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15 January 2012 Discrete fractional Radon transforms and quadratic forms
Lillian B. Pierce
Duke Math. J. 161(1): 69-106 (15 January 2012). DOI: 10.1215/00127094-1507288

Abstract

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 p to 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.

Citation

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Lillian B. Pierce. "Discrete fractional Radon transforms and quadratic forms." Duke Math. J. 161 (1) 69 - 106, 15 January 2012. https://doi.org/10.1215/00127094-1507288

Information

Published: 15 January 2012
First available in Project Euclid: 30 December 2011

zbMATH: 1246.44002
MathSciNet: MR2872554
Digital Object Identifier: 10.1215/00127094-1507288

Subjects:
Primary: 42B20 , 44A12
Secondary: 11E25 , 11P55

Rights: Copyright © 2012 Duke University Press

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Vol.161 • No. 1 • 15 January 2012
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