On a Sobolev type inequality and its applications

Witold Bednorz

Witold Bednorz

Studia Mathematica, Tome 173 (2006), p. 113-137 / Harvested from The Polish Digital Mathematics Library

Résumé

Assume ||·|| is a norm on ℝⁿ and ||·||⁎ its dual. Consider the closed ball $T : = B_{| | \cdot | |} (0, r)$ , r > 0. Suppose φ is an Orlicz function and ψ its conjugate. We prove that for arbitrary A,B > 0 and for each Lipschitz function f on T, $s u p_{s, t \in T} | f (s) - f (t) | \leq 6 A B (\int_{0}^{r} ψ (1 / A ε^{n - 1}) ε^{n - 1} d ε + 1 / (n | B_{| | \cdot | |} (0, 1) |) \int_{T} φ (1 / B | | \nabla f (u) | | ⁎) d u)$ , where |·| is the Lebesgue measure on ℝⁿ. This is a strengthening of the Sobolev inequality obtained by M. Talagrand. We use this inequality to state, for a given concave, strictly increasing function η: ℝ₊ → ℝ with η(0) = 0, a necessary and sufficient condition on φ so that each separable process X(t), t ∈ T, which satisfies $| | X (s) - {X (t) | |}_{φ} \leq η (| | s - t | |)$ for s,t ∈ T is a.s. sample bounded.

Publié le : 2006-01-01

Zbl 1105.60024

EUDML-ID : urn:eudml:doc:286511

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Witold Bednorz. On a Sobolev type inequality and its applications. Studia Mathematica, Tome 173 (2006) pp. 113-137. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm176-2-2/