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Mixed-Norm Estimates for the k-Plane Transform

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Excursions in Harmonic Analysis, Volume 2

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

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Abstract

The Radon transform constitutes a fundamental concept for X-rays in medical imaging, and more generally, in image reconstruction problems from diverse fields. The Radon transform in Euclidean spaces assigns to functions their integrals over affine hyperplanes. This can be extended so that the integration is performed on affine k-dimensional subspaces; the corresponding transform is called k-plane transform. An overview of mixed-norm inequalities for the k-plane transform and related potential-type operators supported on k-planes is presented. Particular attention is given to the action of these operators on classes of radial functions, and applications to bounds for the Kakeya maximal operator are discussed.

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Acknowledgments

Javier Duoandikoetxea’s research is supported in part by grant MTM2007-62186 of MEC (Spain) and FEDER. Virginia Naibo’s research is supported in part by the National Science Foundation under grant DMS 1101327. Virginia Naibo thanks the members of the Organizing Committee (Profs. Radu Balan, John J. Benedetto, Wojtek Czaja, and Kasso Okoudjou) for the invitation to speak in the February Fourier Talks 2009.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad del País Vasco-Euskal Herriko Unibertsitatea, Apartado 644, 48080, Bilbao, Spain

    Javier Duoandikoetxea

  2. Department of Mathematics, Kansas State University, 138 Cardwell Hall, Manhattan, KS, 66506, USA

    Virginia Naibo

Authors
  1. Javier Duoandikoetxea
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  2. Virginia Naibo
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Correspondence to Virginia Naibo .

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Editors and Affiliations

  1. , Department of Mathematics, University of Maryland, Norbert Wiener Center, College Park, 20742-0001, Maryland, USA

    Travis D. Andrews

  2. Norbert Wiener Center, Department of Mathematics, University of Maryland, College Park, 20742, Maryland, USA

    Radu Balan

  3. Department of Mathematics, University of Maryland, College Park, 20742-4015, Maryland, USA

    John J. Benedetto

  4. Department of Mathematics, University of Maryland, College Park, 20742-4015, Maryland, USA

    Wojciech Czaja

  5. , Department of Mathematics, University of Maryland, College Park, 20742, Maryland, USA

    Kasso A. Okoudjou

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Duoandikoetxea, J., Naibo, V. (2013). Mixed-Norm Estimates for the k-Plane Transform. In: Andrews, T., Balan, R., Benedetto, J., Czaja, W., Okoudjou, K. (eds) Excursions in Harmonic Analysis, Volume 2. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8379-5_11

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