Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra (eBook)

Isroil A. Ikromov, Detlef Müller (Autoren)

eBook Download: PDF | EPUB

2016
272 Seiten
Princeton University Press (Verlag)
978-1-4008-8124-6 (ISBN)

Infos

Isroil A. Ikromov is professor of mathematics at Samarkand State University in Uzbekistan. Detlef Müller is professor of mathematics at the University of Kiel in Germany.

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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 Muller 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 Muller 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.

Produkt-Details

• Erscheint lt. Verlag: 24.5.2016
• Reihe/Serie: Annals of Mathematics Studies
• Reihe/Serie: Annals of Mathematics Studies
• Zusatzinfo: 7 line illus.
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre
• Schlagworte: Addition • Airy cone • Airy-type analysis • Airy-type decompositions • algorithm • analytic function • asymptotic analysis • asymptotic expansion • auxiliary results • Big O notation • Borel measure • Bounded set (topological vector space) • Change of variables • coefficient • combination • Compact space • complex interpolation • Convergent series • convex hull • convolution • coordinate system • corollary • critical exponent • Critical point (mathematics) • Degeneracy (mathematics) • Derivative • Dimension • Direct proof • Disjoint sets • Dispersive partial differential equation • Division by zero • dyadic decomposition • dyadic decompositions • dyadic domain decompositions • endpoint estimates • endpoint result • Error Term • estimation • Euclidean space • existential quantification • Factorization • Family of curves • fibration • Flat function • Formal power series • Fourier • Fourier decay • Fourier integral • Fourier inversion theorem • Fourier restriction • Fourier restriction estimate • Fourier restriction problem • Fourier restriction theorem • Fourier transform • frequency domain • Frequency Domain Decomposition • Fubini's Theorem • Greenleaf's restriction • Harmonic Analysis • Hessian Matrix • hypersurface • implicit function theorem • improved estimates • Inequality (mathematics) • Infimum and supremum • Integer • Integration by parts • interpolation arguments • interpolation theorem • invariant description • Iterative Method • Lebesgue Spaces • linear coordinates • linearly adapted coordinates • Line (geometry) • Line–line intersection • Line segment • LittlewoodАaley decomposition • LittlewoodАaley theory • maximal function • Minkowski inequality • Monotonic Function • multiple Integral • Natural number • newton polyhedra • Newton polyhedral • Newton polyhedron • normalized measures • normalized rescale measures • one-dimensional oscillatory integrals • open cases • Operator norm • operator norms • Oscillatory integral • Oscillatory integral operator • Parameter • partial derivative • partition of unity • Phase Functions • polyhedron • polynomial • Power of two • preparatory results • Principal curvature • Principal part • principal root jet • propositions • Puiseux series • Q.E.D. • Quantity • Rational number • real-analytic hypersurface • real interpolation • real number • rectangle • refined Airy-type analysis • Resolution of Singularities • Restriction • restriction estimates • r-height • Scale Parameter • scientific notation • Series expansion • Sign (mathematics) • smooth hypersurface • smooth hypersurfaces • Smoothness • Sobolev inequality • Special case • spectral localization • SteinДomas-type Fourier restriction • stopping-time algorithm • Strichartz estimate • sublevel type • Subset • Summation • Support (mathematics) • Taylor series • tensor product • Theorem • Thin Sets • three dimensions • total variation • transition domains • Twin prime • uniform bounds • Unit circle • Unit square • Upper and lower bounds • van der Corput-type estimates • Variable (mathematics) • Without loss of generality • Zero of a function • Zero set
• ISBN-10: 1-4008-8124-2 / 1400881242
• ISBN-13: 978-1-4008-8124-6 / 9781400881246