## Between Sobolev and Poincaré.(English)Zbl 0986.60017

Milman, V. D. (ed.) et al., Geometric aspects of functional analysis. Proceedings of the Israel seminar (GAFA) 1996-2000. Berlin: Springer. Lect. Notes Math. 1745, 147-168 (2000).
For $$a\in [0,1]$$, put $$r= 2/(2- a)$$ and $$c_r= r/2\Gamma(1/r)$$, and consider the probability measure $$\mu_r(dx)= c^n_r\exp(-|x_1|^r-\cdots-|x_n|^r) dx_1\cdots dx_n$$ on $$\mathbb{R}^n$$. The authors prove that there exists a universal constant $$C$$ such that $E\mu_r f^2- (E\mu_r f^p)^{2/p}\leq C(2- p)^a E_{\mu_r}\|\nabla f\|^2\tag{$$*$$}$ for any smooth function $$f: \mathbb{R}^n\to [0,\infty)$$ and $$p\in [1,2)$$, where $$\|.\|$$ denotes Euclidean norm. Conversely, they also prove that if $$(*)$$ is satisfied, with some probability measure $$\mu$$ on $$\mathbb{R}^n$$ in place of $$\mu_r$$, for any smooth function $$f: \mathbb{R}^n\to [0,\infty)$$ and $$p\in [1,2)$$, then for any $$h: \mathbb{R}^n\to \mathbb{R}$$ such that $$|h(x)- h(y)|\leq\|x- y\|$$, $$x,y\in \mathbb{R}^n$$, there is $E_\mu|h|< \infty\quad\text{and}\quad \mu(h- E_\mu h\geq\sqrt Ct)\leq \exp(- Kt^2)\;\text{or} \leq\exp(- Kt^{2/(2-a)})$ according as $$t\in [0,1]$$ or $$t\geq 1$$, where $$K$$ is a universal constant.
For the entire collection see [Zbl 0949.00025].

### MSC:

 60E15 Inequalities; stochastic orderings 60A10 Probabilistic measure theory
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