## Fourier decay of fractal measures on hyperboloids

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- by Alex Barron, M. Burak Erdoğan and Terence L. J. Harris PDF
- Trans. Amer. Math. Soc.
**374**(2021), 1041-1075 Request permission

## Abstract:

Let $\mu$ be an $\alpha$-dimensional probability measure. We prove new upper and lower bounds on the decay rate of hyperbolic averages of the Fourier transform $\widehat {\mu }$. More precisely, if $\mathbb {H}$ is a truncated hyperbolic paraboloid in $\mathbb {R}^d$ we study the optimal $\beta$ for which \begin{equation*} \int _{\mathbb {H}} |\widehat {\mu }(R\xi )|^2 d \sigma (\xi )\leq C(\alpha , \mu ) R^{-\beta } \end{equation*} for all $R > 1$. Our estimates for $\beta$ depend on the minimum between the number of positive and negative principal curvatures of $\mathbb {H}$; if this number is as large as possible our estimates are sharp in all dimensions.## References

- Juan Antonio Barceló, Jonathan Bennett, Anthony Carbery, and Keith M. Rogers,
*On the dimension of divergence sets of dispersive equations*, Math. Ann.**349**(2011), no. 3, 599–622. MR**2754999**, DOI 10.1007/s00208-010-0529-z - J. A. Barceló, J. M. Bennett, A. Carbery, A. Ruiz, and M. C. Vilela,
*Some special solutions of the Schrödinger equation*, Indiana Univ. Math. J.**56**(2007), no. 4, 1581–1593. MR**2354692**, DOI 10.1512/iumj.2007.56.3016 - A. Barron,
*Restriction estimates for hyperboloids in higher dimensions via bilinear estimates*, Preprint, arXiv:2002.09001. - Jonathan Bennett, Anthony Carbery, and Terence Tao,
*On the multilinear restriction and Kakeya conjectures*, Acta Math.**196**(2006), no. 2, 261–302. MR**2275834**, DOI 10.1007/s11511-006-0006-4 - Jean Bourgain,
*Hausdorff dimension and distance sets*, Israel J. Math.**87**(1994), no. 1-3, 193–201. MR**1286826**, DOI 10.1007/BF02772994 - Jean Bourgain and Ciprian Demeter,
*Decouplings for curves and hypersurfaces with nonzero Gaussian curvature*, J. Anal. Math.**133**(2017), 279–311. MR**3736493**, DOI 10.1007/s11854-017-0034-3 - 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 - Yutae Choi, Seheon Ham, and Sanghyuk Lee,
*Averaged decay estimates for Fourier transforms of measures over curves with nonvanishing torsion*, J. Fourier Anal. Appl.**23**(2017), no. 5, 1028–1061. MR**3704756**, DOI 10.1007/s00041-016-9497-3 - Ciprian Demeter,
*Fourier restriction, decoupling, and applications*, Cambridge Studies in Advanced Mathematics, vol. 184, Cambridge University Press, Cambridge, 2020. MR**3971577**, DOI 10.1017/9781108584401 - X. Du,
*Upper bounds for Fourier decay rates of fractal measures*, J. Lond. Math. Soc. (2020), to appear. - Xiumin Du, Larry Guth, and Xiaochun Li,
*A sharp Schrödinger maximal estimate in $\Bbb R^2$*, Ann. of Math. (2)**186**(2017), no. 2, 607–640. MR**3702674**, DOI 10.4007/annals.2017.186.2.5 - Xiumin Du, Larry Guth, Xiaochun Li, and Ruixiang Zhang,
*Pointwise convergence of Schrödinger solutions and multilinear refined Strichartz estimates*, Forum Math. Sigma**6**(2018), Paper No. e14, 18. MR**3842310**, DOI 10.1017/fms.2018.11 - X. Du, L. Guth, Y. Ou, H. Wang, B. Wilson, and R. Zhang,
*Weighted restriction estimates and application to Falconer distance set problem*, To appear in American Journal of Math., preprint arXiv:1802.10186. - 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 - M. Burak Erdog̃an,
*A bilinear Fourier extension theorem and applications to the distance set problem*, Int. Math. Res. Not.**23**(2005), 1411–1425. MR**2152236**, DOI 10.1155/IMRN.2005.1411 - M. B. Erdoğan, M. Goldberg, and W. Green,
*Strichartz estimates for the Schrodinger equation with a measure-valued potential*, preprint 2019. - M. Burak Erdoğan and Daniel M. Oberlin,
*Restricting Fourier transforms of measures to curves in $\Bbb R^2$*, Canad. Math. Bull.**56**(2013), no. 2, 326–336. MR**3043060**, DOI 10.4153/CMB-2011-171-5 - K. J. Falconer,
*On the Hausdorff dimensions of distance sets*, Mathematika**32**(1985), no. 2, 206–212 (1986). MR**834490**, DOI 10.1112/S0025579300010998 - Michael Goldberg,
*Dispersive estimates for Schrödinger operators with measure-valued potentials in $\Bbb R^3$*, Indiana Univ. Math. J.**61**(2012), no. 6, 2123–2141. MR**3129105**, DOI 10.1512/iumj.2012.61.4786 - 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 - L. Guth, Notes for
*Topics in Analysis: Decoupling*, Lecture 7. Transcribed by J. Tidor, http://math.mit.edu/~lguth/Math118.html. - Larry Guth, Jonathan Hickman, and Marina Iliopoulou,
*Sharp estimates for oscillatory integral operators via polynomial partitioning*, Acta Math.**223**(2019), no. 2, 251–376. MR**4047925**, DOI 10.4310/acta.2019.v223.n2.a2 - Larry Guth, Alex Iosevich, Yumeng Ou, and Hong Wang,
*On Falconer’s distance set problem in the plane*, Invent. Math.**219**(2020), no. 3, 779–830. MR**4055179**, DOI 10.1007/s00222-019-00917-x - Seheon Ham and Sanghyuk Lee,
*Restriction estimates for space curves with respect to general measures*, Adv. Math.**254**(2014), 251–279. MR**3161099**, DOI 10.1016/j.aim.2013.12.017 - T. L. J. Harris,
*Refined Strichartz inequalities for the wave equation*, Preprint 2018, arXiv:1805.07146. - Terence L. J. Harris,
*Improved decay of conical averages of the Fourier transform*, Proc. Amer. Math. Soc.**147**(2019), no. 11, 4781–4796. MR**4011512**, DOI 10.1090/proc/14747 - T. L. J. Harris,
*Improved bounds for restricted projection families via weighted Fourier restriction*, preprint 2019, arXiv:1911.00615v3. - J. Hickman and M. Iliopoulou,
*Sharp $L^p$ estimates for oscillatory integral operators of arbitrary signature*, preprint arXiv:2006.01316 - S. Hofmann and A. Iosevich,
*Circular averages and Falconer/Erdös distance conjecture in the plane for random metrics*, Proc. Amer. Math. Soc.**133**(2005), no. 1, 133–143. MR**2085162**, DOI 10.1090/S0002-9939-04-07603-8 - A. Iosevich and I. Łaba,
*$K$-distance sets, Falconer conjecture, and discrete analogs*, Integers**5**(2005), no. 2, A8, 11. MR**2192086** - Sanghyuk Lee,
*Bilinear restriction estimates for surfaces with curvatures of different signs*, Trans. Amer. Math. Soc.**358**(2006), no. 8, 3511–3533. MR**2218987**, DOI 10.1090/S0002-9947-05-03796-7 - 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 - Bochen Liu,
*Hausdorff dimension of pinned distance sets and the $L^2$-method*, Proc. Amer. Math. Soc.**148**(2020), no. 1, 333–341. MR**4042855**, DOI 10.1090/proc/14740 - Renato Lucà and Keith M. Rogers,
*Average decay of the Fourier transform of measures with applications*, J. Eur. Math. Soc. (JEMS)**21**(2019), no. 2, 465–506. MR**3896208**, DOI 10.4171/JEMS/842 - 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 - Pertti Mattila,
*Hausdorff dimension, projections, and the Fourier transform*, Publ. Mat.**48**(2004), no. 1, 3–48. MR**2044636**, DOI 10.5565/PUBLMAT_{4}8104_{0}1 - Daniel M. Oberlin,
*Packing spheres and fractal Strichartz estimates in $\Bbb R^d$ for $d\geq 3$*, Proc. Amer. Math. Soc.**134**(2006), no. 11, 3201–3209. MR**2231903**, DOI 10.1090/S0002-9939-06-08356-0 - 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 - K. Rogers,
*Falconer’s distance set problem via the wave equation*, Preprint 2018, arXiv:1802.01057. - Keith M. Rogers, Ana Vargas, and Luis Vega,
*Pointwise convergence of solutions to the nonelliptic Schrödinger equation*, Indiana Univ. Math. J.**55**(2006), no. 6, 1893–1906. MR**2284549**, DOI 10.1512/iumj.2006.55.2827 - 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 - Per Sjölin,
*Estimates of averages of Fourier transforms of measures with finite energy*, Ann. Acad. Sci. Fenn. Math.**22**(1997), no. 1, 227–236. MR**1430401** - Ana Vargas,
*Restriction theorems for a surface with negative curvature*, Math. Z.**249**(2005), no. 1, 97–111. MR**2106972**, DOI 10.1007/s00209-004-0691-7 - 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 - T. Wolff,
*Local smoothing type estimates on $L^p$ for large $p$*, Geom. Funct. Anal.**10**(2000), no. 5, 1237–1288. MR**1800068**, DOI 10.1007/PL00001652

## Additional Information

**Alex Barron**- Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
- MR Author ID: 1230904
- ORCID: 0000-0001-9863-7154
- Email: aabarron@illinois.edu
**M. Burak Erdoğan**- Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
- Email: berdogan@illinois.edu
**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): April 14, 2020
- Received by editor(s) in revised form: April 29, 2020
- Published electronically: December 3, 2020
- Additional Notes: The second author was partially supported by the Simons collaboration grant, 634269.
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**374**(2021), 1041-1075 - MSC (2020): Primary 42B37
- DOI: https://doi.org/10.1090/tran/8283
- MathSciNet review: 4196386