Real-Variable Methods in Harmonic Analysis
by Alberto Torchinksy

Chapter 1: Fourier Series

• Fourier Series of Functions
• Fourier Series of Continuous Functions
• Elementary Properties of Fourier Series
• Fourier Series of Functionals
• Notes; Further Results and Problems

Chapter 2: Cesaro Summability

• (C,1) Summability
• Fejer's Kernel
• Characterization of Fourier Series of Functions and Measures
• A.E. Convergence of (C,1) Means of Summable Functions
• Notes; Further Results and Problems

Chapter 3: Norm Convergence of Fourier Series

• The Case L^2(T); Hilbert Space
• Norm Convergence in L^p(T), 1<=p<=oo
• The Conjugate Mapping
• More on Integrable Functions
• Integral Representation of the Conjugate Operator
• The Truncated Hilbert Transform
• Notes; Further Results and Problems

Chapter 4: The Basic Principles

• The Calderon-Zygmund Interval Decomposition
• The Hardy-Littlewood Maximal Function
• The Calderon-Zygmund Decomposition
• The Marcinkiewicz Interpolation Theorem
• Extrapolation and the Zygmund L ln L Class
• The Banach Continuity Principle and a.e. Convergence
• Notes; Further Results and Problems

Chapter 5: The Hilbert Transform and Multipliers

• Existence of the Hilbert Transform of Integrable Functions
• The Hilbert Transform in L^p(T), 1 <= p < oo
• Limiting Results
• Multipliers
• Notes; Further Results and Problems

Chapter 6: Paley's Theorem and Fractional Integration

• Paley's Theorem
• Fractional Integration
• Multipliers
• Notes; Further Results and Problems

Chapter 7: Harmonic and Subharmonic Functions

• Abel Summability, Nontangential Convergence
• The Poisson and Conjugate Poisson Kernels
• Harmonic Functions
• Further Properties of Harmonic Functions and Subharmonic Functions
• Harnack's and Mean Value Inequalities
• Notes; Further Results and Problems

Chapter 8: Oscillation of Functions

• Mean Oscillation of Functions
• The Maximal Operator and BMO
• The Conjugate of Bounded and BMO Functions
• Wk-L^p and K_f. Interpolation
• Lipschitz and Morrey Spaces
• Notes; Further Results and Problems

Chapter 9: A_p Weights

• The Hardy-Littlewood Maximal Theorem for Regular Measures
• A_p Weights and the Hardy-Littlewood Maximal Function
• A_1 Weights
• A_p Weights, p > 1
• Factorization of A_p Weights
• A_p and BMO
• An Extrapolation Result
• Notes; Further Results and Problems

Chapter 10: More about R^n

• Distributions, Fourier Transforms
• Translation Invariant Operators, Multipliers
• The Hilbert and Riesz Transforms
• Sobolev and Poincare Inequalities

Chapter 11: Calderon-Zygmund Singular Integral Operators

• The Benedek-Calderon-Panzone Principle
• A Theorem of Z�
• Convolution Operators
• Cotlar's Lemma
• Calderon-Zygmund Singular Integral Operators
• Maximal Calderon-Zygmund Singular Integral Operators
• Singular Integral Operators in L^oo(R^n)
• Notes; Further Results and Problems

Chapter 12: The Littlewood-Paley Theory

• Vector-Valued Inequalities
• Vector-Valued Singular Integral Operators
• The Littlewood-Paley g Function
• The Lusin Area Function and the Littlewood-Paley g*_lambda Function
• H�mander's Multiplier Theorem
• Notes; Further Results and Problems

Chapter 13: The Good Lambda Principle

• Good Lambda Inequalities
• Weighted Norm Inequalities for Maximal CZ Singular Integral Operators
• Weighted Weak-Type (1,1) Estimates for CZ Singular Integral Operators
• Notes; Further Results and Problems

Chapter 14: Hardy Spaces of Several Real Variables

• Atomic Decomposition
• Maximal Function Characterization of Hardy Spaces
• Systems of Conjugate Functions
• Multipliers
• Interpolation
• Notes; Further Results and Problems

Chapter 15: Carleson Measures

• Carleson Measures
• Duals of Hardy Spaces
• Tent Spaces
• Notes; Further Results and Problems

Chapter 16: Cauchy Integrals on Lipschitz Curves

• Cauchy Integrals on Lipschitz Curves
• Related Operators
• The T1 Theorem
• Notes; Further Results and Problems

Chapter 17: Boundary Value Problems on C^1-Domains

• The Double and Single Layer Potentials on a C^1-Domain
• The Dirichlet and Neumann Problems
• Notes

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See the book Real-Variable Methods in Harmonic Analysis by Alberto Torchinksy at Amazon.com or at Amazon.co.uk.

See also Real and Complex Analysis by Walter Rudin.