## Improved decay of conical averages of the Fourier transform

HTML articles powered by AMS MathViewer

- by Terence L. J. Harris PDF
- Proc. Amer. Math. Soc.
**147**(2019), 4781-4796 Request permission

## Abstract:

An improved lower bound is given for the decay of conical averages of Fourier transforms of measures, for cones of dimension $d \geq 4$. The proof uses a weighted version of the broad restriction inequality, a narrow decoupling inequality for the cone, and some techniques of Du and Zhang originally developed for the Schrödinger equation.## References

- Jean Bourgain and Ciprian Demeter,
*The proof of the $l^2$ decoupling conjecture*, Ann. of Math. (2)**182**(2015), no. 1, 351–389. MR**3374964**, DOI 10.4007/annals.2015.182.1.9 - Chu-Hee Cho, Seheon Ham, and Sanghyuk Lee,
*Fractal Strichartz estimate for the wave equation*, Nonlinear Anal.**150**(2017), 61–75. MR**3584933**, DOI 10.1016/j.na.2016.11.006 - Xiumin Du, Larry Guth, Yumeng Ou, Hong Wang, Bobby Wilson, and Ruixiang Zhang,
*Weighted restriction estimates and application to Falconer distance set problem*, arXiv:1802.10186v1 (2018). - Xiumin Du and Ruixiang Zhang,
*Sharp $L^2$ estimates of the Schrödinger maximal function in higher dimensions*, Ann. of Math. (2)**189**(2019), no. 3, 837–861. MR**3961084**, DOI 10.4007/annals.2019.189.3.4 - M. Burak Erdog̃an,
*A note on the Fourier transform of fractal measures*, Math. Res. Lett.**11**(2004), no. 2-3, 299–313. MR**2067475**, DOI 10.4310/MRL.2004.v11.n3.a3 - Larry Guth,
*Decoupling, Lecture 7*, http://math.mit.edu/%7Elguth/Math118.html (2017). - Larry Guth,
*Restriction estimates using polynomial partitioning II*, Acta Math.**221**(2018), no. 1, 81–142. MR**3877019**, DOI 10.4310/ACTA.2018.v221.n1.a3 - Terence L. J. Harris,
*Refined Strichartz inequalities for the wave equation*, arXiv:1805.07146v3 (2018). - Alex Iosevich and Elijah Liflyand,
*Decay of the Fourier transform*, Birkhäuser/Springer, Basel, 2014. Analytic and geometric aspects. MR**3308120**, DOI 10.1007/978-3-0348-0625-1 - Bochen Liu,
*An $L^2$-identity and pinned distance problem*, Geom. Funct. Anal.**29**(2019), no. 1, 283–294. MR**3925111**, DOI 10.1007/s00039-019-00482-8 - Pertti Mattila,
*Spherical averages of Fourier transforms of measures with finite energy; dimension of intersections and distance sets*, Mathematika**34**(1987), no. 2, 207–228. MR**933500**, DOI 10.1112/S0025579300013462 - Daniel Oberlin and Richard Oberlin,
*Application of a Fourier restriction theorem to certain families of projections in $\Bbb {R}^3$*, J. Geom. Anal.**25**(2015), no. 3, 1476–1491. MR**3358061**, DOI 10.1007/s12220-014-9480-7 - Yumeng Ou and Hong Wang,
*A cone restriction estimate using polynomial partitioning*, arXiv:1704.05485v1 (2017). - Yumeng Ou and Hong Wang,
*Private communication*(2018). - Per Sjölin,
*Estimates of spherical averages of Fourier transforms and dimensions of sets*, Mathematika**40**(1993), no. 2, 322–330. MR**1260895**, DOI 10.1112/S0025579300007087 - Terence Tao,
*Recent progress on the restriction conjecture*, arXiv:math/0311181v1 (2003). - Thomas Wolff,
*Decay of circular means of Fourier transforms of measures*, Internat. Math. Res. Notices**10**(1999), 547–567. MR**1692851**, DOI 10.1155/S1073792899000288

## Additional Information

**Terence L. J. Harris**- Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
- MR Author ID: 1124613
- ORCID: 0000-0003-3174-4320
- Email: terence2@illinois.edu
- Received by editor(s): December 22, 2018
- Received by editor(s) in revised form: January 17, 2019
- Published electronically: August 7, 2019
- Additional Notes: This material is based upon work partially supported by the National Science Foundation under Grant No. DMS-1501041. The author would like to thank Burak Erdoğan for suggesting this problem, for advice on this topic, and for financial support
- Communicated by: Alexander Iosevich
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**147**(2019), 4781-4796 - MSC (2010): Primary 42B37, 42B10
- DOI: https://doi.org/10.1090/proc/14747
- MathSciNet review: 4011512