Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. (AM-63), Volume 63

This work deals with an extension of the classical Littlewood-Paley theory in the context of symmetric diffusion semigroups. In this general setting there are applications to a variety of problems, such as those arising in the study of the expansions coming from second order elliptic operators. A review of background material in Lie groups and martingale theory is included to make the monograph more accessible to the student. Contents Preface Introduction Chapter I. Lie Groups (A Review) §1. Compact groups §2. The Peter-Weyl theorem §3. The Peter-Weyl theorem (Concluded) §4. Lie groups; examples §5. Lie algebras §6. Universal enveloping algebra §7. Laplacian Chapter II. Littlewood-Paley Theory for a Compact Lie Group §1. The heat-diffusion semi-group §2. The Poisson semi-group; the main theorem 3. Proof of Theorem 2 §4. Applications: Riesz transforms, etc Bibliographical remarks Chapter III. General Symmetric Diffusion Semi-groups §1. General setting §2. Analyticity of these semi-groups §3. The maximal theorem §4. A digression: L2 theorems Bibliographical remarks Chapter IV. The General Littlewood-Paley Theory §1. Conditional Expectation and Martingales §2. The inequalities for martingales §3. An additional ‘‘max’’ inequality §4. The link between martingales and semi-groups §5. The Littlewood-Paley inequalities in general §6. Dénouement Bibliographical remarks Chapter V. Further Illustrations §1. Lie groups §2. Semi-simple case §3. Sturm-Liouville §4. Heuristics Bibliographical remarks References Appendix (1985)