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Convolution Powers of Salem MeasuresWith Applications

Published online by Cambridge University Press:  20 November 2018

Xianghong Chen  and
Andreas Seeger
Xianghong Chen
Affiliation:
Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, WI 53211, USA e-mail: chen242@umw.edu
Andreas Seeger
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA e-mail: seeger@math.wisc.edu
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Abstract

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We study the regularity of convolution powers for measures supported on Salem sets, and prove related results on Fourier restriction and Fourier multipliers. In particular we show that for $\alpha $ of the form $d\,/\,n,\,n\,=\,2,3,...$ there exist $\alpha $-Salem measures for which the ${{L}^{2}}$ Fourier restriction theorem holds in the range $p\,\le \,\frac{2d}{2d\,-\,\alpha }$. The results rely on ideas of Körner. We extend some of his constructions to obtain upper regular $\alpha $-Salem measures, with sharp regularity results for $n$-fold convolutions for all $n\,\in \,\mathbb{N}$.

Keywords

42A8542B9942B1542A61convolution powersFourier restrictionSalem setsSalem measuresrandom sparse setsFourier multipliers of Bochner-Riesz type.
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 69 , Issue 2 , 01 April 2017 , pp. 284 - 320
Copyright
Copyright © Canadian Mathematical Society 2017

References

[1] Bak, J.-G and Seeger, A., Extensions of the Stein-Tomas theorem. Math. Res. Lett. 18(2011), no. 4,767–781. http://dx.doi.Org/10.4310/MRL.2011.v18.n4.a14 Google Scholar
[2] Bluhm, C., Random recursive construction of Salem sets. Ark. Mat. 34(1996), no. 1, 51–63. http://dx.doi.org/10.1007/BF02559506 Google Scholar
[3] Bluhm, C., On a theorem of Kaufman: Cantor-type construction of linear fractal Salem sets. Ark. Mat. 36(1998), no. 2, 307–316. http://dx.doi.org/10.1007/BF02384771 Google Scholar
[4] Bourgain, J., Bounded orthogonal systems and the A(p)-set problem. Acta Math. 162(1989), no. 3-4, 227–245. http://dx.doi.org/10.1007/BF02392838 Google Scholar
[5] Chan, V., On convolution squares of singular measures. Master's Thesis, University of Waterloo, 2010.Google Scholar
[6] Chen, X., A Fourier restriction theorem based on convolution powers. Proc. Amer. Math. Soc. 142(2014), no. 11, 3897–3901. http://dx.doi.org/10.1090/S0002-9939-2014-12148-4 Google Scholar
[7] Chen, X.,Restriction of the Fourier transform to Salem sets. Ph.D. Thesis, University of Wisconsin-Madison 2015.Google Scholar
[8] Chen, X., Sets of Salem type and sharpness of the L2-Fourier restriction theorem. Trans. Amer.Math. Soc. 368(2016), no.3, 1959–1977. http://dx.doi.Org/10.1090/tran/6396 Google Scholar
[9] Erdös, P. and Rényi, A., A probabilistic approach to problems of diophantine approximation. Illinois J. Math. 1(1957), 303–315.Google Scholar
[10] Fefferman, C., Inequalities for strongly singular convolution operators. Acta Math. 124(1970), 9–36. http://dx.doi.org/10.1007/BF02394567 Google Scholar
[11] Hambrook, K., Explicit Salem sets and applications to metrical Diophantine approximation. arxiv:1604.00411.Google Scholar
[12] Hambrook, K. and Laba, I., On the sharpness of Mockenhaupt's restriction theorem. Geom. Funct.Anal. 23(2013), no. 4, 1262–1277. http://dx.doi.org/10.1007/s00039-013-0240-9 Google Scholar
[13] Hambrook, K.,Sharpness of the Mockenhaupt-Mitsis-Bak-Seeger restriction theorem in higher dimensions.arxiv:5O9.O5912.Google Scholar
[14] Hoeffding, W., Probability inequalities for sums of bounded random variables. J. Amer. Statist.Assoc. 58(1963), 13–30. http://dx.doi.org/10.1080/01621459.1963.10500830 Google Scholar
[15] Kahane, J.-P., Some random series of functions. 2nd edition. Cambridge, Cambridge University Press, 1985.Google Scholar
[16] Kaufman, R., On the theorem ofjarnik and Besicovitch. Acta Arith. 39(1981), no. 3, 265–267.Google Scholar
[17] Körner, T. W., On the theorem of Ivasev-Musatov. III. Proc. London Math. Soc. (3) 53(1986), no. 1,143–192. http://dx.doi.Org/10.1112/plms/s3-53.1.143 Google Scholar
[18] Körner, T. W., On a theorem ofSaeki concerning convolution squares of singular measures. Bull. Soc. Math. France 136(2008), 439–464.Google Scholar
[19] Körner, T. W.,Baire category, probabilistic constructions and convolution squares. Lecture notes, University of Cambridge, 2009.Google Scholar
[20] Körner, T. W., Hausdorff and Fourier dimension. Studia Math. 206 (2011), no. 1, 37–50.http://dx.doi.org/10.4064/sm206-1-3 Google Scholar
[21] łaba, I., Harmonic analysis and the geometry of fractals, Proceedings of the 2014 International Congress of Mathematicians, to appear.Google Scholar
[22] łaba, I. and Pramanik, M., Arithmetic progressions in sets of fractional dimension. Geom. Funct.Anal. 19(2009), no. 2, 429–456.http://dx.doi.Org/10.1007/s00039-009-0003-9 Google Scholar
[23] Mattila, P., Hausdorff dimension, projections, and the Fourier transform. Publ. Mat. 48(2004), no. 1, 3–48. http://dx.doi.org/10.5565/PUBLMAT_48104_01 Google Scholar
[24] Mitsis, T., Stein-Tomas, A restriction theorem for general measures. Publ. Math. Debrecen 60(2002), no. 1-2, 89–99.Google Scholar
[25] Mockenhaupt, G., Salem sets and restriction properties of Fourier transform. Geom. Funct. Anal. 10(2000), no. 6, 1579–1587. http://dx.doi.org/10.1007/PL00001662 Google Scholar
[26] Rudin, W., Trigonometric series with gaps. J. Math. Mech. 9(1960), 203–227.Google Scholar
[27] Saeki, S., On convolution powers of singular measures. Illinois J. Math. 24(1980), 225–232.Google Scholar
[28] Salem, R.,On singular monotonie functions whose spectrum has a given Hausdorff dimension. Ark.Mat. 1(1951), no. 4, 353–365. http://dx.doi.org/10.1007/BF02591372 Google Scholar
[29] Senthil Raani, K. S., Lp asymptotics of Fourier transform of fractal measures. Ph.D. Thesis, Indian Institute of Science, Bangalore, 2015.Google Scholar
[30] Shmerkin, P. and Suomala, V., Spatially independent martingales, intersections, and applications. Memoirs of the American Mathematical Society, to appear. arxiv:1409.6707.Google Scholar
[31] Stein, E. M. and Shakarchi, R., Functional analysis. Princeton, Princeton University Press, 2011.Google Scholar
[32] Strichartz, R., Fourier asymptotics of fractal measures. J. Funct. Anal. 89(1990), no. 1,154–187.http://dx.doi.Org/10.1016/0022-1236(90)90009-A Google Scholar
[33] Tao, T., Some recent progress on the restriction conjecture. Fourier analysis and convexity. Appl. Numer. Harmon. Anal., Birkhäuser, Boston, MA, 2004. pp. 217–243.Google Scholar
[34] Tomas, P. A., A restriction theorem for the Fourier transform. Bull. Amer. Math. Soc. 81(1975), 477–478. http://dx.doi.org/10.1090/S0002-9904-1975-13790-6 Google Scholar
[35] Tomas, P. A., Restriction theorems for the Fourier transform. In: Harmonic analysis in Euclidean spaces. Proc. Symp. Pure Math. American Mathematical Society, Providence, RI, 1979,pp. 111–114.Google Scholar
[36] Wiener, N. and Wintner, A., Fourier-Stieltjes transforms and singular infinite convolutions. Amer. J. Math 60(1938), 513–522.http://dx.doi.org/10.2307/2371591 Google Scholar
[37] Wolff, T. H., Lectures on harmonic analysis. I. ł aba and C. Shubin, eds. University Lecture Series 29. American Mathematical Society, Providence, RI, 2003.Google Scholar