| |

| |

Foreword | |

| |

| |

Preface | |

| |

| |

| |

L<sup>p</sup> Spaces and Banach Spaces | |

| |

| |

| |

L<sup>p</sup> spaces | |

| |

| |

| |

The Hï¿½lder and Minkowski inequalities | |

| |

| |

| |

Completeness of L<sup>p</sup> | |

| |

| |

| |

Further remarks | |

| |

| |

| |

The case p = ∞ | |

| |

| |

| |

Banach spaces | |

| |

| |

| |

Examples | |

| |

| |

| |

Linear functionals and the dual of a Banach space | |

| |

| |

| |

The dual space of L<sup>p</sup> when 1 ≤ p < ∞ | |

| |

| |

| |

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 | |

| |

| |

| |

Calderï¿½n-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 = ∞ | |

| |

| |

| |

Behavior of SN as N → ∞, 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 Schrï¿½dinger equation | |

| |

| |

| |

Another dispersion equation | |

| |

| |

| |

The non-homogeneous Schrï¿½dinger 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 | |