On a Sobolev type inequality and its applications
Witold Bednorz
Studia Mathematica, Tome 173 (2006), p. 113-137 / Harvested from The Polish Digital Mathematics Library

Assume ||·|| is a norm on ℝⁿ and ||·||⁎ its dual. Consider the closed ball T:=B||·||(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, sups,tT|f(s)-f(t)|6AB(0rψ(1/Aεn-1)εn-1dε+1/(n|B||·||(0,1)|)Tφ(1/B||f(u)||)du), 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)||φη(||s-t||) for s,t ∈ T is a.s. sample bounded.

Publié le : 2006-01-01
EUDML-ID : urn:eudml:doc:286511
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     title = {On a Sobolev type inequality and its applications},
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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/