Duke Mathematical Journal

Discrete fractional Radon transforms and quadratic forms

Lillian B. Pierce

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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.

Article information

Duke Math. J., Volume 161, Number 1 (2012), 69-106.

First available in Project Euclid: 30 December 2011

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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

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  • [1] M. Christ, Endpoint bounds for singular fractional integral operators, unpublished manuscript, 1988.
  • [2] A. D. Ionescu and S. Wainger, Lp boundedness of discrete singular Radon transforms, J. Amer. Math. Soc. 19 (2006), 357–383.
  • [3] H. Iwaniec and E. Kowalski, Analytic Number Theory, Amer. Math. Soc. Colloq. Publ. 53, Amer. Math. Soc., Providence, 2004.
  • [4] D. M. Oberlin, Two discrete fractional integrals, Math. Res. Lett. 8 (2001), 1–6.
  • [5] L. B. Pierce, On discrete fractional integral operators and mean values of Weyl sums, Bull. London Math. Soc. 43 (2011), 597–612.
  • [6] E. M. Stein and S. Wainger, Discrete analogues in harmonic analysis, II: Fractional integration, J. Anal. Math. 80 (2000), 335–355.
  • [7] E. M. Stein and S. Wainger, Two discrete fractional integral operators revisited, J. Anal. Math. 87 (2002), 451–479.
  • [8] A. Walfisz, Gitterpunkte in mehrdimensionalen Kugeln, Monografie Matematyczne 33, Polish Scientific Publishers, Warsaw, 1957.