99-195 Johnsen, J.

ON THE SPECTRAL PROPERTIES OF WITTEN-LAPLACIANS, THEIR RANGE PROJECTIONS AND BRASCAMP--LIEB'S INEQUALITY (508K, Post Script) May 26, 99

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Abstract. A study is made of a recent integral identity of B.~Helffer and J.~Sj{\"o}strand, which for a not yet fully determined class of probability measures yields a formula for the covariance of two functions (of a stochastic variable); in comparison with the Brascamp--Lieb inequality, this formula is a more flexible and in some contexts stronger means for the analysis of correlation asymptotics in statistical mechanics. Using a fine version of the Closed Range Theorem, the identity's validity is shown to be equivalent to some explicitly given spectral properties of Witten--Laplacians on Euclidean space, and the formula is moreover deduced from the obtained abstract expression for the range projection. As a corollary, a generalised version of Brascamp--Lieb's inequality is obtained. For a certain class of measures occuring in statistical mechanics, explicit criteria for the Witten-Laplacians are found from the Persson--Agmon formula, from compactness of embeddings and from the Weyl calculus, which give results for closed range, strict positivity, essential self-adjointness and domain characterisations.

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