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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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