Functional Analysis Introduction to Further Topics in Analysis

ISBN-10: 0691113874
ISBN-13: 9780691113876
Edition: 2011
List price: $74.95 Buy it from $68.98
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Book details

List price: $74.95
Copyright year: 2011
Publisher: Princeton University Press
Publication date: 9/11/2011
Binding: Hardcover
Pages: 448
Size: 6.50" wide x 9.50" long x 1.25" tall
Weight: 1.892
Language: English

Foreword
Preface
L<sup>p</sup> Spaces and Banach Spaces
L<sup>p</sup> spaces
The H�lder and Minkowski inequalities
Completeness of L<sup>p</sup>
Further remarks
The case p = &#8734;
Banach spaces
Examples
Linear functionals and the dual of a Banach space
The dual space of L<sup>p</sup> when 1 &#8804; p < &#8734;
More about linear functionals
Separation of convex sets
The Hahn-Banach Theorem
Some consequences
The problem of measure
Complex L<sup>p</sup> and Banach spaces
Appendix: The dual of C(X)
The case of positive linear functionals
The main result
An extension
Exercises
Problems
L<sup>p</sup> Spaces in Harmonic Analysis
Early Motivations
The Riesz interpolation theorem
Some examples
The L<sup>p</sup> theory of the Hilbert transform
The L<sup>2</sup> formalism
The L<sup>p</sup> theorem
Proof of Theorem 3.2
The maximal function and weak-type estimates
The L<sup>p</sup> inequality
The Hardy space H<sub>r</sub><sup>1</sup>
Atomic decomposition of H<sub>r</sub><sup>1</sup>
An alternative definition of H<sub>r</sub><sup>1</sup>
Application to the Hilbert transform
The space H<sub>r</sub><sup>1</sup> and maximal functions
The space BMO
Exercises
Problems
Distributions: Generalized Functions
Elementary properties
Definitions
Operations on distributions
Supports of distributions
Tempered distributions
Fourier transform
Distributions with point supports
Important examples of distributions
The Hilbert transform and pv(1/x)
Homogeneous distributions
Fundamental solutions
Fundamental solution to general partial differential equations with constant coefficients
Parametrices and regularity for elliptic equations
Calder�n-Zygmund distributions and L<sup>p</sup> estimates
Defining properties
The L<sup>p</sup> theory
Exercises
Problems
Applications of the Baire Category Theorem
The Baire category theorem
Continuity of the limit of a sequence of continuous functions
Continuous functions that are nowhere differentiable
The uniform boundedness principle
Divergence of Fourier series
The open mapping theorem
Decay of Fourier coefficients of L<sup>1</sup>-functions
The closed graph theorem
Grothendieck's theorem on closed subspaces of L<sup>p</sup>
Besicovitch sets
Exercises
Problems
Rudiments of Probability Theory
Bernoulli trials
Coin flips
The case N = &#8734;
Behavior of SN as N &#8594; &#8734;, first results
Central limit theorem
Statement and proof of the theorem
Random series
Random Fourier series
Bernoulli trials
Sums of independent random variables
Law of large numbers and ergodic theorem
The role of martingales
The zero-one law
The central limit theorem
Random variables with values in Rd
Random walks
Exercises
Problems
An Introduction to Brownian Motion
The Framework
Technical Preliminaries
Construction of Brownian motion
Some further properties of Brownian motion
Stopping times and the strong Markov property
Stopping times and the Blumenthal zero-one law
The strong Markov property
Other forms of the strong Markov Property
Solution of the Dirichlet problem
Exercises
Problems
A Glimpse into Several Complex Variables
Elementary properties
Hartogs' phenomenon: an example
Hartogs' theorem: the inhomogeneous Cauchy-Riemann equations
A boundary version: the tangential Cauchy-Riemann equations
The Levi form
A maximum principle
Approximation and extension theorems
Appendix: The upper half-space
Hardy space
Cauchy integral
Non-solvability
Exercises
Problems
Oscillatory Integrals in Fourier Analysis
An illustration
Oscillatory integrals
Fourier transform of surface-carried measures
Return to the averaging operator
Restriction theorems
Radial functions
The problem
The theorem
Application to some dispersion equations
The Schr�dinger equation
Another dispersion equation
The non-homogeneous Schr�dinger equation
A critical non-linear dispersion equation
A look back at the Radon transform
A variant of the Radon transform
Rotational curvature
Oscillatory integrals
Dyadic decomposition
Almost-orthogonal sums
Proof of Theorem 7.1
Counting lattice points
Averages of arithmetic functions
Poisson summation formula
Hyperbolic measure
Fourier transforms
A summation formula
Exercises
Problems
Notes and References
Bibliography
Symbol Glossary
Index

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