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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 | |