Preface During the second half of the XXth century one could observe an interesting development in the infinite dimensional analysis slowly moving it from the "exper- imental science" based on interesting examples to a brink of becoming something more than that. In particular during its last decade we have experienced a con- siderable progress in the domain of coercive inequalities and isoperimetry as a starting point to explore the related literature we suggest [A-B-C-F-G-M-R-S], [G-Z] and [L5], as they contain an extensive overview and a comprehensive list of references. For the purposes of this work we would like to recall two important inequalities: the so called Logarithmic Sobolev Inequality of [Gl] and Isoperimetric Functional Inequality introduced in [B2]. The first one in plain language is simply an estimate of the relative entropy in terms of the Fisher entropy. Alternatively, from the perspective of application to control the ergodicity of Markov semigroup generated by the Dirichlet operator, we could think of the relative density as given by a square of a function divided by a normalisation constant and in this setting the upper bound in question is given by the Dirichlet form as follows (LS2) Ent(|/| 2 )c/'X |V i /| 2 d/i. J i The Isoperimetric Functional Inequality is a bound on the value of an isoperimetric function X computed at the point equal to the expectation of a positive bounded by 1 function / by an expectation of the length of a vector which first component equals to the composition of X with / and the other components are given by the components of the gradient of / scaled by a constant (IFI2) X (/x/) j /l(/) 2 + c J ] | V i / | 2 d / i . By an appropriate approximation of a characteristic function the right hand side converges to the surface measure of the set whereas the limit of the left hand side is simply equal to the value of the isoperimetric function at the volume of the set in question this provides justification for the name of the inequality. The first key point is that both of these inequalities have tensorisation property, (i.e. if they hold in two measure spaces, then the product of the measures also satisfies them). The second important common point is that both of them involve the same /2-norm on the tangent space. Naturally given a variety of probability measures one may like to ask for an optimal information that is for inequalities reflecting possibly precisely properties of a given measure. It is certainly well known that in finite dimensions the isoperimetric functions may be different for different families of the measures. In our work we would like to emphasise the role of the metric on the vii
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