Radon transformation on reductive symmetric spaces:Support theorems

Job Jacob Kuit

doi:10.1016/j.aim.2013.03.010

Radon transformation on reductive symmetric spaces:Support theorems

Job Jacob Kuit

Institut for Matematiske Fag

2 Citationer (Scopus)

Abstract

We introduce a class of Radon transforms for reductive symmetric spaces, including the horospherical transforms, and derive support theorems for these transforms.

A reductive symmetric space is a homogeneous space G/H for a reductive Lie group G of the Harish-Chandra class, where H is an open subgroup of the fixed-point subgroup for an involution σ on G. Let P be a parabolic subgroup such that σ(P) is opposite to P and let NP be the unipotent radical of P. For a compactly supported smooth function ϕ on G/H, we define RP(ϕ)(g) to be the integral of NP∋n↦ϕ(gn⋅H) over NP. The Radon transform RP thus obtained can be extended to a large class of distributions containing the rapidly decreasing smooth functions and the compactly supported distributions.

For these transforms we derive support theorems in which the support of ϕ is (partially) characterized in terms of the support of RPϕ. The proof is based on the relation between the Radon transform and the Fourier transform on G/H, and a Paley–Wiener-shift type argument. Our results generalize the support theorem of Helgason for the Radon transform on a Riemannian symmetric space.

Originalsprog	Engelsk
Tidsskrift	Advances in Mathematics
Vol/bind	240
Sider (fra-til)	427-483
ISSN	0001-8708
DOI	https://doi.org/10.1016/j.aim.2013.03.010
Status	Udgivet - jun. 2013

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10.1016/j.aim.2013.03.010

Citationsformater

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title = "Radon transformation on reductive symmetric spaces:Support theorems",

abstract = "We introduce a class of Radon transforms for reductive symmetric spaces, including the horospherical transforms, and derive support theorems for these transforms.A reductive symmetric space is a homogeneous space G / H for a reductive Lie group G of the Harish-Chandra class, where H is an open subgroup of the fixed-point subgroup for an involution σ on G. Let P be a parabolic subgroup such that σ (P) is opposite to P and let N P be the unipotent radical of P. For a compactly supported smooth function φ on G / H, we define RP(φ)(g) to be the integral of N P ∋ n → φ (g n {dot operator} H) over N P. The Radon transform RP thus obtained can be extended to a large class of distributions containing the rapidly decreasing smooth functions and the compactly supported distributions.For these transforms we derive support theorems in which the support of φ is (partially) characterized in terms of the support of RPφ. The proof is based on the relation between the Radon transform and the Fourier transform on G / H, and a Paley-Wiener-shift type argument. Our results generalize the support theorem of Helgason for the Radon transform on a Riemannian symmetric space.",

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T1 - Radon transformation on reductive symmetric spaces:Support theorems

AU - Kuit, Job Jacob

PY - 2013/6

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N2 - We introduce a class of Radon transforms for reductive symmetric spaces, including the horospherical transforms, and derive support theorems for these transforms.A reductive symmetric space is a homogeneous space G / H for a reductive Lie group G of the Harish-Chandra class, where H is an open subgroup of the fixed-point subgroup for an involution σ on G. Let P be a parabolic subgroup such that σ (P) is opposite to P and let N P be the unipotent radical of P. For a compactly supported smooth function φ on G / H, we define RP(φ)(g) to be the integral of N P ∋ n → φ (g n {dot operator} H) over N P. The Radon transform RP thus obtained can be extended to a large class of distributions containing the rapidly decreasing smooth functions and the compactly supported distributions.For these transforms we derive support theorems in which the support of φ is (partially) characterized in terms of the support of RPφ. The proof is based on the relation between the Radon transform and the Fourier transform on G / H, and a Paley-Wiener-shift type argument. Our results generalize the support theorem of Helgason for the Radon transform on a Riemannian symmetric space.

AB - We introduce a class of Radon transforms for reductive symmetric spaces, including the horospherical transforms, and derive support theorems for these transforms.A reductive symmetric space is a homogeneous space G / H for a reductive Lie group G of the Harish-Chandra class, where H is an open subgroup of the fixed-point subgroup for an involution σ on G. Let P be a parabolic subgroup such that σ (P) is opposite to P and let N P be the unipotent radical of P. For a compactly supported smooth function φ on G / H, we define RP(φ)(g) to be the integral of N P ∋ n → φ (g n {dot operator} H) over N P. The Radon transform RP thus obtained can be extended to a large class of distributions containing the rapidly decreasing smooth functions and the compactly supported distributions.For these transforms we derive support theorems in which the support of φ is (partially) characterized in terms of the support of RPφ. The proof is based on the relation between the Radon transform and the Fourier transform on G / H, and a Paley-Wiener-shift type argument. Our results generalize the support theorem of Helgason for the Radon transform on a Riemannian symmetric space.

U2 - 10.1016/j.aim.2013.03.010

DO - 10.1016/j.aim.2013.03.010

M3 - Journal article

SN - 0001-8708

VL - 240

SP - 427

EP - 483

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -

Radon transformation on reductive symmetric spaces:Support theorems

Abstract

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