Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals

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Princeton University Press, 1993 - Mathematics - 695 pages

This book contains an exposition of some of the main developments of the last twenty years in the following areas of harmonic analysis: singular integral and pseudo-differential operators, the theory of Hardy spaces, L\sup\ estimates involving oscillatory integrals and Fourier integral operators, relations of curvature to maximal inequalities, and connections with analysis on the Heisenberg group.

 

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Contents

PROLOGUE
3
Examples
9
Generalization of the CalderonZygmund decomposition
16
Examples of the general theory
23
Truncation of singular integrals
30
Further results
37
MORE ABOUT MAXIMAL FUNCTIONS
49
HARDY SPACES
87
Averages over a fcdimensional submanifold of finite type
476
Averages on variable hypersurfaces
493
Further results
511
INTRODUCTION TO THE HEISENBERG GROUP
527
Geometry of the complex ball and the Heisenberg group
528
The CauchySzego integral
532
Formalism of quantum mechanics and the Heisenberg group
547
Weyl correspondence and pseudodifferential operators
553

Hl AND BMO
139
PseudoDifferential and Singular Integral
228
An L2 theorem
234
Singular integral realization of pseudodifferential operators
241
Estimates in Lp Sobolev and Lipschitz spaces
250
Compound symbols
258
PseudoDifferential and Singular Integral
269
Almost orthogonality
278
L2 theory of operators with CalderonZygmund kernels
289
The Cauchy integral
310
Further results
317
Oscillatory Integrals of the First Kind
329
Oscillatory Integrals of the Second Kind
375
Some Examples
433
MAXIMAL AVERAGES AND OSCILLATORY INTEGRALS
467
Maximal averages and square functions
469
Twisted convolution and singular integrals on Hn
557
Representations of the Heisenberg group
568
Further results
574
MORE ABOUT THE HEISENBERG GROUP
587
The CauchyRiemann complex and its boundary analogue
589
The operators Bb and Db on the Heisenberg group
594
Applications of the fundamental solution
605
The Lewy operator
611
Homogeneous groups
618
The dNeumann problem
627
Further results
632
BIBLIOGRAPHY
645
AUTHOR INDEX
679
SUBJECT INDEX
685
Copyright

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About the author (1993)

Elias M. Stein is Professor of Mathematics at Princeton University.

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