Rodrigo Bañuelos

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


Preface