LECTURE NOTES IN ANALYSIS

• 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
• §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