LECTURE NOTES IN ANALYSIS

TABLE OF CONTENTS

• PREFACE

• CHAPTER I. DIFFERENTIATION

  • §1. Covering Lemmas

  • §2. Monotone Functions

  • §3. Functions of Bounded Variation

  • §4. Absolute Continuity

• CHAPTER II. SIGNED MEASURES AND APPLICATIONS

  • §1. Signed Measures

  • §2. The Radon-Nikodym Theorem

  • §3. The Riesz Representation Theorem for Lp

• CHAPTER III. PRODUCT MEASURES

  • §1. Product Measures

  • §2. Fubini's Theorem

• CHAPTER IV. CONVOLUTIONS AND APPROXIMATIONS TO THE IDENTITY

  • §1. Minkowski's Integral Inequality

  • §2. Convolution Operator

  • §3. Approximations to the Identity

• CHAPTER V. THE HARDY-LITTLEWOOD MAXIMAL FUNCTION

  • §1. Hardy-Littlewood Maximal Function

  • §2. The Calderón-Zygmund Decomposition

  • §3. Applications to BMO

  • §4. Interpolation Theorems

• CHAPTER VI. THE FOURIER TRANSFORM

  • §1. The Fourier transform on L1

  • §2. The Fourier transform on L2

  • §3. Applications

• CHAPTER VII. SINGULAR INTEGRALS

  • §1. Singular Integrals on L1

  • §2. Singular Integrals on Lp

  • §3. Singular Integrals and BMO

  • §4. Some Vector Valued Inequalities

• CHAPTER VIII. THE RIESZ TRANSFORMS

  • §1. Hilbert Transform

  • §2. Riesz Transforms

  • §3. The Cauchy-Riemann Equations

  • §4. Beurling-Ahlfors Transform

• CHAPTER IX. FRACTIONAL INTEGRATION

  • §1. Definitions and boundedness

  • §2. Inequalities of Sobolev and Nash

• CHAPTER X. LITTLEWOOD-PALEY AND LUSIN SQUARE FUNCTIONS

  • §1. Definitions, L2-properties, and pointwise comparisons

  • §2. Lp-properties

  • §3. The Hörmander multiplier theorem

• REFERENCES

• INDEX

• NOTATION