## Real-Variable Methods in Harmonic Analysis"A very good choice." — MathSciNet, American Mathematical SocietyAn exploration of the unity of several areas in harmonic analysis, this self-contained text emphasizes real-variable methods. Appropriate for advanced undergraduate and graduate students, it starts with classical Fourier series and discusses summability, norm convergence, and conjugate function. An examination of the Hardy-Littlewood maximal function and the Calderón-Zygmund decomposition is followed by explorations of the Hilbert transform and properties of harmonic functions. Additional topics include the Littlewood-Paley theory, good lambda inequalities, atomic decomposition of Hardy spaces, Carleson measures, Cauchy integrals on Lipschitz curves, and boundary value problems. 1986 edition. |

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### Contents

1 | |

9 | |

13 | |

17 | |

22 | |

28 | |

29 | |

34 | |

Subharmonic Functions 1 | 182 |

Harnacks and Mean Value Inequalities | 187 |

Notes Further Results and Problems | 191 |

Oscillation of Functions QQ 1 Mean Oscillation of Functions | 199 |

The Maximal Operator and BMO | 204 |

The Conjugate of Bounded and BMO Functions | 206 |

WkL and A Interpolation | 209 |

Lipschitz and Morrey Spaces | 213 |

A E Convergence of C 1 Means of Summable Functions | 41 |

Notes Further Results and Problems | 43 |

Norm Convergence of Fourier Series 1 The Case LT Hilbert Space | 48 |

Norm Convergence in LрT I p t | 51 |

The Conjugate Mapping | 52 |

More on Integrable Functions | 54 |

Integral Representation of the Conjugate Operator | 59 |

The Truncated Hilbert Transform | 65 |

The Basic Principles 1 The CalderónZygmund Interval Decomposition | 74 |

The HardyLittlewood Maximal Function | 76 |

The CalderónZygmund Decomposition | 84 |

The Marcinkiewicz Interpolation Theorem | 87 |

Extrapolation and the Zygmund L In L Class 6 The Banach Continuity Principle and a e Convergence | 97 |

Notes Further Results and Problems | 109 |

The Hubert Transform and Multipliers 1 Existence of the Hilbert Transform of Integrable Functions | 110 |

The Hübet i Transform in LT 1 p 3 Limiting Results | 121 |

Multipliers 5 Notes Further Results and Problems 13 | 132 |

Paleys Theorem and Fractional Integration 1 Paleys Theorem | 142 |

Fractional Integration | 144 |

Multipliers | 157 |

Notes Further Results and Problems | 159 |

Harmonic and Subharmonic Functions 1 Abel Summability Nontangential Convergence | 167 |

The Poisson and Conjugate Poisson Kernels | 173 |

Harmonic Functions | 176 |

Further Properties of Harmonic Functions | 181 |

Notes Further Results and Problems | 216 |

Measures | 223 |

Ap Weights p 1 | 233 |

A and BMO | 240 |

Notes Further Results and Problems | 247 |

More about | 259 |

The Hübet i and Riesz Transforms | 266 |

CalderónZygmund Singular | 280 |

CalderónZygmund Singular Integral Operators | 286 |

Singular Integral Operators in IR | 294 |

The LittlewoodPaley Theory | 303 |

The LittlewoodPaley g Function | 309 |

Hormanders Multiplier Theorem | 319 |

The Good X Principle | 328 |

Weighted Norm Inequalities for Maximal CZ Singular | 330 |

Maximal Function Characterization of Hardy Spaces | 350 |

Interpolation | 363 |

Tent Spaces | 381 |

Related Operators | 408 |

Notes Further Results and Problems | 416 |

Boundary Value Problems on CDomains | 424 |

The Dirichlet and Neumann Problems | 438 |

457 | |

### Common terms and phrases

Assume atoms Banach Banach space Calderón Calderón-Zygmund decomposition Carleson measure Cesàro means Chapter conclusion follows condition conjugate consequently consider continuous function converges convolution Corollary corresponding defined denote disjoint estimate exists finite follows at once Fourier series Furthermore given Hardy spaces harmonic function Hilbert transform Hint Holder's inequality holds implies independent integrable function interval invoke Lebesgue Lebesgue measure Lemma locally integrable function mapping Marcinkiewicz interpolation theorem maximal function maximal operator Mf(x Moreover multiplier norm notation numbers observe obtain open cube Poisson kernel precisely Proof properties Proposition prove readily follows readily seen Riesz right-hand side sequence singular integral operators space subharmonic subintervals sublinear operator suffices to show summable Suppose Tf(x Theorem 1.1 trigonometric polynomial uniformly verifies weak-type Whence тг

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