Linear problems and the existence of ϕε

Part II. Proofs
Part of the Lecture Notes in Physics book series (LNP, volume 74)


Sobolev Inequality Hermite Polynomial Spectral Projection Logarithmic Sobolev Inequality Particle Number Operator 
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The main result, the existence of ϕɛ has been proved before by Bleher and Sinai in their fundamental paper [16], cf the “Remarks on Section 3”. The functional analytic apparatus we are using here can be found for questions of topology in

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    DUNFORD-SCHWARTZ. Linear operators. Part I:General theory; Part II:Spectral theory. New York Interscience 1958, 1963Google Scholar

while a good reference for the perturbation theory is

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    T. KATO. Perturbation theory for linear operators. Berlin-Heidelberg-New York. Springer, 1966.Google Scholar

The hypercontractive estimates were first given by Glimm in a special case and later formulated and proved in full generality by Nelson in

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    E. NELSON. The Free Markoff Field. J. Functional Anal. 12, 211–227 (1973).CrossRefGoogle Scholar

A nice proof which gives connections to Orlitz-Spaces has been given in

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    L. GROSS. Logarithmic Sobolev Inequalities. Amer. J. Math. 97, 1061 (1975).Google Scholar

The fact that the inequality follows from the ordinary Sobolev inequalities has been shown by Sénéor (private communication), by using the bounds given by

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    T. AUBIN. Problèmes isopérimétriques et espaces de Sobolev. C.R. Acad. Sc. Paris 280, A 279 (1975).Google Scholar

A very elegant new proof can be found in

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    H.J. BRASCAMP, E.H. LIEB. Best constants in Young's inequality, its converse and its generalization to more than three functions. Adv. Math. 20, 151 (1976).CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1978

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