Coercive Inequalities and U-Bounds

Z dλ is the Gaussian measure. In a setup of a more general metric space, a natural question would be to try to find similar inequalities with different measures of the form dμ = e Z dλ, where U is a function of a metric d, and where the Euclidean gradient is replaced by a more general sub-gradient in R. Aside from their theoretical importance, such inequalities are needed because of their applications, some of which will be discussed briefly. L.Gross also pointed out ([13]) the importance of the inequality (1.1) in the sense that it can be extended to infinite dimensions with additional useful results. (See also works: [14, 5, 30, 27, 6, 34, 29].) He proved that if L is the non-positive self-adjoint operator on L (μ) such that (−Lf, f)L2(μ) = ∫

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