Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra (AM-194)

Nonfiction, Science & Nature, Mathematics, Topology, Geometry
Cover of the book Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra (AM-194) by Detlef Müller, Isroil A. Ikromov, Princeton University Press
View on Amazon View on AbeBooks View on Kobo View on B.Depository View on eBay View on Walmart
Author: Detlef Müller, Isroil A. Ikromov ISBN: 9781400881246
Publisher: Princeton University Press Publication: May 24, 2016
Imprint: Princeton University Press Language: English
Author: Detlef Müller, Isroil A. Ikromov
ISBN: 9781400881246
Publisher: Princeton University Press
Publication: May 24, 2016
Imprint: Princeton University Press
Language: English

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.

View on Amazon View on AbeBooks View on Kobo View on B.Depository View on eBay View on Walmart

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.

More books from Princeton University Press

Cover of the book Fly Me to the Moon by Detlef Müller, Isroil A. Ikromov
Cover of the book I Hear My People Singing by Detlef Müller, Isroil A. Ikromov
Cover of the book Saints and Their Miracles in Late Antique Gaul by Detlef Müller, Isroil A. Ikromov
Cover of the book The Rebbe by Detlef Müller, Isroil A. Ikromov
Cover of the book Imperfect Garden by Detlef Müller, Isroil A. Ikromov
Cover of the book The Rise of Russia and the Fall of the Soviet Empire by Detlef Müller, Isroil A. Ikromov
Cover of the book Raptors of Mexico and Central America by Detlef Müller, Isroil A. Ikromov
Cover of the book From Higher Aims to Hired Hands by Detlef Müller, Isroil A. Ikromov
Cover of the book Information Choice in Macroeconomics and Finance by Detlef Müller, Isroil A. Ikromov
Cover of the book Bound by Recognition by Detlef Müller, Isroil A. Ikromov
Cover of the book The Travels and Adventures of Serendipity by Detlef Müller, Isroil A. Ikromov
Cover of the book Positive Definite Matrices by Detlef Müller, Isroil A. Ikromov
Cover of the book Life Exposed by Detlef Müller, Isroil A. Ikromov
Cover of the book Radioactive Starlings by Detlef Müller, Isroil A. Ikromov
Cover of the book Gifted Tongues by Detlef Müller, Isroil A. Ikromov
We use our own "cookies" and third party cookies to improve services and to see statistical information. By using this website, you agree to our Privacy Policy