Between sobolev and poincaré

Abstract

Let a a ∈ [0, 1] and r ∈ [1, 2] satisfy relation r = 2/(2 − a). Let μ(dx)=c ⁿ _r exp(-(|x ₁|^r+|x ₂|^r+...+|x _n |^r))dx ₁ dx ₂...dx _n be a probability measure on the Euclidean space (R ⁿ, ‖ · ‖). We prove that there exists a universal constant C such that for any smooth real function f on R ⁿ and any p ∈ [1,2)

$$E_\mu f^2 - (E_\mu \left| f \right|^p )^{2/p} \leqslant C(2 - p)^a E_\mu \left\| {\nabla f} \right\|^2$$

. We prove also that if for some probabilistic measure μ on R ⁿ the above inequality is satisfied for any p ∈ [1, 2) and any smooth f then for any h : R ⁿ → R such that |h(x)-h(y)|≤∥x-y∥ there is E _μ |h| < ∞ and

$$\mu (h - E_\mu h > \sqrt C \cdot t) \leqslant e^{ - Kt^r }$$

for t > 1, where K > 0 is some universal constant.

Research partially supported by KBN Grant 2 P03A 043 15.