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A spectral Paley-Wiener theorem for the Heisenberg group and a support theorem for the twisted spherical means on n

E. K. Narayanan, S. Thangavelu (2006)

Annales de l’institut Fourier

We prove a spectral Paley-Wiener theorem for the Heisenberg group by means of a support theorem for the twisted spherical means on n . If f ( z ) e 1 4 | z | 2 is a Schwartz class function we show that f is supported in a ball of radius B in n if and only if f × μ r ( z ) = 0 for r > B + | z | for all z n . This is an analogue of Helgason’s support theorem on Euclidean and hyperbolic spaces. When n = 1 we show that the two conditions f × μ r ( z ) = μ r × f ( z ) = 0 for r > B + | z | imply a support theorem for a large class of functions with exponential growth. Surprisingly enough,this latter...

A Wiener type theorem for (U(p,q),Hₙ)

Linda Saal (2010)

Colloquium Mathematicae

It is well known that (U(p,q),Hₙ) is a generalized Gelfand pair. Applying the associated spectral analysis, we prove a theorem of Wiener Tauberian type for the reduced Heisenberg group, which generalizes a known result for the case p = n, q = 0.

Admissibility for quasiregular representations of exponential solvable Lie groups

Vignon Oussa (2013)

Colloquium Mathematicae

Let N be a simply connected, connected non-commutative nilpotent Lie group with Lie algebra of dimension n. Let H be a subgroup of the automorphism group of N. Assume that H is a commutative, simply connected, connected Lie group with Lie algebra . Furthermore, assume that the linear adjoint action of on is diagonalizable with non-purely imaginary eigenvalues. Let τ = I n d H N H 1 . We obtain an explicit direct integral decomposition for τ, including a description of the spectrum as a submanifold of (+)*, and a...

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

D. Müller, E. Prestini (2010)

Colloquium Mathematicae

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 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 → ∞.

A.e. convergence of spectral sums on Lie groups

Christopher Meaney, Detlef Müller, Elena Prestini (2007)

Annales de l’institut Fourier

Let be a right-invariant sub-Laplacian on a connected Lie group G , and let S R f : = 0 R d E λ f , R 0 , denote the associated “spherical partial sums,” where = 0 λ d E λ is the spectral resolution of . We prove that S R f ( x ) converges a.e. to f ( x ) as R under the assumption log ( 2 + ) f L 2 ( G ) .

Analysis of joint spectral multipliers on Lie groups of polynomial growth

Alessio Martini (2012)

Annales de l’institut Fourier

We study the problem of L p -boundedness ( 1 < p < ) of operators of the form m ( L 1 , , L n ) for a commuting system of self-adjoint left-invariant differential operators L 1 , , L n on a Lie group G of polynomial growth, which generate an algebra containing a weighted subcoercive operator. In particular, when G is a homogeneous group and L 1 , , L n are homogeneous, we prove analogues of the Mihlin-Hörmander and Marcinkiewicz multiplier theorems.

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