Modified logarithmic Sobolev inequalities in null curvature
نویسندگان
چکیده
We present a logarithmic Sobolev inequality adapted to a log-concave measure. Assume that Φ is a symmetric convex function on R satisfying (1 + ε)Φ(x) 6 xΦ(x) 6 (2 − ε)Φ(x) for x > 0 large enough and with ε ∈]0, 1/2]. We prove that the probability measure on R μΦ(dx) = e /ZΦdx satisfies a modified and adapted logarithmic Sobolev inequality : there exist three constant A,B,D > 0 such that for all smooth f > 0, EntμΦ ( f ) 6 A ∫ HΦ ( f ′ f ) fdμΦ, with HΦ(x) = { Φ(Bx) if |x| > D, x if |x| 6 D. Mathematics Subject Classification 2000: 26D10, 60E15.
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