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Discrete fractional radon transforms and quadratic forms

Publication ,  Journal Article
Pierce, LB
Published in: Duke Mathematical Journal
2012

We consider discrete analogues of fractional Radon transforms involving integration over paraboloids defined by positive definite quadratic forms. We prove sharp results for this class of discrete operators in all dimensions, providing necessary and sufficient conditions for them to extend to bounded operators from l p to l q. The method involves an intricate spectral decomposition according to major and minor arcs, motivated by ideas from the circle method of Hardy and Littlewood. Techniques from harmonic analysis, in particular Fourier transform methods and oscillatory integrals, as well as the number theoretic structure of quadratic forms, exponential sums, and theta functions, play key roles in the proof.

Duke Scholars

Published In

Duke Mathematical Journal

DOI

ISSN

0012-7094

Publication Date

2012

Volume

161

Issue

1

Start / End Page

69 / 106

Related Subject Headings

  • General Mathematics
  • 4904 Pure mathematics
  • 0101 Pure Mathematics
 

Citation

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Pierce, L. B. (2012). Discrete fractional radon transforms and quadratic forms. Duke Mathematical Journal, 161(1), 69–106. https://doi.org/10.1215/00127094-1507288
Pierce, L. B. “Discrete fractional radon transforms and quadratic forms.” Duke Mathematical Journal 161, no. 1 (2012): 69–106. https://doi.org/10.1215/00127094-1507288.
Pierce LB. Discrete fractional radon transforms and quadratic forms. Duke Mathematical Journal. 2012;161(1):69–106.
Pierce, L. B. “Discrete fractional radon transforms and quadratic forms.” Duke Mathematical Journal, vol. 161, no. 1, 2012, pp. 69–106. Scival, doi:10.1215/00127094-1507288.
Pierce LB. Discrete fractional radon transforms and quadratic forms. Duke Mathematical Journal. 2012;161(1):69–106.
Journal cover image

Published In

Duke Mathematical Journal

DOI

ISSN

0012-7094

Publication Date

2012

Volume

161

Issue

1

Start / End Page

69 / 106

Related Subject Headings

  • General Mathematics
  • 4904 Pure mathematics
  • 0101 Pure Mathematics