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