This book is intended to be a detailed and carefully written account of various versions of the Littlewood-Paley theorem and of some of its applications, together with indications of its general significance in Fourier multiplier theory. We have striven to make the presentation self-contained and unified, and adapted primarily for use by graduate students and established mathematicians who wish to begin studies in these areas: it is certainly not intended for experts in the subject. It has been our experience, and the experience of many of our students and colleagues, that this is an area poorly served by existing books. Their accounts of the subject tend to be either ill-suited to the needs of a beginner, or fragmentary, or, in one or two instances, obscure. We hope that our book will go some way towards filling this gap in the literature. Our presentation of the Littlewood-Paley theorem proceeds along two main lines, the first relating to singular integrals on locally com pact groups, and the second to martingales. Both classical and modern versions of the theorem are dealt with, appropriate to the classical n groups IRn, ?L , Tn and to certain classes of disconnected groups. It is for the disconnected groups of Chapters 4 and 5 that we give two separate accounts of the Littlewood-Paley theorem: the first Fourier analytic, and the second probabilistic.

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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 × Tm × ?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: FirstApproach.- 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.