CHAPTER 2 Poincare-type inequalities Here, in the setting of a probability metric space (fl, d, /x), we deduce from (1.2) a corresponding Poincare-type inequality: Theorem 2.1. Under LSq-inequality with constant c, for any locally Lipschitz function f on Q, we have the following Poincare type inequality called later on SGg-inequality (2.0.1) Jlf-jfdJdpKJjLjlVfpd^ More precisely, as soon as |V/| G Lq{n), we have f G Lq(fi), and (2.1) holds true. Recall that |V/(x)| = limsupd(a.j2/)_0+ dixy) *n Particular, this definition is applied on the real line, so / does not need to be different iable everywhere (correspondingly, we understand partial derivatives *£y' for functions defined on the product space fl x R). For the proof of (2.1), we need a general Lemma 2.2. For any non-negative measurable function g on Q, (2.0.2) Ent (g) - log fi{g 0} f g dfi. Proof. The distribution of g under /i represents a probability measure on the positive half-axis [0, +oo) and may be written as (l a)So-\-ai/ where a = n{g 0} and where v is concentrated on (0,+oo). In terms of a random variable, say £, having the distribution i/, we thus may write / /+oo gdfi= xd((l a)5o + av) aE£, / /•+oo g\oggdfi= / x\ogxd((l - a)50 + av) = aE^log^, so, Ent(g) = a E ^ l o g ^ - ( a E 0 l o g ( a E 0 = a Ent (f) - a log a - l o g a ( a E ^ ) . Proof of Theorem 2.1. Let / be a locally Lipschitz function from Lg(fl,/i). Without loss of generality, assume that zero is a median of / , i.e., fi{f 0} | and /i{/ 0} \. First assume that fi{f 0} 0. Set / + = max(/, 0), / ~ = max(—/, 0). These functions are also locally Lip- schitz, and we can apply to them the log-Sobolev inequality (1.2). The moduli 5
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