On a Sobolev type inequality and its applications

• Autorzy: Witold Bednorz
• Języki publikacji: EN

Abstrakty

EN

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,
$sup_{s,t∈ T} |f(s)-f(t)| ≤ 6AB(∫_{0}^{r} ψ(1/Aε^{n-1})ε^{n-1} dε + 1/(n|B_{||·||}(0,1)|) ∫_{T} φ(1/B ||∇f(u)||⁎)du)$,
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
$||X(s)-X(t)||_{φ} ≤ η(||s-t||)$ for s,t ∈ T
is a.s. sample bounded.

Kategorie tematyczne

• 28A99: None of the above, but in this section
• 60G17: Sample path properties

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2006
• Tom: 176
• Numer: 2
• Strony: 113-137

Daty

wydano

2006

Twórcy

autor

Witold Bednorz

• Department of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland

Identyfikatory

DOI

10.4064/sm176-2-2