## Topics in Harmonic Analysis, Related to the Littlewood-Paley TheoryThis 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. |

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### Contents

LIE GROUPS A REVIEW | |

LITTLEWOODPALEY THEORY FOR A COMPACT LIE GROUP | 31 |

GENERAL SYMMETRIC DIFFUSION SEMIGROUPS | 59 |

THE GENERAL LITTLEWOODPALEY THEORY | 83 |

FURTHER ILLUSTRATIONS | 117 |

### Other editions - View all

Topics in Harmonic Analysis, Related to the Littlewood-Paley Theory Elias M. Stein Limited preview - 1970 |

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analysis analytic argument bi-invariant bounded operator Chapter commutes compact groups compact Lie group completes the proof conditional expectation converges convexity theorem defined denotes differential operator diffusion semigroups direct sum eigenfunction equation ergodic theorem fact finite finite-dimensional follows formally self-adjoint Fourier function f Haar measure Hence Hilbert Hölder's inequality identity interpolation irreducible representations isomorphism Laplacian left-invariant lemma Let f Lie algebra Lie group G Littlewood-Paley inequalities Littlewood-Paley theory Lp(G Lp(M M,dx Marcinkiewicz martingale Math matrix maximal function maximal theorem measure space norm obtain orthogonal Peter-Weyl theorem Poisson integral positive proof of Theorem properties prove reader regular representation representation of G result Riesz transforms ſ ſ satisfies Section semigroup sequence sigma-field ſº SO(n spectral theorem subset Suppose symmetric diffusion Theorem 9 tion unitary representation universal enveloping algebra vector field vector space