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This is the first book to present a complete characterization of Stein-Tomas type Fourier restriction estimates for large classes of smooth hypersurfaces in three dimensions, including all real-analytic hypersurfaces. The range of Lebesgue spaces for which these estimates are valid is described in terms of Newton polyhedra associated to the given surface.Isroil Ikromov and Detlef Müller begin with Elias M. Stein's concept of Fourier restriction and some relations between the decay of the Fourier transform of the surface measure and Stein-Tomas type restriction estimates. Varchenko's ideas relating Fourier decay to associated Newton polyhedra are briefly explained, particularly the concept of adapted coordinates and the notion of height. It turns out that these classical tools essentially suffice already to treat the case where there exist linear adapted coordinates, and thus Ikromov and Müller concentrate on the remaining case. Here the notion of r-height is introduced, which proves to be the right new concept. They then describe decomposition techniques and related stopping time algorithms that allow to partition the given surface into various pieces, which can eventually be handled by means of oscillatory integral estimates. Different interpolation techniques are presented and used, from complex to more recent real methods by Bak and Seeger.Fourier restriction plays an important role in several fields, in particular in real and harmonic analysis, number theory, and PDEs. This book will interest graduate students and researchers working in such fields.

Hypersurfaces. --- Polyhedra. --- Surfaces, Algebraic. --- Fourier analysis. --- Analysis, Fourier --- Mathematical analysis --- Polyhedral figures --- Polyhedrons --- Geometry, Solid --- Shapes --- Algebraic surfaces --- Geometry, Algebraic --- Hyperspace --- Surfaces --- Airy cone. --- Airy-type analysis. --- Airy-type decompositions. --- Fourier decay. --- Fourier integral. --- Fourier restriction estimate. --- Fourier restriction problem. --- Fourier restriction theorem. --- Fourier restriction. --- Fourier transform. --- Greenleaf's restriction. --- Lebesgue spaces. --- LittlewoodАaley decomposition. --- LittlewoodАaley theory. --- Newton polyhedra. --- Newton polyhedral. --- Newton polyhedron. --- SteinДomas-type Fourier restriction. --- auxiliary results. --- complex interpolation. --- dyadic decomposition. --- dyadic decompositions. --- dyadic domain decompositions. --- endpoint estimates. --- endpoint result. --- improved estimates. --- interpolation arguments. --- interpolation theorem. --- invariant description. --- linear coordinates. --- linearly adapted coordinates. --- normalized measures. --- normalized rescale measures. --- one-dimensional oscillatory integrals. --- open cases. --- operator norms. --- phase functions. --- preparatory results. --- principal root jet. --- propositions. --- r-height. --- real interpolation. --- real-analytic hypersurface. --- refined Airy-type analysis. --- restriction estimates. --- restriction. --- smooth hypersurface. --- smooth hypersurfaces. --- spectral localization. --- stopping-time algorithm. --- sublevel type. --- thin sets. --- three dimensions. --- transition domains. --- uniform bounds. --- van der Corput-type estimates.

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Harmonic analysis. --- Analyse harmonique