Pointwise convergence for the Schr\"odinger equation [After Xiumin Du and Ruixiang Zhang]

This expository essay accompanied the author's presentation at the S\'eminaire Bourbaki on 01 April 2023. It describes the breakthrough work of Du--Zhang on the Carleson problem for the Schr\"odinger equation, together with background material in multilinear harmonic analysis.

[1]  L. Pierce On Bourgain’s Counterexample for the Schrödinger Maximal Function , 2019, The Quarterly Journal of Mathematics.

[2]  Hong Wang,et al.  On Falconer’s distance set problem in the plane , 2019, Inventiones mathematicae.

[3]  Hong Wang,et al.  Lower bounds for estimates of the Schrödinger maximal function , 2019, 1902.01430.

[4]  L. Guth,et al.  On Falconer’s distance set problem in the plane , 2018, Inventiones mathematicae.

[5]  Ruixiang Zhang,et al.  Sharp L2 estimates of the Schrödinger maximal function in higher dimensions , 2018, Annals of Mathematics.

[6]  L. Guth,et al.  POINTWISE CONVERGENCE OF SCHRÖDINGER SOLUTIONS AND MULTILINEAR REFINED STRICHARTZ ESTIMATES , 2018, Forum of Mathematics, Sigma.

[7]  K. Rogers,et al.  Coherence on Fractals Versus Pointwise Convergence for the Schrödinger Equation , 2017 .

[8]  J. Bourgain A note on the Schrödinger maximal function , 2016, 1609.05744.

[9]  L. Guth Restriction estimates using polynomial partitioning II , 2016, 1603.04250.

[10]  L. Guth A short proof of the multilinear Kakeya inequality , 2014, Mathematical Proceedings of the Cambridge Philosophical Society.

[11]  Jonathan Bennett,et al.  Aspects of Multilinear Harmonic Analysis Related to Transversality , 2014, 1405.5369.

[12]  J. Bourgain,et al.  The proof of the $l^2$ Decoupling Conjecture , 2014, 1403.5335.

[13]  J. Bourgain Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces , 2011, 1107.1129.

[14]  J. Bourgain,et al.  Bounds on Oscillatory Integral Operators Based on Multilinear Estimates , 2010, 1012.3760.

[15]  Terence Tao,et al.  On the multilinear restriction and Kakeya conjectures , 2005, math/0509262.

[16]  T. Tao,et al.  A bilinear approach to the restriction and Kakeya conjectures , 1998, math/9807163.

[17]  G. Mockenhaupt A restriction theorem for the Fourier transform , 1991 .

[18]  Elias M. Stein,et al.  OSCILLATORY INTEGRALS IN FOURIER ANALYSIS , 1987 .

[19]  Robert S. Strichartz,et al.  Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations , 1977 .

[20]  Charles Fefferman,et al.  Inequalities for strongly singular convolution operators , 1970 .

[21]  L. Guth,et al.  On the Erdős distinct distances problem in the plane , 2015 .

[22]  Sanghyuk Lee,et al.  On pointwise convergence of the solutions to Schrödinger equations in ℛ2 , 2006 .

[23]  T. Wol A sharp bilinear cone restriction estimate , 2001 .

[24]  Luis Vega,et al.  Oscillatory integrals and regularity of dispersive equations , 1991 .

[25]  C. Kenig,et al.  A note on the almost everywhere behavior of solutions to the Schrödinger equation , 1982 .

[26]  Lennart Carleson,et al.  Some analytic problems related to statistical mechanics , 1980 .