Fourier restriction for hypersurfaces in three dimensions and Newton polyhedra (Q2813039)

Let \(S\) be a smooth hypersurface in \(\mathbb R^3\) of finite type. The central problem in this book is the determination of the range of exponents \(p\) for which a Fourier restriction estimate NEWLINE\[NEWLINE\left(\int_S |\widehat{f}|\,d\mu\right)^{1/2}\leq C\| f\|_{L^p(\mathbb R^3)}NEWLINE\]NEWLINE with \(f\in\mathcal S(\mathbb R^3)\) holds true. Though this is a special case of the more general Fourier restriction problem, it is rich and important enough. By localizing the estimate, \(S\) is represented as \((x_1,x_2,\phi(x_1,x_2))\). Analogously to Varchenko's notion of height, the authors introduce the notion of \textit{linear height} of \(\phi\), written as \(h_{\mathrm{lin}}(\phi)\). In fact, Varchenko's ideas are crucial in the approach the authors undertake. One of the key ingredients is the existence of adapted coordinate systems, which produce new coordinates related to the Newton polyhedron. The mentioned range of \(p\) is described in terms of these Newton polyhedra associated to the given surface. The other important machinery consists in different interpolation techniques. The two papers [the authors, Trans. Am. Math. Soc. 363, No. 6, 2821--2848 (2011; Zbl 1222.35219)] and [\textit{I. A. Ikromov} et al., Acta Math. 204, No. 2, 151--271 (2010; Zbl 1223.42011)] are the main sources for the results, techniques and references in this book.NEWLINENEWLINENEWLINEThere are nine chapters in the book. The titles of six of them, from Chapter 3 to Chapter 8, give a picture of how and along which lines the study goes through. They areNEWLINENEWLINEChapter 3. Reduction to Restriction Estimates near the Principal Root Jet;NEWLINENEWLINEChapter 4. Restriction for Surfaces with Linear Height below 2;NEWLINENEWLINEChapter 5. Improved Estimates by Means of Airy-Type AnalysisNEWLINENEWLINEChapter 6. The Case When \(h_{\mathrm{lin}}(\phi)\geq2\). Preparatory ResultsNEWLINENEWLINEChapter 7. How to Go beyond the Case \(h_{\mathrm{lin}}(\phi)\geq5\)NEWLINENEWLINEChapter 8. The Remaining CasesNEWLINENEWLINEThe restriction problem is an active area in harmonic analysis, and the book under review is an essential and important contribution to it. For the first time, a major part of the restriction problem has appeared as a book.