Between Sobolev and Poincaré

摘要: Let a ∈ [0, 1] and r [1, 2] satisfy relation = 2/(2 − a). μ(dx)=c n exp(-(|x1| +|x2| +...+|x | ))dx1dx2...dx be probability measure on the Euclidean space (R , ‖ · ‖). We prove that there exists universal constant C such for any smooth real function f R p [1,2) $$E_\mu f^2 - (E_\mu \left| \right|^p )^{2/p} \leqslant C(2 p)^a E_\mu \left\| {\nabla f} \right\|^2$$ . also if some probabilistic μ above inequality is satisfied 2) then h : → |h(x)-h(y)|≤∥x-y∥ E |h| < ∞ $$\mu (h > \sqrt \cdot t) e^{ Kt^r }$$ for t 1, where K 0 constant.