Home Mathematics Restricted averaging operators to cones over finite fields
Article
Licensed
Unlicensed Requires Authentication

Restricted averaging operators to cones over finite fields

  • Doowon Koh , Chun-Yen Shen EMAIL logo and Seongjun Yeom
Published/Copyright: June 26, 2018

Abstract

We investigate the sharp LpLr estimates for the restricted averaging operator AC over the cone C of the d-dimensional vector space 𝔽qd over the finite field 𝔽q with q elements. The restricted averaging operator AC for the cone C is defined by the relation ACf=fσ|C, where σ denotes the normalized surface measure on the cone C, and f is a complex-valued function on the space 𝔽qd with the normalized counting measure dx. In the previous work [D. Koh, C.-Y. Shen and I. Shparlinski, Averaging operators over homogeneous varieties over finite fields, J. Geom. Anal. 26 2016, 2, 1415–1441], the sharp boundedness of AC was obtained in odd dimensions d3, but only partial results were given in even dimensions d4. In this paper we prove the optimal estimates in even dimensions d6 in the case when the cone C𝔽qd contains a d/2-dimensional subspace.

MSC 2010: 42B05; 11T23

Communicated by Christopher D. Sogge


Award Identifier / Grant number: NRF-2015R1A1A1A05001374

Funding statement: The first author was supported by the research grant of Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2015R1A1A1A05001374) and the second author was supported by MOST, through grant 104-2628-M-002-015-MY4.

References

[1] B. Barceló, The restriction of the Fourier transform to some curves and surfaces, Studia Math. 84 (1986), no. 1, 39–69. 10.4064/sm-84-1-39-69Search in Google Scholar

[2] J. Bourgain, Besicovitch type maximal operators and applications to Fourier analysis, Geom. Funct. Anal. 1 (1991), no. 2, 147–187. 10.1007/BF01896376Search in Google Scholar

[3] J. Bourgain, On the restriction and multiplier problems in 𝐑3, Geometric Aspects of Functional Analysis (1989–90), Lecture Notes in Math. 1469, Springer, Berlin (1991), 179–191. 10.1007/BFb0089225Search in Google Scholar

[4] A. Carbery, Harmonic analysis on vector spaces over finite fields, Lecture note, http://www.maths.ed.ac.uk/~carbery/analysis/notes/fflpublic.pdf. Search in Google Scholar

[5] A. Carbery, B. Stones and J. Wright, Averages in vector spaces over finite fields, Math. Proc. Cambridge Philos. Soc. 144 (2008), no. 1, 13–27. 10.1017/S0305004107000680Search in Google Scholar

[6] J. S. Ellenberg, R. Oberlin and T. Tao, The Kakeya set and maximal conjectures for algebraic varieties over finite fields, Mathematika 56 (2010), no. 1, 1–25. 10.1112/S0025579309000400Search in Google Scholar

[7] C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9–36. 10.1007/BF02394567Search in Google Scholar

[8] D.-A. Geba, A. Greenleaf, A. Iosevich, E. Palsson and E. Sawyer, Restricted convolution inequalities, multilinear operators and applications, Math. Res. Lett. 20 (2013), no. 4, 675–694. 10.4310/MRL.2013.v20.n4.a6Search in Google Scholar

[9] L. Guth, A restriction estimate using polynomial partitioning, J. Amer. Math. Soc. 29 (2016), no. 2, 371–413. 10.1090/jams827Search in Google Scholar

[10] L. Guth, A restriction estimate using polynomial partitioning. II, preprint (2016), https://arxiv.org/abs/1603.04250. 10.4310/ACTA.2018.v221.n1.a3Search in Google Scholar

[11] A. Iosevich and D. Koh, Extension theorems for spheres in the finite field setting, Forum Math. 22 (2010), no. 3, 457–483. 10.1515/forum.2010.025Search in Google Scholar

[12] D. Koh, Averaging operators over nondegenerate quadratic surfaces in finite fields, Forum Math. 27 (2015), no. 2, 1227–1247. 10.1515/forum-2012-0135Search in Google Scholar

[13] D. Koh, Sharp Lp-Lr estimates of restricted averaging operators over curves on planes in finite fields, J. Chungcheong Math. Soc. 28 (2015), no. 2, 251–259. 10.14403/jcms.2015.28.2.251Search in Google Scholar

[14] D. Koh and C.-Y. Shen, Extension and averaging operators for finite fields, Proc. Edinb. Math. Soc. (2) 56 (2013), no. 2, 599–614. 10.1017/S0013091512000326Search in Google Scholar

[15] D. Koh, C.-Y. Shen and I. Shparlinski, Averaging operators over homogeneous varieties over finite fields, J. Geom. Anal. 26 (2016), no. 2, 1415–1441. 10.1007/s12220-015-9595-5Search in Google Scholar

[16] D. Koh and S. Yeom, Restriction of averaging operators to algebraic varieties over finite fields, Taiwanese J. Math. 21 (2017), no. 1, 211–229. 10.11650/tjm.21.2017.7743Search in Google Scholar

[17] A. Lewko and M. Lewko, Endpoint restriction estimates for the paraboloid over finite fields, Proc. Amer. Math. Soc. 140 (2012), no. 6, 2013–2028. 10.1090/S0002-9939-2011-11444-8Search in Google Scholar

[18] M. Lewko, Finite field restriction estimates based on Kakeya maximal operator estimates, preprint (2014), https://arxiv.org/abs/1401.8011. 10.4171/JEMS/910Search in Google Scholar

[19] M. Lewko, New restriction estimates for the 3-d paraboloid over finite fields, Adv. Math. 270 (2015), 457–479. 10.1016/j.aim.2014.11.008Search in Google Scholar

[20] G. Mockenhaupt and T. Tao, Restriction and Kakeya phenomena for finite fields, Duke Math. J. 121 (2004), no. 1, 35–74. 10.1215/S0012-7094-04-12112-8Search in Google Scholar

[21] T. Tao, A sharp bilinear restrictions estimate for paraboloids, Geom. Funct. Anal. 13 (2003), no. 6, 1359–1384. 10.1007/s00039-003-0449-0Search in Google Scholar

[22] T. Tao, Some recent progress on the restriction conjecture, Fourier Analysis and Convexity, Appl. Numer. Harmon. Anal., Birkhäuser, Boston (2004), 217–243. 10.1007/978-0-8176-8172-2_10Search in Google Scholar

[23] P. A. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975), 477–478. 10.1090/S0002-9904-1975-13790-6Search in Google Scholar

[24] L. A. Vinh, Maximal sets of pairwise orthogonal vectors in finite fields, Canad. Math. Bull. 55 (2012), no. 2, 418–423. 10.4153/CMB-2011-160-xSearch in Google Scholar

[25] T. Wolff, A sharp bilinear cone restriction estimate, Ann. of Math. (2) 153 (2001), no. 3, 661–698. 10.2307/2661365Search in Google Scholar

[26] T. H. Wolff, Lectures on Harmonic Analysis, Univ. Lecture Ser. 29, American Mathematical Society, Providence, 2003. 10.1090/ulect/029Search in Google Scholar

[27] A. Zygmund, On Fourier coefficients and transforms of functions of two variables, Studia Math. 50 (1974), 189–201. 10.4064/sm-50-2-189-201Search in Google Scholar

Received: 2017-02-28
Revised: 2018-03-11
Published Online: 2018-06-26
Published in Print: 2018-11-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 7.3.2026 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2017-0042/html
Scroll to top button