Assume ||·|| is a norm on ℝⁿ and ||·||⁎ its dual. Consider the closed ball , 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, , 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 for s,t ∈ T is a.s. sample bounded.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm176-2-2, author = {Witold Bednorz}, title = {On a Sobolev type inequality and its applications}, journal = {Studia Mathematica}, volume = {173}, year = {2006}, pages = {113-137}, zbl = {1105.60024}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm176-2-2} }
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/