Littlewood-Paley and Multiplier Theory

Prologue.- 1. Introduction.- 1.1. Littlewood-Paley Theory for T.- 1.2. The LP and WM Properties.- 1.3. Extension of the LP and R Properties to Product Groups.- 1.4 Intersections of Decompositions Having the LP Property.- 2. Convolution Operators (Scalar-Valued Case).- 2.1. Covering Families.- 2.2. The Covering Lemma.- 2.3. The Decomposition Theorem.- 2.4. Bounds for Convolution Operators.- 3. Convolution Operators (Vector-Valued Case).- 3.1. Introduction.- 3.2. Vector-Valued Functions.- 3.3. Operator-Valued Kernels.- 3.4. Fourier Transforms.- 3.5. Convolution Operators.- 3.6. Bounds for Convolution Operators.- 4. The Littlewood-Paley Theorem for Certain Disconnected Groups.- 4.1. The Littlewood-Paley Theorem for a Class of Totally Disconnected Groups.- 4.2. The Littlewood-Paley Theorem for a More General Class of Disconnected Groups?.- 4.3. A Littlewood-Paley Theorem for Decompositions of ? Determined by a Decreasing Sequence of Subgroups.- 5. Martingales and the Littlewood-Paley Theorem.- 5.1. Conditional Expectations.- 5.2. Martingales and Martingale Difference Series.- 5.3. The Littlewood-Paley Theorem.- 5.4. Applications to Disconnected Groups.- 6. The Theorems of M. Riesz and Steckin for ?, Tand ?.- 6.1. Introduction.- 6.2. The M. Riesz, Conjugate Function, and Ste?kin Theorems for ?.- 6.3. The M. Riesz, Conjugate Function, and Ste?kin Theorems for T.- 6.4. The M. Riesz, Conjugate Function, and Ste?kin Theorems for ?.- 6.5. The Vector Version of the M. Riesz Theorem for ?, Tand ?.- 6.6. The M. Riesz Theorem for ?k x Tm x ?n.- 6.7. The Hilbert Transform.- 6.8. A Characterisation of the Hilbert Transform.- 7. The Littlewood-Paley Theorem for ?, Tand ?: Dyadic Intervals.- 7.1. Introduction.- 7.2. The Littlewood-Paley Theorem: First Approach.- 7.3. The Littlewood-Paley Theorem: Second Approach.- 7.4. The Littlewood-Paley Theorem for Finite Products of ?, Tand ?: Dyadic Intervals.- 7.5. Fournier's Example.- 8. Strong Forms of the Marcinkiewicz Multiplier Theorem and Littlewood-Paley Theorem for ?, Tand ?.- 8.1. Introduction.- 8.2. The Strong Marcinkiewicz Multiplier Theorem for T.- 8.3. The Strong Marcinkiewicz Multiplier Theorem for ?.- 8.4. The Strong Marcinkiewicz Multiplier Theorem for ?.- 8.5. Decompositions which are not Hadamard.- 9. Applications of the Littlewood-Paley Theorem.- 9.1. Some General Results.- 9.2. Construction of ?(p) Sets in ?.- 9.3. Singular Multipliers.- Appendix A. Special Cases of the Marcinkiewicz Interpolation Theorem.- A.1. The Concepts of Weak Type and Strong Type.- A.2. The Interpolation Theorems.- A.3. Vector-Valued Functions.- Appendix B. The Homomorphism Theorem for Multipliers...- B.1. The Key Lemmas.- B.2. The Homomorphism Theorem.- Appendix D. Bernstein's Inequality.- D.1. Bernstein's Inequality for ?.- D.2. Bernstein's Inequality for T.- D.3. Bernstein's Inequality for LCA Groups.- Historical Notes.- References.- Terminology.- Index of Notation.- Index of Authors and Subjects.