Note: Supplemental materials are not guaranteed with Rental or Used book purchases.
Purchase Benefits
What is included with this book?
Foreword | p. vii |
Preface | p. xvii |
L^{p} Spaces and Banach Spaces | p. 1 |
L^{p} spaces | p. 2 |
The Hölder and Minkowski inequalities | p. 3 |
Completeness of L^{p} | p. 5 |
Further remarks | p. 7 |
The case p = ∞ | p. 7 |
Banach spaces | p. 9 |
Examples | p. 9 |
Linear functionals and the dual of a Banach space | p. 11 |
The dual space of L^{p} when 1 ≤ p < ∞ | p. 13 |
More about linear functionals | p. 16 |
Separation of convex sets | p. 16 |
The Hahn-Banach Theorem | p. 20 |
Some consequences | p. 21 |
The problem of measure | p. 23 |
Complex L^{p} and Banach spaces | p. 27 |
Appendix: The dual of C(X) | p. 28 |
The case of positive linear functionals | p. 29 |
The main result | p. 32 |
An extension | p. 33 |
Exercises | p. 34 |
Problems | p. 43 |
L^{p} Spaces in Harmonic Analysis | p. 47 |
Early Motivations | p. 48 |
The Riesz interpolation theorem | p. 52 |
Some examples | p. 57 |
The L^{p} theory of the Hilbert transform | p. 61 |
The L^{2} formalism | p. 61 |
The L^{p} theorem | p. 64 |
Proof of Theorem 3.2 | p. 66 |
The maximal function and weak-type estimates | p. 70 |
The L^{p} inequality | p. 71 |
The Hardy space H_{r}^{1} | p. 73 |
Atomic decomposition of H_{r}^{1} | p. 74 |
An alternative definition of H_{r}^{1} | p. 81 |
Application to the Hilbert transform | p. 82 |
The space H_{r}^{1} and maximal functions | p. 84 |
The space BMO | p. 86 |
Exercises | p. 90 |
Problems | p. 94 |
Distributions: Generalized Functions | p. 98 |
Elementary properties | p. 99 |
Definitions | p. 100 |
Operations on distributions | p. 102 |
Supports of distributions | p. 104 |
Tempered distributions | p. 105 |
Fourier transform | p. 107 |
Distributions with point supports | p. 110 |
Important examples of distributions | p. 111 |
The Hilbert transform and pv(1/x) | p. 111 |
Homogeneous distributions | p. 115 |
Fundamental solutions | p. 125 |
Fundamental solution to general partial differential equations with constant coefficients | p. 129 |
Parametrices and regularity for elliptic equations | p. 131 |
Calderón-Zygmund distributions and L^{p} estimates | p. 134 |
Defining properties | p. 134 |
The L^{p} theory | p. 138 |
Exercises | p. 145 |
Problems | p. 153 |
Applications of the Baire Category Theorem | p. 157 |
The Baire category theorem | p. 158 |
Continuity of the limit of a sequence of continuous functions | p. 160 |
Continuous functions that are nowhere differentiable | p. 163 |
The uniform boundedness principle | p. 166 |
Divergence of Fourier series | p. 167 |
The open mapping theorem | p. 170 |
Decay of Fourier coefficients of L^{1}-functions | p. 173 |
The closed graph theorem | p. 174 |
Grothendieck's theorem on closed subspaces of L^{p} | p. 174 |
Besicovitch sets | p. 176 |
Exercises | p. 181 |
Problems | p. 185 |
Rudiments of Probability Theory | p. 188 |
Bernoulli trials | p. 189 |
Coin flips | p. 189 |
The case N = ∞ | p. 191 |
Behavior of SN as N → ∞, first results | p. 194 |
Central limit theorem | p. 195 |
Statement and proof of the theorem | p. 197 |
Random series | p. 199 |
Random Fourier series | p. 202 |
Bernoulli trials | p. 204 |
Sums of independent random variables | p. 205 |
Law of large numbers and ergodic theorem | p. 205 |
The role of martingales | p. 208 |
The zero-one law | p. 215 |
The central limit theorem | p. 215 |
Random variables with values in Rd | p. 220 |
Random walks | p. 222 |
Exercises | p. 227 |
Problems | p. 235 |
An Introduction to Brownian Motion | p. 238 |
The Framework | p. 239 |
Technical Preliminaries | p. 241 |
Construction of Brownian motion | p. 246 |
Some further properties of Brownian motion | p. 251 |
Stopping times and the strong Markov property | p. 253 |
Stopping times and the Blumenthal zero-one law | p. 254 |
The strong Markov property | p. 258 |
Other forms of the strong Markov Property | p. 260 |
Solution of the Dirichlet problem | p. 264 |
Exercises | p. 268 |
Problems | p. 273 |
A Glimpse into Several Complex Variables | p. 276 |
Elementary properties | p. 276 |
Hartogs' phenomenon: an example | p. 280 |
Hartogs' theorem: the inhomogeneous Cauchy-Riemann equations | p. 283 |
A boundary version: the tangential Cauchy-Riemann equations | p. 288 |
The Levi form | p. 293 |
A maximum principle | p. 296 |
Approximation and extension theorems | p. 299 |
Appendix: The upper half-space | p. 307 |
Hardy space | p. 308 |
Cauchy integral | p. 311 |
Non-solvability | p. 313 |
Exercises | p. 314 |
Problems | p. 319 |
Oscillatory Integrals in Fourier Analysis | p. 321 |
An illustration | p. 322 |
Oscillatory integrals | p. 325 |
Fourier transform of surface-carried measures | p. 332 |
Return to the averaging operator | p. 337 |
Restriction theorems | p. 343 |
Radial functions | p. 343 |
The problem | p. 345 |
The theorem | p. 345 |
Application to some dispersion equations | p. 348 |
The Schrödinger equation | p. 348 |
Another dispersion equation | p. 352 |
The non-homogeneous Schrödinger equation | p. 355 |
A critical non-linear dispersion equation | p. 359 |
A look back at the Radon transform | p. 363 |
A variant of the Radon transform | p. 363 |
Rotational curvature | p. 365 |
Oscillatory integrals | p. 367 |
Dyadic decomposition | p. 370 |
Almost-orthogonal sums | p. 373 |
Proof of Theorem 7.1 | p. 374 |
Counting lattice points | p. 376 |
Averages of arithmetic functions | p. 377 |
Poisson summation formula | p. 379 |
Hyperbolic measure | p. 384 |
Fourier transforms | p. 389 |
A summation formula | p. 392 |
Exercises | p. 398 |
Problems | p. 405 |
Notes and References | p. 409 |
Bibliography | p. 413 |
Symbol Glossary | p. 417 |
Index | p. 419 |
Table of Contents provided by Ingram. All Rights Reserved. |