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Riesz potentials and orthogonal radon transforms on affine Grassmannians

  • Boris Rubin EMAIL logo and Yingzhan Wang
Published/Copyright: May 9, 2021
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 24 Issue 2

Abstract

We establish intertwining relations between Riesz potentials associated with fractional powers of minus-Laplacian and orthogonal Radon transforms 𝓡j,k of the Gonzalez-Strichartz type. The latter take functions on the Grassmannian of j-dimensional affine planes in ℝn to functions on a similar manifold of k-dimensional planes by integration over the set of all j-planes that meet a given k-plane at a right angle. The main results include sharp existence conditions of 𝓡j,kf on Lp-functions, Fuglede type formulas connecting 𝓡j,k with Radon-John k-plane transforms and Riesz potentials, and explicit inversion formulas for 𝓡j,kf under the assumption that f belongs to the range of the j-plane transform. The method extends to another class of Radon transforms defined on affine Grassmannians by inclusion.

Key Words and Phrases: Riesz potentials; Erdélyi–Kober fractional integrals; Radon transforms; Grassmann manifolds
MSC 2010: Primary 44A12; Secondary 47G10

Dedicated to Professor Stefan Grigorievich Samko on the occasion of his 80th anniversary

Acknowledgements

The first-named author is thankful to Fulton Gonzalez, Todd Quinto and Sigurdur Helgason for useful discussions during his visit to Tufts University in April, 2006. The second-named author was supported by National Natural Science Foundation of China, 11971125.

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Received: 2020-10-04
Published Online: 2021-05-09
Published in Print: 2021-04-27

© 2021 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. Anniversary of Prof. S.G. Samko, FC Events (FCAA–Volume 24–2–2021)
  4. Research Paper
  5. Operational calculus for the general fractional derivative and its applications
  6. Riesz potentials and orthogonal radon transforms on affine Grassmannians
  7. Characterizations of variable martingale Hardy spaces via maximal functions
  8. Survey Paper
  9. Contributions on artificial potential field method for effective obstacle avoidance
  10. Research Paper
  11. Self-similar cauchy problems and generalized Mittag-Leffler functions
  12. Asymptotic behavior of solutions of fractional differential equations with Hadamard fractional derivatives
  13. A boundary value problem for a partial differential equation with fractional derivative
  14. Operational calculus for the Riemann–Liouville fractional derivative with respect to a function and its applications
  15. Duality theory of fractional resolvents and applications to backward fractional control systems
  16. Kinetic solutions for nonlocal stochastic conservation laws
  17. A fractional analysis in higher dimensions for the Sturm-Liouville problem
  18. Averaging theory for fractional differential equations
https://doi.org/10.1515/fca-2021-0017
Keywords for this article
Riesz potentials; Erdélyi–Kober fractional integrals; Radon transforms; Grassmann manifolds

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. Anniversary of Prof. S.G. Samko, FC Events (FCAA–Volume 24–2–2021)
  4. Research Paper
  5. Operational calculus for the general fractional derivative and its applications
  6. Riesz potentials and orthogonal radon transforms on affine Grassmannians
  7. Characterizations of variable martingale Hardy spaces via maximal functions
  8. Survey Paper
  9. Contributions on artificial potential field method for effective obstacle avoidance
  10. Research Paper
  11. Self-similar cauchy problems and generalized Mittag-Leffler functions
  12. Asymptotic behavior of solutions of fractional differential equations with Hadamard fractional derivatives
  13. A boundary value problem for a partial differential equation with fractional derivative
  14. Operational calculus for the Riemann–Liouville fractional derivative with respect to a function and its applications
  15. Duality theory of fractional resolvents and applications to backward fractional control systems
  16. Kinetic solutions for nonlocal stochastic conservation laws
  17. A fractional analysis in higher dimensions for the Sturm-Liouville problem
  18. Averaging theory for fractional differential equations
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