On a Sobolev type inequality and its applications

Witold Bednorz. "On a Sobolev type inequality and its applications." Studia Mathematica 176.2 (2006): 113-137. <http://eudml.org/doc/286511>.

@article{WitoldBednorz2006,
abstract = {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.},
author = {Witold Bednorz},
journal = {Studia Mathematica},
keywords = {Sobolev inequalities; sample boundedness},
language = {eng},
number = {2},
pages = {113-137},
title = {On a Sobolev type inequality and its applications},
url = {http://eudml.org/doc/286511},
volume = {176},
year = {2006},
}

TY - JOUR
AU - Witold Bednorz
TI - On a Sobolev type inequality and its applications
JO - Studia Mathematica
PY - 2006
VL - 176
IS - 2
SP - 113
EP - 137
AB - 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.
LA - eng
KW - Sobolev inequalities; sample boundedness
UR - http://eudml.org/doc/286511
ER -