of K-finite functions of type δ on K/M. For a function f on X let f̃(λ,b), λ ∈ ℂ, be the Helgason Fourier transform. Let $h_{t}$ be the heat kernel associated to the Laplace-Beltrami operator and let $Q_{δ} (i λ + ϱ)$ be the Kostant polynomials. We establish the following version...

Hardy-type inequalities related to degenerate elliptic differential operators

Lorenzo D’Ambrosio (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We prove some Hardy-type inequalities related to quasilinear second-order degenerate elliptic differential operators $L_{p} u : = - \nabla_{L}^{*} ({|\nabla_{L} u|}^{p - 2} \nabla_{L} u)$ . If $φ$ is a positive weight such that $- L_{p} φ \geq 0$ , then the Hardy-type inequalityholds. We find an explicit value of the constant involved, which, in most cases, results optimal. As particular case we derive Hardy inequalities for subelliptic operators on Carnot Groups.

Harmonic analysis and unitary group representations : the development from 1927 to 1950

George W. Mackey (1992)

Cahiers du séminaire d'histoire des mathématiques

Harmonic analysis for spinors on real hyperbolic spaces

Roberto Camporesi, Emmanuel Pedon (2001)

Colloquium Mathematicae

We develop the L² harmonic analysis for (Dirac) spinors on the real hyperbolic space Hⁿ(ℝ) and give the analogue of the classical notions and results known for functions and differential forms: we investigate the Poisson transform, spherical function theory, spherical Fourier transform and Fourier transform. Very explicit expressions and statements are obtained by reduction to Jacobi analysis on L²(ℝ). As applications, we describe the exact spectrum of the Dirac operator, study the Abel transform...

Harmonic analysis in one-parameter metabelian nilmanifolds.

Ghorbel, Amira (2011)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

Harmonic analysis in weighted $L_{2}$ -spaces

Jens Franke (1998)

Annales scientifiques de l'École Normale Supérieure

Harmonic analysis of Abelian inner automorphism groups of von Neumann algebras.

Herbert Halpern (1982)

Mathematica Scandinavica

Harmonic analysis on $S U (n, n) / S L (n, ℂ) \times ℝ_{+}$ .

Andersen, Nils Byrial, Unterberger, Jérémie M. (2000)

Journal of Lie Theory

Harmonic analysis on semisimple $p$ -adic Lie algebras.

Kottwitz, Robert E. (1998)

Documenta Mathematica

Harmonic functions on homogeneous spaces of commutator induced extensions of compact groups

R. E. Lewkowicz (1988)

Colloquium Mathematicae

Harmonic induction on Lie groups.

Ronald L. Lipsman (1983)

Journal für die reine und angewandte Mathematik

Harmonic maps and representations of non-uniform lattices of $PU (m, 1)$

Vincent Koziarz, Julien Maubon (2008)

Annales de l’institut Fourier

We study representations of lattices of $PU (m, 1)$ into $PU (n, 1)$ . We show that if a representation is reductive and if $m$ is at least 2, then there exists a finite energy harmonic equivariant map from complex hyperbolic $m$ -space to complex hyperbolic $n$ -space. This allows us to give a differential geometric proof of rigidity results obtained by M. Burger and A. Iozzi. We also define a new invariant associated to representations into $PU (n, 1)$ of non-uniform lattices in $PU (1, 1)$ , and more generally of fundamental groups of orientable...

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Hardy spaces on affine symmetric spaces.

Hardy spaces on SU(2)

Hardy spaces on two-sheeted covering semigroups.

Hardy-Littlewood theory on unimodular groups

Hardy's inequality for compact Lie groups.

Hardy's theorem for the helgason Fourier transform on noncompact rank one symmetric spaces