0 Between Sobolev and Poincaré ∗
نویسنده
چکیده
Let a ∈ [0, 1] and r ∈ [1, 2] satisfy relation r = 2/(2− a). Let μ(dx) = cr exp(−(|x1|+|x2|+. . .+|xn|))dx1dx2 . . . dxn be a probability measure on the Euclidean space (R, ‖ · ‖). We prove that there exists a universal constant C such that for any smooth real function f on R and any p ∈ [1, 2) Eμf 2 − (Eμ|f |) ≤ C(2− p)Eμ‖∇f‖. We prove also that if for some probabilistic measure μ on R the above inequality is satisfied for any p ∈ [1, 2) and any smooth f then for any h : R −→ R such that |h(x)− h(y)| ≤ ‖x− y‖ there is Eμ|h| < ∞ and μ(h−Eμh > √ C · t) ≤ e r for t > 1, where K > 0 is some universal constant. Let us begin with few definitions. Definition 1 Let (Ω, μ) be a probability space and let f be a measurable, square integrable non-negative function on Ω. For p ∈ [1, 2) we define the p-variance of f by
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