CHAPTER 1 Introduction and notations Let (Q, /i) denote a probability space, and assume there is an operator T defined on some algebra A of measurable functions on Q with the following properties: 1) for any f E A, I \ / ) is a non-negative measurable function on ft 2) for all / G A and smooth functions u on R, u(f) G A and T(u(f)) = K(/)iim Introduce the entropy functional: Ent( 5 ) = / 5 l o g ^ - / ^ l o g / ^ . It is well defined for all measurable g 0 and then it is finite if and only if so is the integral J g log(l + g) d\i. One says that (fi, /i, V) satisfies a logarithmic Sobolev inequality, for short LSI with constant c 0 if, for all / G A, (i-o.i) Ent(|/ncyr(/)^.•jnff The definition is similar to the one considered in [A-M-S]. For example, when O is a metric space with metric d, the "modulus of the gradient" comes naturally via the identity r ( / ) (x) = \Vf(x)\ = limsup l ^ ) - y with the convention that T(/) (x) = 0 at isolated points x in 17. In this case, we may define T on the class A of all Lipschitz functions / , i.e., such that ||/||Lip °o, or for a larger class of all locally Lipschitz functions (i.e., Lipschitz in a neighbourhood of any point). Moreover, in case of the Euclidean space Q = R n with the usual Euclidean metric d(x,y) \x y\, we clearly have T(f)(x) = |V/(x)| at each point x G R n where / is different iable and has gradient V/(x). Since locally Lipschitz functions are differentiable almost everywhere (with respect to Lebesgue measure), we then arrive in (1.1) at the usual definition of logarithmic Sobolev inequalities. More generally, given a number p G (1, +oo], we may equip R n with ^-metric d(x,y) = \\x y||p = (]CILi \xi ~ Vi\P) ? a n d then we obtain another important modulus of gradient, rtf)(x) = \vf(x)\ q = iJ2 [fa. fto °Received by the editor October 1, 2002. 1
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