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Published online by Cambridge University Press:  12 December 2019

Ciprian Demeter
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Indiana University, Bloomington
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[1] Arkhipov, G. I., Chubarikov, V. N. and Karatsuba, A. A. Trigonometric sums in number theory and analysis, Translated from the 1987 Russian original. De Gruyter Expositions in Mathematics, 39. Walter de Gruyter GmbH & Co. KG, Berlin, 2004Google Scholar
[2] Bak, Jong-Guk and Seeger, A. Extensions of the Stein–Tomas theorem, Math. Res. Lett. 18 (2011), no. 4, 767–81Google Scholar
[3] Barcelo, B. On the restriction of the Fourier transform to a conical surface, Trans. Amer. Math. Soc. 292 (1985), 321–33Google Scholar
[4] Bejenaru, I. Optimal bilinear restriction estimates for general hypersurfaces and the role of the shape operator, Int. Math. Res. Not. IMRN (2017), no. 23, 7109–47Google Scholar
[5] Bejenaru, I. Optimal multilinear restriction estimates for a class of surfaces with curvature, Anal. PDE 12 (2019), no. 4, 1115–48Google Scholar
[6] Bejenaru, I. The multilinear restriction estimate: a short proof and a refinement, Math. Res. Lett. 24 (2017), no. 6, 15851603Google Scholar
[7] Bejenaru, I. The optimal trilinear restriction estimate for a class of hypersurfaces with curvature, Adv. Math. 307 (2017), 1151–83Google Scholar
[8] Bennett, J. Aspects of multilinear harmonic analysis related to transversality, Harmonic analysis and partial differential equations, Edited by Cifuentes, Patricio, García-Cuerva, José, Garrigós, Gustavo, et al., 1–28, Contemp. Math., 612, Amer. Math. Soc., Providence, RI, 2014Google Scholar
[9] Bennett, J. A trilinear restriction problem for the paraboloid in R3, Electron. Res. Announc. Amer. Math. Soc. 10 (2004), 97102Google Scholar
[10] Bennett, J., Bez, N., Flock, T. and Lee, S. Stability of Brascamp–Lieb constant and applications, Amer. J. Math. 140 (2018), no. 2, 543–69Google Scholar
[11] Bennett, J., Carbery, A., Christ, M. and Tao, T. Finite bounds for Hölder– Brascamp–Lieb multilinear inequalities, Math. Res. Lett. 17 (2010), no. 4, 647–66Google Scholar
[12] Bennett, J., Carbery, A., Christ, M. and Tao, T. The Brascamp–Lieb inequalities: finiteness, structure and extremals, Geom. Funct. Anal. 17 (2008), no. 5, 13431415Google Scholar
[13] Bennett, J., Carbery, A. and Tao, T. On the multilinear restriction and Kakeya conjectures, Acta Math. 196 (2006), no. 2, 261302Google Scholar
[14] Bennett, J., Carbery, A. and Wright, J. A non-linear generalisation of the Loomis–Whitney inequality and applications, Math. Res. Lett. 12 (2005), no. 4, 443–57Google Scholar
[15] Biswas, C., Gilula, M., Li, L., Schwend, J. and Xi, Y. l2 decoupling in R2 for curves with vanishing curvature, available on arXivGoogle Scholar
[16] Blomer, V. and Brüdern, J. The number of integer points on Vinogradov’s quadric, Monatsh. Math. 160 (2010), no. 3, 243–56Google Scholar
[17] Bombieri, E. and Bourgain, J. A problem on sums of two squares, Int. Math. Res. Not. IMRN (2015), no. 11, 3343–407Google Scholar
[18] Bombieri, E. and Zannier, U. Note on squares in arithmetic progressions II, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9), Mat. Appl. 13 (2002), no. 2, 6975Google Scholar
[19] Bourgain, J. On the Vinogradov integral (Russian) English version published in Proc. Steklov Inst. Math. 296 (2017), no. 1, 30–40. Tr. Mat. Inst. Steklova 296 (2017)Google Scholar
[20] Bourgain, J. Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces, Israel J. Math. 193 (2013), no. 1, 441–58Google Scholar
[21] Bourgain, J. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3 (1993), no. 2, 107–56Google Scholar
[22] Bourgain, J. Besicovitch type maximal operators and applications to Fourier analysis, Geom. Funct. Anal. 1 (1991), no. 2, 147–87Google Scholar
[23] Bourgain, J. Eigenfunction bounds for the Laplacian on the n-torus, Internat. Math. Res. Not. (1993), no. 3, 61–6Google Scholar
[24] Bourgain, J. On (p)-subsets of squares, Israel J. Math. 67 (1989), no. 3, 291311Google Scholar
[25] Bourgain, J., Demeter, C. and Kemp, D. Decouplings for real analytic surfaces of revolution, to appear in GAFA seminar notesGoogle Scholar
[26] Bourgain, J., Demeter, C. and Guo, S. Sharp bounds for the cubic Parsell–Vinogradov system in two dimensions, Adv. Math. 320 (2017), 827–75Google Scholar
[27] Bourgain, J. and Demeter, C. Decouplings for curves and hypersurfaces with nonzero Gaussian curvature, Journal d’Analyse Mathematique 133 (2017), 279311Google Scholar
[28] Bourgain, J. and Demeter, C. Mean value estimates for Weyl sums in two dimensions, J. Lond. Math. Soc. (2) 94 (2016), no. 3, 814–38Google Scholar
[29] Bourgain, J. and Demeter, C. The proof of the l2 decoupling conjecture, Ann. of Math. 182 (2015), no. 1, 351–89Google Scholar
[30] Bourgain, J. and Demeter, C. New bounds for the discrete Fourier restriction to the sphere in 4D and 5D, Int. Math. Res. Not. IMRN (2015), no. 11, 3150–84Google Scholar
[31] Bourgain, J. and Demeter, C. Improved estimates for the discrete Fourier restriction to the higher dimensional sphere, Illinois J. Math. 57 (2013), no. 1, 213–27Google Scholar
[32] Bourgain, J. and Guth, L. Bounds on oscillatory integral operators based on multilinear estimates, Geom. Funct. Anal. 21 (2011), no. 6, 1239–95Google Scholar
[33] Bourgain, J., Demeter, C. and Guth, L., Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three, Ann. of Math. (2) 184 (2016), no. 2, 633–82Google Scholar
[34] Brandolini, L., Gigante, G., Greenleaf, A., Iosevich, A., Seeger, A. and Travaglini, G. Average decay estimates for Fourier transforms of measures supported on curves, J. Geom. Anal. 17 (2007), no. 1, 1540Google Scholar
[35] Carbery, A. A remark on reverse Littlewood–Paley, restriction and Kakeya, available on arXiv at https://arxiv.org/pdf/1507.02515.pdfGoogle Scholar
[36] Carleson, L. On the Littlewood–Paley theorem, Report, Mittag-Leffler Inst. (1967)Google Scholar
[37] Christ, M. On the restriction of the Fourier transform to curves: endpoint results and the degenerate case, Trans. Amer. Math. Soc. 287 (1985), no. 1, 223–38Google Scholar
[38] Cilleruelo, J. and Granville, A. Lattice points on circles, squares in arithmetic progressions and sumsets of squares, Additive combinatorics, Edited by Granville, Andrew, Nathanson, Melvyn B., and Solymosi, József, 241–62, CRM Proc. Lecture Notes, 43, Amer. Math. Soc., Providence, RI, 2007CrossRefGoogle Scholar
[39] Córdoba, A. Vector valued inequalities for multipliers, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II, 295–305 (Chicago, Il, 1981), Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983Google Scholar
[40] Córdoba, A. Geometric Fourier analysis, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 3, vii, 215–26Google Scholar
[41] Córdoba, A. The Kakeya maximal function and the spherical summation multipliers, Amer. J. Math. 99 (1977), no. 1, 122Google Scholar
[42] Davies, R. O. Some remarks on the Kakeya problem, Proc. Cambridge Philos. Soc. 69 (1971), 417–21Google Scholar
[43] Demeter, C. A decoupling for Cantor-like sets, Proc. Amer. Math. Soc. 147 (2019), no. 3, 1037–50Google Scholar
[44] Demeter, C. On the restriction theorem for paraboloid in R4, Colloq. Math. 156 (2019), no. 2, 301–11Google Scholar
[45] Drury, S. W. Restrictions of Fourier transforms to curves, Ann. Inst. Fourier (Grenoble) 35 (1985), no. 1, 117–23Google Scholar
[46] Dvir, Z. On the size of Kakeya sets in finite fields, J. Amer. Math. Soc. 22 (2009), no. 4, 1093–7Google Scholar
[47] Fefferman, C. A note on spherical summation multipliers, Israel J. Math. 15 (1973), 4452Google Scholar
[48] Fefferman, C. The multiplier problem for the ball, Ann. of Math. (2) 94 (1971), 330–6Google Scholar
[49] Fefferman, C. Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 936Google Scholar
[50] Foschi, D. Global maximizers for the sphere adjoint Fourier restriction inequality, J. Funct. Anal. 268 (2015), no. 3, 690702Google Scholar
[51] Garrigós, G. and Seeger, A. A mixed norm variant of Wolff’s inequality for paraboloids, Harmonic analysis and partial differential equations, Contemporary Mathematics, Vol. 505, Edited by Cifuentes, Patricio, García-Cuerva, José, Garrigós, Gustavo, et al., Amer. Math. Soc., Providence, RI, 2010, 179–97Google Scholar
[52] Garrigós, G. and Seeger, A. On plate decompositions of cone multipliers, Proc. Edinb. Math. Soc. (2) 52 (2009), no. 3, 631–51Google Scholar
[53] Greenleaf, A. Principal curvature and harmonic analysis, Indiana Univ. Math. J. 30 (1981), no. 4, 519–37Google Scholar
[54] Grosswald, E. Representations of integers as sums of squares, Springer-Verlag, New York, NY, 1985Google Scholar
[55] Guo, S. and Zhang, R. On integer solutions of Parsell–Vinogradov systems, available on arXivGoogle Scholar
[56] Guth, L. Restriction estimates using polynomial partitioning II, available on arXivGoogle Scholar
[57] Guth, L. A restriction estimate using polynomial partitioning, J. Amer. Math. Soc. 29 (2016), no. 2, 371413Google Scholar
[58] Guth, L. Polynomial methods in combinatorics, University Lecture Series, 64. Amer. Math. Soc., Providence, RI, 2016Google Scholar
[59] Guth, L. A short proof of the multilinear Kakeya inequality, Math. Proc. Cambridge Philos. Soc. 158 (2015), no. 1, 147–53Google Scholar
[60] Guth, L. The endpoint case of the Bennett–Carbery–Tao multilinear Kakeya conjecture, Acta Math. 205 (2010), no. 2, 263–86Google Scholar
[61] Guth, L. and Katz, N. H. On the Erdös distinct distances problem in the plane, Ann. of Math. (2) 181 (2015), no. 1, 155–90Google Scholar
[62] Guth, L. and Katz, N. H. Algebraic methods in discrete analogs of the Kakeya problem, Adv. Math. 225 (2010), no. 5, 2828–39Google Scholar
[63] Guth, L. and Zahl, J. Polynomial Wolff axioms and Kakeya-type estimates in R4, available on arXivGoogle Scholar
[64] Guth, L., Iosevich, A., Ou, Y. and Wang, H. On Falconer’s distance set problem in the plane, available on arXivGoogle Scholar
[65] Hardy, G. H. and Littlewood, J. E. Some problems of diophantine approximation, Acta Math. 37 (1914), no. 1, 193239Google Scholar
[66] Heo, Y., Nazarov, F. and Seeger, A. Radial Fourier multipliers in high dimensions, Acta Math. 206 (2011), no. 1, 5592Google Scholar
[67] Hickman, J. and Rogers, K., M. Improved Fourier restriction estimates in higher dimensions, available on arXivGoogle Scholar
[68] Hörmander, L. Oscillatory integrals and multipliers on FLp, Ark. Mat. 11, (1973), 111Google Scholar
[69] Hua, L. K. On Waring’s problem, Q. J. Math 9. 199–202Google Scholar
[70] Kaplan, H., Sharir, M. and Shustin, E. On lines and joints, Discrete Comput. Geom. 44 (2010), no. 4, 838–43Google Scholar
[71] Katznelson, Y., An introduction to harmonic analysis. Third edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2004Google Scholar
[72] Katz, N. H. and Rogers, K. On the polynomial Wolff axioms, available on arXivGoogle Scholar
[73] Katz, N. H. and Zahl, J. An improved bound on the Hausdorff dimension of Besicovitch sets in R3, available on arXivGoogle Scholar
[74] Katz, N. H. and Tao, T. New bounds for Kakeya problems. Dedicated to the memory of Thomas H. Wolff, J. Anal. Math. 87 (2002), 231–63Google Scholar
[75] Killip, R. and Visan, M. Scale invariant Strichartz estimates on tori and applications, Math. Res. Lett. 23 (2016), no. 2, 445–72Google Scholar
[76] Łaba, I. and Pramanik, M. Wolff’s inequality for hypersurfaces, Collect. Math. 2006, Vol. Extra, 293326Google Scholar
[77] Łaba, I. and Tao, T. An improved bound for the Minkowski dimension of Besicovitch sets in medium dimension, Geom. Funct. Anal. 11 (2001), no. 4, 773806Google Scholar
[78] Łaba, I. and Tao, T. An X-ray transform estimate in R3, Rev. Mat. Iberoamericana 17 (2001), no. 2, 375407Google Scholar
[79] Łaba, I. and Wolff, T. A local smoothing estimate in higher dimensions, dedicated to the memory of Tom Wolff. J. Anal. Math. 88 (2002), 149–71Google Scholar
[80] Lacey, M. T. Issues related to Rubio de Francia’s Littlewood–Paley inequality, New York Journal of Mathematics. NYJM Monographs, 2. State University of New York, University at Albany (2007)Google Scholar
[81] Lee, S. and Vargas, A. On the cone multiplier in R3, J. Funct. Anal. 263 (2012), no. 4, 925–40Google Scholar
[82] Lee, S. and Vargas, A. Restriction estimates for some surfaces with vanishing curvatures, J. Funct. Anal. 258 (2010), no. 9, 2884–909Google Scholar
[83] Mattila, P. Fourier analysis and Hausdorff dimension, Cambridge Studies in Advanced Mathematics, 150. Cambridge University Press, Cambridge, 2015Google Scholar
[84] Miyachi, A. On some estimates for the wave equation in Lp and Hp, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), no. 2, 331–54Google Scholar
[85] Mockenhaupt, G., Seeger, A. and Sogge, D. Wave front sets, local smoothing and Bourgain’s circular maximal theorem, Ann. of Math. (2) 136 (1992), no. 1, 2072018Google Scholar
[86] Müller, D., Ricci, F. and Wright, J. A maximal restriction theorem and Lebesgue points of functions in F(Lp), available on arXivGoogle Scholar
[87] Nicola, F. Slicing surfaces and the Fourier restriction conjecture, Proc. Edinb. Math. Soc. (2) 52 (2009), no. 2, 515–27Google Scholar
[88] Ou, Y. and Wang, H. A cone restriction estimate using polynomial partitioning, available on arXivGoogle Scholar
[89] Pach, J. and Sharir, M. Repeated angles in the plane and related problems, J. Combin. Theory Ser. A, 59 (1992), 1222Google Scholar
[90] Peral, Juan, C. Lp estimates for the wave equation, J. Funct. Anal. 36 (1980), no. 1, 114–45Google Scholar
[91] Pramanik, M. and Seeger, A. Lp regularity of averages over curves and bounds for associated maximal operators, Amer. J. Math. 129 (2007), no. 1, 61103Google Scholar
[92] Prestini, E. A restriction theorem for space curves, Proc. Amer. Math. Soc. 70 (1978), no. 1, 810Google Scholar
[93] Prestini, E. Restriction theorems for the Fourier transform to some manifolds in Rn, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, MA, 1978), Part 1, 101–9, Proc. Sympos. Pure Math., XXXV, Part I, Amer. Math. Soc., Providence, RI, 1979Google Scholar
[94] Quilodrán, R. The joints problem in R3, SIAM J. Discrete Math. 23 (2009/10), no. 4, 2211–13Google Scholar
[95] Ramos, J. The trilinear restriction estimate with sharp dependence on the transversality, available on arXivGoogle Scholar
[96] Rubio de Francia, J, L. A Littlewood–Paley inequality for arbitrary intervals, Rev. Mat. Iberoamericana 1 (1985), no. 2, 114Google Scholar
[97] Rudin, W. Trigonometric series with gaps, Journal of Mathematics and Mechanics, 9 (1960), no. 2, 203–27Google Scholar
[98] Sanders, T. The structure theory of set addition revisited, Bull. Amer. Math. Soc. (N.S.) 50 (2013), no. 1, 93127Google Scholar
[99] Sogge, C., D. Improved critical eigenfunction estimates on manifolds of nonpositive curvature, Math. Res. Lett. 24 (2017), no. 2, 549–70Google Scholar
[100] Sogge, C., D. Propagation of singularities and maximal functions in the plane, Invent. Math. 104 (1991), no. 2, 349–76Google Scholar
[101] Sogge, C., D. Concerning the Lp norm of spectral clusters for second-order elliptic operators on compact manifolds, J. Funct. Anal. 77 (1988), no. 1, 123–38Google Scholar
[102] Stein, E. M. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. With the assistance of Timothy S. Murphy, Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993Google Scholar
[103] Stein, E. M. Oscillatory integrals in Fourier analysis. Beijing lectures in harmonic analysis (Beijing, 1984), 307–355, Ann. of Math. Stud., 112, Princeton University Press, Princeton, NJ, 1986Google Scholar
[104] Stone, A., and Tukey, J., Generalized sandwich theorems, Duke Math. J. 9, (1942), 356–59Google Scholar
[105] Strichartz, R. S. Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), no. 3, 705–14Google Scholar
[106] Székely, L., A. Crossing numbers and hard Erdös problems in discrete geometry, Combin. Probab. Comput. 6 (1997), no. 3, 353–8Google Scholar
[107] Tao, T. Lectures Notes 5, www.math.ucla.edu~tao/254b.1.99s/Google Scholar
[108] Tao, T. A sharp bilinear restrictions estimate for paraboloids, Geom. Funct. Anal. 13 (2003), no. 6, 1359–84Google Scholar
[109] Tao, T. Endpoint bilinear restriction theorems for the cone, and some sharp null form estimates, Math. Z. 238 (2001), no. 2, 215–68Google Scholar
[110] Tao, T. The Bochner–Riesz conjecture implies the restriction conjecture, Duke Math. J. 96 (1999), no. 2, 363–75Google Scholar
[111] Tao, T. The weak-type endpoint Bochner–Riesz conjecture and related topics, Indiana Univ. Math. J. 47 (1998), no. 3, 10971124Google Scholar
[112] Tao, T. and Vargas, A. A bilinear approach to cone multipliers I. Restriction estimates, Geom. Funct. Anal. 10 (2000), no. 1, 185215Google Scholar
[113] Tao, T. and Vu, V. Additive combinatorics, Cambridge Studies in Advanced Mathematics, 105. Cambridge University Press, Cambridge, 2006Google Scholar
[114] Tao, T., Vargas, A. and Vega, L. A bilinear approach to the restriction and Kakeya conjectures, J. Amer. Math. Soc. 11 (1998), no. 4, 9671000Google Scholar
[115] Tomas, P. A. A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975), 477–8Google Scholar
[116] Wang, H. A restriction estimate in R3 using brooms, available on arXivGoogle Scholar
[117] Wolff, T. Lectures on harmonic analysis. With a foreword by Charles Fefferman and preface by Izabella Łaba, Edited by Łaba and Shubin, Carol. University Lecture Series, 29, Amer. Math. Soc., Providence, RI, 2003Google Scholar
[118] Wolff, T. A sharp bilinear cone restriction estimate, Ann. of Math. (2) 153 (2001), no. 3, 661–98Google Scholar
[119] Wolff, T. Local smoothing type estimates on Lp for large p, Geom. Funct. Anal. 10 (2000), no. 5, 1237–88Google Scholar
[120] Wolff, T. A mixed norm estimate for the X-ray transform, Rev.Mat.Iberoamericana 14 (1998), no. 3, 561600Google Scholar
[121] Wolff, T. An improved bound for Kakeya type maximal functions, Rev. Mat. Iberoamericana 11 (1995), no. 3, 651–74Google Scholar
[122] Wongkew, R. Volumes of tubular neighbourhoods of real algebraic varieties, Pacific J. Math. 159 (1993), no. 1, 177–84Google Scholar
[123] Wooley, T. Nested efficient congruencing and relatives of Vinogradov’s mean value theorem, available on arXivGoogle Scholar
[124] Wooley, T. D., The cubic case of the main conjecture in Vinogradov’s mean value theorem, Adv. Math. 294 (2016), 532–61Google Scholar
[125] Wooley, T. D., Translation invariance, exponential sums, and Warings problem, Proceedings of the International Congress of Mathematicians–Seoul 2014, Vol. II, 505–529, Kyung Moon Sa, Seoul, 2014Google Scholar
[126] Zahl, J. A discretized Severi-type theorem with applications to harmonic analy- sis, available on arXivGoogle Scholar
[127] Zhang, R. The endpoint perturbed Brascamp–Lieb inequality with examples, available on arXivGoogle Scholar
[128] Zygmund, A. On Fourier coefficients and transforms of functions of two variables, Studia Math. 50 (1974), 189201Google Scholar

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  • References
  • Ciprian Demeter, Indiana University, Bloomington
  • Book: Fourier Restriction, Decoupling, and Applications
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