Duke Mathematical Journal
- Duke Math. J.
- Volume 161, Number 1 (2012), 69-106.
Discrete fractional Radon transforms and quadratic forms
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 to . 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 Math. J., Volume 161, Number 1 (2012), 69-106.
First available in Project Euclid: 30 December 2011
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 44A12: Radon transform [See also 92C55] 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 11P55: Applications of the Hardy-Littlewood method [See also 11D85] 11E25: Sums of squares and representations by other particular quadratic forms
Pierce, Lillian B. Discrete fractional Radon transforms and quadratic forms. Duke Math. J. 161 (2012), no. 1, 69--106. doi:10.1215/00127094-1507288. https://projecteuclid.org/euclid.dmj/1325264706