Sets of Salem type and sharpness of the 𝐿²-Fourier restriction theorem

Chen, Xianghong

doi:10.1090/tran/6396

Sets of Salem type and sharpness of the $L^2$-Fourier restriction theorem
HTML articles powered by AMS MathViewer

by Xianghong Chen PDF

Trans. Amer. Math. Soc. 368 (2016), 1959-1977 Request permission

Abstract:

We construct Salem sets on the real line with endpoint Fourier decay and near-endpoint regularity properties. This complements a result of Łaba and Pramanik, who obtained near-endpoint Fourier decay and endpoint regularity properties. We then modify the construction to extend a theorem of Hambrook and Łaba to show sharpness of the $L^2$-Fourier restriction estimate by Mockenhaupt and Bak-Seeger, including the case where the Hausdorff and Fourier dimension do not coincide.

References

Jong-Guk Bak and Andreas Seeger, Extensions of the Stein-Tomas theorem, Math. Res. Lett. 18 (2011), no. 4, 767–781. MR 2831841, DOI 10.4310/MRL.2011.v18.n4.a14
Christian Bluhm, Random recursive construction of Salem sets, Ark. Mat. 34 (1996), no. 1, 51–63. MR 1396622, DOI 10.1007/BF02559506
Xianghong Chen, A Fourier restriction theorem based on convolution powers, Proc. Amer. Math. Soc. 142 (2014), no. 11, 3897–3901. MR 3251729, DOI 10.1090/S0002-9939-2014-12148-4
Xianghong Chen and A. Seeger, manuscript in preparation.
Rick Durrett, Probability: theory and examples, 4th ed., Cambridge Series in Statistical and Probabilistic Mathematics, vol. 31, Cambridge University Press, Cambridge, 2010. MR 2722836, DOI 10.1017/CBO9780511779398
Kyle Hambrook and Izabella Łaba, On the sharpness of Mockenhaupt’s restriction theorem, Geom. Funct. Anal. 23 (2013), no. 4, 1262–1277. MR 3077913, DOI 10.1007/s00039-013-0240-9
Wassily Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 58 (1963), 13–30. MR 144363
Jean-Pierre Kahane, Some random series of functions, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 5, Cambridge University Press, Cambridge, 1985. MR 833073
R. Kaufman, On the theorem of Jarník and Besicovitch, Acta Arith. 39 (1981), no. 3, 265–267. MR 640914, DOI 10.4064/aa-39-3-265-267
T. W. Körner, On the theorem of Ivašev-Musatov. III, Proc. London Math. Soc. (3) 53 (1986), no. 1, 143–192. MR 842159, DOI 10.1112/plms/s3-53.1.143
Thomas William Körner, Hausdorff and Fourier dimension, Studia Math. 206 (2011), no. 1, 37–50. MR 2845614, DOI 10.4064/sm206-1-3
Izabella Łaba and Malabika Pramanik, Arithmetic progressions in sets of fractional dimension, Geom. Funct. Anal. 19 (2009), no. 2, 429–456. MR 2545245, DOI 10.1007/s00039-009-0003-9
Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890, DOI 10.1017/CBO9780511623813
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
Themis Mitsis, A Stein-Tomas restriction theorem for general measures, Publ. Math. Debrecen 60 (2002), no. 1-2, 89–99. MR 1882456
G. Mockenhaupt, Salem sets and restriction properties of Fourier transforms, Geom. Funct. Anal. 10 (2000), no. 6, 1579–1587. MR 1810754, DOI 10.1007/PL00001662
R. Salem, On singular monotonic functions whose spectrum has a given Hausdorff dimension, Ark. Mat. 1 (1951), 353–365. MR 43249, DOI 10.1007/BF02591372
Peter A. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975), 477–478. MR 358216, DOI 10.1090/S0002-9904-1975-13790-6
Peter A. Tomas, Restriction theorems for the Fourier transform, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 111–114. MR 545245
Thomas H. Wolff, Lectures on harmonic analysis, University Lecture Series, vol. 29, American Mathematical Society, Providence, RI, 2003. With a foreword by Charles Fefferman and a preface by Izabella Łaba; Edited by Łaba and Carol Shubin. MR 2003254, DOI 10.1090/ulect/029