Developments in Mathematics

Nonelliptic Partial Differential Equations

Analytic Hypoellipticity and the Courage to Localize High Powers of T

Authors: Tartakoff, David S.

  • The exposition is generous and relaxed, allowing the reader to come to terms to the technique at their own pace. The main difficulty, localization, is approached directly from the beginning with simple examples
  • Numerous applications are included
  • There is no similar book
  • There are other techniques for proving analytic hypoellipticity, but each has its own difficulties. While this is elementary but not simple, once the few basic formulas are established the rest is combinatorial in nature, and not conceptually difficult
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About this book

This book fills a real gap in the analytical literature. After many years and many results of analytic regularity for partial differential equations, the only access to the technique known as $(T^p)_\phi$ has remained embedded in the research papers themselves, making it difficult for a graduate student or a mature mathematician in another discipline to master the technique and use it to advantage. This monograph takes a particularly non-specialist approach, one might even say gentle, to smoothly bring the reader into the heart of the technique and its power, and ultimately to show many of the results it has been instrumental in proving. Another technique developed simultaneously by F. Treves is developed and compared and contrasted to ours.

 

The techniques developed here are tailored to proving real analytic regularity to solutions of sums of squares of vector fields with symplectic characteristic variety and others, real and complex. The motivation came from the field of several complex variables and the seminal work of J. J. Kohn. It has found application in non-degenerate (strictly pseudo-convex) and degenerate situations alike, linear and non-linear, partial and pseudo-differential equations, real and complex analysis. The technique is utterly elementary, involving powers of vector fields and carefully chosen localizing functions. No knowledge of advanced techniques, such as the FBI transform or the theory of hyperfunctions is required. In fact analyticity is proved using only $C^\infty$ techniques.

 

The book is intended for mathematicians from graduate students up, whether in analysis or not, who are curious which non-elliptic partial differential operators have the property that all solutions must be real analytic. Enough background is provided to prepare the reader with it for a clear understanding of the text, although this is not, and does not need to be, very extensive. In fact, it is very nearly true that if the reader is willing to accept the fact that pointwise bounds on the derivatives of a function are equivalent to bounds on the $L^2$ norms of its derivatives locally, the book should read easily.

Reviews

From the reviews:

“The present book deals with the analytic and Gevrey local hypoellipticity of certain nonelliptic partial differential operators. … this nice book is mostly addressed to Ph.D. students and researchers in harmonic analysis and partial differential equations, the reader being supposed to be familiar with the basic facts of pseudodifferential calculus and several complex variables. It represents the first presentation, in book form, of the challenging and still open problem of analytic and Gevrey hypoellipticity of sum-of-squares operators.” (Fabio Nicola, Mathematical Reviews, Issue 2012 h)


Table of contents (15 chapters)

Buy this book

eBook $109.00
price for USA (gross)
  • ISBN 978-1-4419-9813-2
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $139.00
price for USA
  • ISBN 978-1-4419-9812-5
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $139.00
price for USA
  • ISBN 978-1-4614-2969-2
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Nonelliptic Partial Differential Equations
Book Subtitle
Analytic Hypoellipticity and the Courage to Localize High Powers of T
Authors
Series Title
Developments in Mathematics
Series Volume
22
Copyright
2011
Publisher
Springer-Verlag New York
Copyright Holder
Springer Science+Business Media, LLC
eBook ISBN
978-1-4419-9813-2
DOI
10.1007/978-1-4419-9813-2
Hardcover ISBN
978-1-4419-9812-5
Softcover ISBN
978-1-4614-2969-2
Series ISSN
1389-2177
Edition Number
1
Number of Pages
VIII, 203
Topics