The main goal of these lectures is to introduce the Fourier restriction phenomenon in the most classical setting of S being a compact hypersurface with non-vanishing Gaussian curvature and discuss some of its connections to other topics in Analysis and PDE. In particular, we will focus on the classical but fundamental Stein–Tomas theorem [8] and the recent celebrated sharp decoupling inequalities of Bourgain and Demeter [3].
We will formulate the Fourier restriction conjecture for hypersurfaces of non-vanishing Gaussian curvature via a careful discussion of the Knapp example. We will show some first non-trivial estimates in the presence of curvature, and will discuss the proof of the theorem in the 2-dimensional case (the only dimension in which the conjecture is fully settled). We will review the Fourier decay for surface carried measures via stationary phase to later proceed with the proof of the Stein–Tomas theorem via a TT* argument. Connections with PDE, Bochner–Riesz summability and the Kakeya problem will be discussed too.
Standard references are [6, Chapter VIII] and the surveys [7, 1].
Decoupling (or Wolff- type) inequalities were introduced by Wolff [9] to deduce sharp local smoothing estimates for the solution of the wave equation. These consist in controlling in a sharp way the $L^p$ norm of a function (with certain specific Fourier support properties) through the $L^p$ norm of its fundamental frequency localised pieces. The breakthrough result of Bourgain and Demeter in 2015 [3] established sharp $\ell^2(L^p)$ versions for all $2\leq p<\infty$ and any dimension of the aforementioned decoupling inequalities for functions Fourier supported in a neighborhood of the paraboloid. In this course, we will discuss the slightly simpler case of the parabola in $\mathbb R^2$, which already contains the main ideas, and some of its applications if time permits.
The main references are [3] and the study guide [4], although it is remarked that the multilinear theory of [2] and the higher-dimensional Bourgain–Guth argument [5] will not be needed in the 2-dimensional case.
[1] J. Bennett. Aspects of multilinear harmonic analysis related to transversality. In Harmonic analysis and partial differential equations, volume 612 of Contemp. Math., pages 1–28. Amer. Math. Soc., Providence, RI, 2014.
[2] J. Bennett, A. Carbery, and T. Tao. On the multilinear restriction and Kakeya conjectures. Acta Math., 196(2):261–302, 2006.
[3] J. Bourgain and C. Demeter. The proof of the l2 decoupling conjecture. Ann. of Math. (2), 182(1):351–389, 2015.
[4] J. Bourgain and C. Demeter. A study guide for the l2 decoupling theorem. Chin. Ann. Math. Ser. B, 38(1):173–200, 2017.
[5] J. Bourgain and L. Guth. Bounds on oscillatory integral operators based on multilinear esti- mates. Geom. Funct. Anal., 21(6):1239–1295, 2011.
[6] E. M. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory inte- grals, volume 43 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III.
[7] T. Tao. Some recent progress on the restriction conjecture. In Fourier analysis and convexity, Appl. Numer. Harmon. Anal., pages 217–243. Birkhäuser Boston, Boston, MA, 2004.
[8] P. A. Tomas. A restriction theorem for the Fourier transform. Bull. Amer. Math. Soc., 81:477– 478, 1975.
[9] T. Wolff. Local smoothing type estimates on Lp for large p. Geom. Funct. Anal., 10(5):1237– 1288, 2000.
The following topics of study for the working groups are suggested and will be coordinated by David Beltran.
The Stein–Tomas Fourier restriction theorem in Euclidean space $\mathbb R^d$ has a generalisation in the context of Riemannian manifolds due to Sogge [3]. More precisely, if $(M,g)$ is a smooth compact Riemannian manifold, $\Delta g$ denotes its associated Laplace–Beltrami operator, $\{\lambda_j\}$ denote the eigenvalues of $\Delta_g$ and $E_j$ the corresponding eigenspaces, this consists in determining the sharp mapping properties of the projection operator associated to $E_j$ for functions in $L^2(M)$.
The result in the specific case of the $M=S^{d-1}$ consists in the projection of $ into the spherical harmonics and is also due to Sogge [2]. As a corollary, this yields a simple proof for a sharp unique continuation theorem for the Laplacian in $\mathbb R^d$. The goal of this study group is to investigate this discrete Stein–Tomas theorem and moreover, to review the work of Kenig, Ruiz and Sogge [1] in which the classical Euclidean Stein–Tomas theorem for the sphere is used to prove certain Sobolev estimates and unique continuation theorems for certain second order differential operators (in particular, the general result of Sogge [3] for Riemannian manifolds follows from a Sobolev estimate generalising the ones of Kenig–Ruiz–Sogge [1]).
[1] C. E. Kenig, A. Ruiz, and C. D. Sogge. Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators. Duke Math. J., 55(2):329–347, 1987.
[2] C. D. Sogge. Oscillatory integrals and spherical harmonics. Duke Math. J., 53(1):43–65, 1986.
[3] C. D. Sogge. Concerning the Lp norm of spectral clusters for second-order elliptic operators on compact manifolds. J. Funct. Anal., 77(1):123–138, 1988.
Polynomial partitioning has become a fundamental tool for the recent progress in Fourier restriction theory. It was first used in this setting by Guth [3, 2], and subsequently used in the latter related works. A prior evidence of the use of polynomials in contexts related to Fourier restriction was the work of Dvir [1] for the resolution of the Kakeya conjecture in finite fields or the work of Guth and Katz [4] on the Erdös distance problem.
The goal of this study group is that the students get familiar with the use of polynomial partitioning in Fourier restriction theory as in [3, 2] and to study the techniques used to bound the contribution coming from the cells and the wall after the partitioning.
[1] Z. Dvir. On the size of Kakeya sets in finite fields. J. Amer. Math. Soc., 22(4):1093–1097, 2009.
[2] L. Guth. Restriction estimates using polynomial partitioning II. Preprint available at arXiv:1603.04250.
[3] L. Guth. A restriction estimate using polynomial partitioning. J. Amer. Math. Soc., 29(2):371– 413, 2016.
[4] L. Guth and N. H. Katz. On the Erdös distinct distances problem in the plane. Ann. of Math. (2), 181(1):155–190, 2015.
In this course we will review some results concerning the bilinear restriction theory. We will begin with a proof of the restriction conjecture in two dimensions using the bilinear approach, and then continue with the proof of the sharp bilinear restriction estimate for the paraboloid [2]. In addition, we will consider some other surfaces such as the cone [5] and the hyperboloid [1], [4]. Finally, we will explain how to use bilinear estimates to prove linear ones in all dimensions [3].
[1] S. Lee, Bilinear restriction estimates for surfaces with curvatures of different signs, Trans. Amer. Math. Soc. 358 (8) (2006) 3511–3533
[2] T. Tao, A sharp bilinear restrictions estimate for paraboloids, Geom. Funct. Anal. 13 (2003), no. 6, 1359–1384.
[3] T. Tao, A. Vargas and L. Vega, A bilinear approach to the restriction and Kakeya conjectures, J. Amer. Math. Soc. 11 (1998), 967–1000.
[4] A. Vargas, Restriction theorems for a surface with negative curvature, Math. Z. 249 (1) (2005) 97–111.
[5] T. Wolff, A sharp bilinear cone restriction estimate, Ann. of Math. 153 (2001), 661–698.
The following topics of study for the working groups are suggested and will be coordinated by Javier Ramos.
The Fourier restriction conjecture [4] affirms that we can meaningfully restrict the Fourier transform to the sphere $S^d−1$ for $L^p(\mathbb R^d)$ functions with $p < 2d/(d+1)$. Although the conjecture has attracted the attention of some of the greatest mathematicians, it is still open in dimensions d ≥ 3. In recent years, the multilinear theory initiated by Bennett-Carbery-Tao [1] and the polynomial method [2] have drastically improved the state of the art of the theory and related problems. This topic will focus on the multilinear restriction estimate. In particular, we will overview the proof of the multilinear Kakeya estimate given by Guth [3], and the induction argument to prove that the multilinear Kakeya estimate implies the multilinear restriction estimate (section 2 of [1]).
[1] J. Bennett, A. Carbery, T. Tao, On the multilinear restriction and Kakeya conjectures, Acta Math. 196 (2006) 261-302.
[2] Z. Dvir, On the size of Kakeya sets in finite fields, J. Amer. Math. Soc. 22 (4): 1093–1097.
[3] L. Guth, A short proof of the multilinear Kakeya inequality, Math. Proc. Camb. Phil. Soc. 158 (2015), 147–153
[4] E. M. Stein, Some problems in harmonic analysis, Harmonic analysis in Euclidean spaces, Proc. Sympos. Pure Math. Part 1 (1978), 3–20.
For some years it was unclear how to use the multilinear restriction estimate proven by Bennett-Carbery-Tao [1], in order to prove new linear restriction estimates. Bourgain-Guth [4] were able to use the multilinear estimate to get partial results on the linear restriction conjecture (it was the best result towards the conjecture until 2014) thanks to an argument that exploits a dichotomy between transversality and good $L^4$-orthogonality. Furthermore, this argument is one of the main steps in the proof of some sharp decoupling estimates (see [2], [3] or [5]), which have revealed deep connections with problems in a variety of different fields of mathematics such as dispersive equations (by means of Strichartz type estimates), combinatorics (improved bounds for additive energies, incidence geometry) or number theory (counting solutions to Diophantine equations or progress on the Lindelöf hypothesis). We will focus on the argument for the tridimensional case (sections 2 and 4 of [4]).
[1] J. Bennett, A. Carbery, T. Tao, On the multilinear restriction and Kakeya conjectures, Acta Math. 196 (2006) 261-302.
[2] J. Bourgain Decoupling, exponential sums and the Riemann zeta function, to appear in J. Amer. Math. Soc.
[3] J. Bourgain, C. Demeter, The proof of the $\ell^2$-decoupling conjecture, Annals of Math. 182 (2015), no. 1, 351-389.
[4] J. Bourgain, L. Guth, Bounds on oscillatory integral operators based on multilinear estimates, Geometric And Functional Analysis, Volume 21, Number 6, 1239–1295.
[5] J. Bourgain, C. Demeter, L. Guth Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three, Annals of Math. 184 (2016) 633-682
The goal of the lecture is to review the basic methods used to study extremizers for Fourier restriction inequalities, that are the starting points of all current research on this topic. The course will be split in two kinds of approaches (that are not completely disjoint). Ideally, each approach will be devoted two lectures.
The first approach consists in trying to understand when extremizers exist, in particular when extremizing sequences converge in some sense. Here, the goal is to exactly determine the possible loss of compactness of such sequences, in the spirit of the concentration-compactness of Lions. The strength of this method is that it is very robust and can be applied to a large class of restriction inequalities, and to many other optimization problems. Its weakness is that it does not say anything about the exact form of optimizer, a natural question in the context of restriction inequalities.
This part of the course will be illustrated on the two most simple examples: restriction of the sphere (Stein-Tomas), and on the paraboloid (Strichartz). The reference that we will use for this part of the class is [4].
The second approach concerns particular cases of restriction inequalities (or rather, extension inequalities), for which the Lebesgue exponent is an even integer. In this case, Plancherel theorem allows to take away the oscillations in the inequality and state it in terms of a multilinear convolution inequalities. One can then prove sharp form of the inequality by tracking at each step the cases of equality. This strategy has been initiated by Foschi [2, 3] in the case of the paraboloid, the cone, and the sphere. These articles will be the basis for this part of the course.
The two topics proposed to the students will match the two parts of the lecture: one which is related to concentration-compactness, one which is related to convolution identities. The following topics of study for the working groups are suggested and will be coordinated by Julien Sabin.
The goal of this first topic is to understand the generality of the concentration-compactness method by applying it to the restriction problem on the cubic curve in $\mathbb R^2$, as done in [5]. Following this article, the students will see which harmonic analysis tools (bilinear estimates, smoothing estimates, Brezis-Lieb lemma) are useful in this approach and how to adapt them from the ones seen in the course.
This second topic follows the method of convolution identities to study the extremizer problem for the hyperboloid. Interestingly, it can lead to the proof of non-existence of extremizers for some exponents, following [6, 1]. If time permits, the students can also look at the description of extremizing sequences in this case.
[1] E. Carneiro, D. O. e Silva, and M. Sousa, Extremizers for fourier restriction on hyperboloids, in Ann. Inst. Henri Poincaré C, Anal. non lin., 2018.
[2] D. Foschi, Maximizers for the Strichartz inequality, J. Eur. Math. Soc. (JEMS), 9 (2007), pp. 739–774.
[3] D. Foschi, Global maximizers for the sphere adjoint Fourier restriction inequality, J. Funct. Anal., 268 (2015), pp. 690–702.
[4] R. L. Frank, E. H. Lieb, and J. Sabin, Maximizers for the Stein-Tomas inequality, Geom. Funct. Anal., 26 (2016), pp. 1095–1134.
[5] R. L. Frank and J. Sabin, Extremizers for the Airy–Strichartz inequality, Math. Ann., (2018), pp. 1–46.
[6] R. Quilodrán, Nonexistence of extremals for the adjoint restriction inequality on the hyperboloid, J. Anal. Math., 125 (2015), pp. 37–70.