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Laguerre Calculus and Its Applications on the Heisenberg Group
Carlos Berenstein, University of Maryland, College Park, MD, Der-Chen Chang, Georgetown University, Washington, DC, and Jingzhi Tie, University of Georgia, Athens, GA
A co-publication of the AMS and International Press.
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AMS/IP Studies in Advanced Mathematics
2001; 319 pp; hardcover
Volume: 22
ISBN-10: 0-8218-2761-8
ISBN-13: 978-0-8218-2761-1
List Price: US$74
Member Price: US$59.20
Order Code: AMSIP/22
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For nearly two centuries, the relation between analytic functions of one complex variable, their boundary values, harmonic functions, and the theory of Fourier series has been one of the central topics of study in mathematics. The topic stands on its own, yet also provides very useful mathematical applications.

This text provides a self-contained introduction to the corresponding questions in several complex variables: namely, analysis on the Heisenberg group and the study of the solutions of the boundary Cauchy-Riemann equations. In studying this material, readers are exposed to analysis in non-commutative compact and Lie groups, specifically the rotation group and the Heisenberg groups--both fundamental in the theory of group representations and physics.

Introduced in a concrete setting are the main ideas of the Calderón-Zygmund-Stein school of harmonic analysis. Also considered in the book are some less conventional problems of harmonic and complex analysis, in particular, the Morera and Pompeiu problems for the Heisenberg group, which relates to questions in optics, tomography, and engineering.

The book was borne of graduate courses and seminars held at the University of Maryland (College Park), the University of Toronto (ON), Georgetown University (Washington, DC), and the University of Georgia (Athens). Readers should have an advanced undergraduate understanding of Fourier analysis and complex analysis in one variable.

Titles in this series are co-published with International Press, Cambridge, MA.

Readership

Graduate students and physicists interested in mathematics and the physical sciences; advanced undergraduates and graduate students with knowledge of Fourier analysis and complex analysis in one variable.

Table of Contents

  • The Laguerre calculus
  • Estimates for powers of the sub-Laplacian
  • Estimates for the spectrum projection operators of the sub-Laplacian
  • The inverse of the operator \(\square_{\alpha} = {\sum}^n_{j=1}({X^2_j} - {X^2_{j+n}}) - 2i{\alpha}\)T
  • The explicit solution of the \(\bar{\partial}\)-Neumann problem in a non-isotropic Siegel domain
  • Injectivity of the Pompeiu transform in the isotropic H\(_n\)
  • Morera-type theorems for holomorphic \(\mathcal H^p\) spaces in H\(_n\) (I)
  • Morera-type theorems for holomorphic \(\mathcal H^p\) spaces in H\(_n\) (II)
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