On a Sobolev type inequality and its applications
Studia Mathematica, Tome 176 (2006) no. 2, p. 113.
Voir la notice de l'article dans European 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,
s
u
p
s
,
t
∈
T
|
f
(
s
)
-
f
(
t
)
|
≤
6
A
B
(
∫
0
r
ψ
(
1
/
A
ε
n
-
1
)
ε
n
-
1
d
ε
+
1
/
(
n
|
B
|
|
·
|
|
(
0
,
1
)
|
)
∫
T
φ
(
1
/
B
|
|
∇
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
≤
η
(
|
|
s
-
t
|
|
)
for s,t ∈ T
is a.s. sample bounded.
@article{STUMA_2006__176_2_286511,
author = {Witold Bednorz},
title = {On a {Sobolev} type inequality and its applications},
journal = {Studia Mathematica},
pages = {113},
publisher = {mathdoc},
volume = {176},
number = {2},
year = {2006},
zbl = {1105.60024},
language = {en},
url = {https://geodesic-test.mathdoc.fr/item/STUMA_2006__176_2_286511/}
}
Witold Bednorz. On a Sobolev type inequality and its applications. Studia Mathematica, Tome 176 (2006) no. 2, p. 113. https://geodesic-test.mathdoc.fr/item/STUMA_2006__176_2_286511/