Fourier decay of fractal measures on hyperboloids

Barron, Alex; Erdoğan, M.; Harris, Terence

doi:10.1090/tran/8283

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