Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-13T04:21:05.178Z Has data issue: false hasContentIssue false
  • English
  • Français

Restricted Radon Transforms and Projections of Planar Sets

Published online by Cambridge University Press:  20 November 2018

Daniel M. Oberlin
Daniel M. Oberlin*
Affiliation:
Department of Mathematics, Florida State University, Tallahassee, FL 32306, USAe-mail: oberlin@math.fsu.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We establish a mixed norm estimate for the Radon transform in ${{\mathbb{R}}^{2}}$ when the set of directions has fractional dimension. This estimate is used to prove a result about an exceptional set of directions connected with projections of planar sets. That leads to a conjecture analogous to a well-known conjecture of Furstenberg.

Keywords

44A1228A78
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 4 , 01 December 2012 , pp. 815 - 820
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Bergh, J. and Löfström, J., Interpolation Spaces. Grundlehren der mathematischenWissenschaften, 223, Springer-Verlag, Berlin, 1976.Google Scholar
[2] Kaufman, R., On Hausdorff dimension of projections. Mathematika 15(1968), 153–155. http://dx.doi.org/10.1112/S0025579300002503 Google Scholar
[3] Marstrand, J., Some fundamental geometrical properties of plane sets of fractional dimension. Proc. London. Math. Soc. 4(1954), no. 3, 257–302. http://dx.doi.org/10.1112/plms/s3-4.1.257 Google Scholar
[4] Stocke, B.-M., Differentiability properties of Bessel potentials and Besov spaces. Arkiv Mat. 22(1984), no. 2, 269–286. http://dx.doi.org/10.1007/BF02384383 Google Scholar
[5] Wolff, T., Recent work connected with the Kakeya problem. In: Prospects in Mathematics, American Mathematical Society, Providence, RI, 1999, pp. 129–162.Google Scholar