On the Radon and Riesz transforms in real hyperbolic spaces

Berenstein, Carlos A.; Tarabusi, Enrico Casadio

doi:10.1090/conm/140/1197583

Cited by 10 publications

(13 citation statements)

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“…The case j = 0 corresponds to the totally geodesic transform R k : h → ζ h that takes functions on H n to functions on G H (n, k); cf. [1,2,20,21,26,35,50]. The corresponding dual transform is defined by…”

Section: Radon Transforms Of Zonal Functionsmentioning

confidence: 99%

“…The group G = SO 0 (n, 1) acts on Γ H (n, d) transitively. We consider the d-dimensional hyperboloidH d = H n ∩ E d,1 as the origin (the base element) of Γ H (n, d). The group H d in (4.9) is the stabilizer of H d in G. Hence Γ H (n, d) can be regarded as the quotient space G/H d and each d-geodesic t ∈ Γ H (n, d) has the form t = gH d for some g ∈ G.We denote by ||t|| the geodesic distance from t ∈ Γ H (n, d) to the origin õ = (0, .…”

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confidence: 99%

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Higher-Rank Radon Transforms on Constant Curvature Spaces

Rubin¹

2021

Preprint

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We study higher-rank Radon transforms of the form f (τ ) → τ ⊂ζ f (τ ), where τ is a j-dimensional totally geodesic submanifold in the n-dimensional real constant curvature space and ζ is a similar submanifold of dimension k > j. The corresponding dual transforms are also considered. The transforms are explored the Euclidean case (affine Grassmannian bundles), the elliptic case (compact Grassmannians), and the hyperbolic case (the hyperboloid model, the Beltrami-Klein model, and the projective model). The main objectives are sharp conditions for the existence and injectivity of the Radon transforms in Lebesgue spaces, transition from one model to another, support theorems, and inversion formulas. Conjectures and open problems are discussed.

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Section: Radon Transforms Of Zonal Functionsmentioning

confidence: 99%

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confidence: 99%

Higher-Rank Radon Transforms on Constant Curvature Spaces

Rubin¹

2021

Preprint

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“…7, 188]. 1 The main topics of the paper can be seen in the that Contents. Explicit transition formulas (2.28), (3.19), (5.14), (5.17), and (6.13) from one model or setting to another play a key role in our work.…”

Section: Introductionmentioning

confidence: 99%

Higher-Rank Radon Transforms on Constant Curvature Spaces

Rubin

2022

J Math Sci

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We study higher-rank Radon transforms of the form f (𝜏) → ∫ 𝜏⊂𝜁 f (𝜏) , where is a j-dimensional totally geodesic submanifold in the n-dimensional real constant curvature space and is a similar submanifold of dimension k > j . The corresponding dual transforms are also considered. The transforms are explored in the Euclidean case (affine Grassmannian bundles), the elliptic case (compact Grassmannians), and the hyperbolic case (the hyperboloid model, the Beltrami-Klein model, and the projective model). The main objectives are sharp conditions for the existence and injectivity of the Radon transforms in Lebesgue spaces, transition from one model to another, support theorems, and inversion formulas. Conjectures and open problems are discussed.

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“…How to invert operators (1.2) and (1. Berenstein and Casadio Tarabusi [2,3] suggested to solve Problem 1 as follows: f ¼ QðDÞSR n Rf ; where S is a certain convolution operator on X and the polynomial QðDÞ is understood in the sense of distributions. Helgason [18, pp.…”

Section: Introductionmentioning

confidence: 99%

“…We hope this problem can be tackled using the results of Semyanistyi [28] as in [27] for X ¼ S n : If a is an even integer, K a f can be inverted by a polynomial of D as in [18, p. 92]. For a odd, K a f was inverted in [2,3] by making use of the Fourier analysis and distribution theory. This argument was reproduced with slight changes in [18, pp.…”

Section: Introductionmentioning

confidence: 99%