Sharp restriction theory (Q6592000)

A remarkable observation by Stein from 1967 [\textit{E. M. Stein}, Ann. Math. Stud. 112, 307--355 (1986; Zbl 0618.42006)] is that the Fourier transform of an \(L^p(\mathbb R^n)\) function (for a certain range of \(p\)) has a meaningful restriction to the sphere (which has measure zero). Moreover, this leads to a bound of the form \N\[\N\| \widehat{f}|_{S^{n-1}}\|_2 \leq C_p \|f\|_p.\N\]\NResults of this kind are called restriction theorems. Since Stein's observation, this subject has expanded in various directions, bringing forth the role of curvature in harmonic analysis. Additionally, these results can also be applied to obtain sharp results in related fields, like the Bochner-Riesz conjecture, estimates for solutions to certain PDEs, decoupling phenomena, etc.\N\NThe present set of lecture notes is an excellent exposition of recent results on restriction theorems and related topics with a focus on maximizers and optimal constants for sharp forms of restrictions and Strichartz inequalities. The author starts with a sharpened version of Hausdorff-Young inequality and goes on to introduce the restriction theory and its connection to other fundamental problems in harmonic analysis. The author then moves on to the Agmon-Hörmander inequality and describes the author's work on the maximisers. It is emphasised that the maximisers are not necessarily constants, which is somewhat surprising. This is followed by a result on sharp mixed norm spherical restriction. Next, a sharp spherical restriction is taken up in detail. First, the method towards the unconditional existence of endpoint Stein-Tomas maximizers when additional symmetries (for example, symmetry under the action of \(O(d-k+1) \times O(k)\)) is present and explained. Next, a bootstrapping method to fully characterize the real-valued maximizers of \(L^2(\sigma) \rightarrow L^{2k}\) extension inequalities is described. Complex valued maximizers are also discussed. After this, Strichartz estimates are dealt with. Using the scale invariance of the Stein-Tomas argument, Strichartz proved [\textit{R. S. Strichartz}, Duke Math. J. 44, 705--714 (1977; Zbl 0372.35001)] that \N\[\N \|e^{it\Delta}f\|_{ L^2(2 + \frac{4}{d}) (\mathbb R^{1+d})} \leq S_d~\|f \|_{L^2(\mathbb R^d)}.\N\]\NThe fundamental conjectures of \textit{E. H. Lieb} [Invent. Math. 102, No. 1, 179--208 (1990; Zbl 0726.42005)] and \textit{D. Foschi} [J. Funct. Anal. 268, No. 3, 690--702 (2015; Zbl 1311.42019)] about maximizers for Strichartz estimates are open in full generality. The author describes the known results in this setting. The article ends with a similar discussion of the cone and hyperboloids. This is a well-written article by someone who has made non-trivial contributions to these topics, which is a must-read for anyone wanting to work in this active and exciting area.