Real-Variable Methods in Harmonic Analysis

PrefaceChapter I Fourier Series 1. Fourier Series of Functions 2. Fourier Series of Continuous Functions 3. Elementary Properties of Fourier Series 4. Fourier Series of Functionals 5. Notes; Further Results and ProblemsChapter II Cesàro Summability 1. (C, 1) Summability 2. Fejér's Kernel 3. Characterization of Fourier Series of Functions and Measures 4. A.E. Convergence of (C, 1) Means of Summable Functions 5. Notes; Further Results and ProblemsChapter III Norm Convergence of Fourier Series 1. The Case L2(T); Hilbert Space 2. Norm Convergence in Lp(T), 1 ≤ p ≤ ∞ 3. The Conjugate Mapping 4. More on Integrable Functions 5. Integral Representation of the Conjugate Operator 6. The Truncated Hilbert Transform 7. Notes; Further Results and ProblemsChapter IV The Basic Principles 1. The Calderón-Zygmund Interval Decomposition 2. The Hardy-Littlewood Maximal Function 3. The Calderón-Zygmund Decomposition 4. The Marcinkiewicz Interpolation Theorem 5. Extrapolation and the Zygmund L in L Class 6. The Banach Continuity Principle and a.e. Convergence 7. Notes; Further Results and ProblemsChapter V The Hilbert Transform and Multipliers 1. Existence of the Hilbert Transform of Integrable Functions 2. The Hilbert Transform in Lp(T), 1 ≤ p ≤ ∞ 3. Limiting Results 4. Multipliers 5. Notes; Further Results and ProblemsChapter VI Paley's Theorem and Fractional Integration 1. Paley's Theorem 2. Fractional Integration 3. Multipliers 4. Notes; Further Results and ProblemsChapter VII Harmonic and Subharmonic Functions 1. Abel Summability, Nontangential Convergence 2. The Poisson and Conjugate Poisson Kernels 3. Harmonic Functions 4. Further Properties of Harmonic Functions and Subharmonic Functions 5. Harnack's and Mean Value Inequalities 6. Notes; Further Results and ProblemsChapter VIII Oscillation of Functions 1. Mean Oscillation of Functions 2. The Maximal Operator and BMO 3. The Conjugate of Bounded and BMO Functions 4. Wk-Lp and Kf. Interpolation 5. Lipschitz and Morrey Spaces 6. Notes; Further Results and ProblemsChapter IX Ap Weights 1. The Hardy-Littlewood Maximal Theorem for Regular Measures 2. Ap Weights and the Hardy-Littlewood Maximal Function 3. A1 Weights 4. Ap Weights, p > 1 5. Factorization of Ap Weights 6. Ap and BMO 7. An Extrapolation Result 8. Notes; Further Results and ProblemsChapter X More about Rn 1. Distributions. Fourier Transforms 2. Translation Invariant Operators. Multipliers 3. The Hilbert and Riesz Transforms 4. Sobolev and Poincare InequalitiesChapter XI Calderón-Zygmund Singular Integral Operators 1. The Benedek-Calderón-Panzone Principle 2. A Theorem of Zó 3. Convolution Operators 4. Cotlar's Lemma 5. Calderón-Zygmund Singular Integral Operators 6. Maximal Calderón-Zygmund Singular Integral Operators 7. Singular Integral Operators in L∞ (Rn) 8. Notes; Further Results and ProblemsChapter XII The Littlewood-Paley Theory 1. Vector-Valued Inequalities 2. Vector-Valued Singular Integral Operators 3. The Littlewood-Paley g Function 4. The Lusin Area Function and the Littlewood-Paley gλ Function 5. Hormander's Multiplier Theorem 6. Notes; Further Results and ProblemsChapter XIII The Good ƒλ Principle 1. Good λ Inequalities 2. Weighted Norm Inequalities for Maximal CZ Singular Integral Operators 3. Weighted Weak-Type (1,1) Estimates for CZ Singular Integral Operators 4. Notes; Further Results and ProblemsChapter XIV Hardy Spaces of Several Real Variables 1. Atomic Decomposition 2. Maximal Function Characterization of Hardy Spaces 3. Systems of Conjugate Functions 4. Multipliers 5. Interpolation 6. Notes; Further Results and ProblemsChapter XV Carleson Measures 1. Carleson Measures 2. Duals of Hardy Spaces 3. Tent Spaces 4. Notes; Further Results and ProblemsChapter XVI Cauchy Integrals on Lipschitz Curves 1. Cauchy Integrals on Lipschitz Curves 2. Related Operators 3. The T1 Theorem 4. Notes; Further Results and ProblemsChapter XVII Boundary Value Problems on C1-Domains 1. The Double and Single Layer Potentials on a C1 -Domain 2. The Dirichlet and Neumann Problems 3. NotesBibliographyIndex