A sharp form of an embedding into multiple exponential spaces

Černý, Robert, and Mašková, Silvie. "A sharp form of an embedding into multiple exponential spaces." Czechoslovak Mathematical Journal 60.3 (2010): 751-782. <http://eudml.org/doc/38040>.

@article{Černý2010,
abstract = {Let $\Omega $ be a bounded open set in $\mathbb \{R\}^n$, $n \ge 2$. In a well-known paper Indiana Univ. Math. J., 20, 1077–1092 (1971) Moser found the smallest value of $K$ such that \[ \sup \bigg \lbrace \int \_\{\Omega \} \exp \Big (\Big (\frac\{\left|f(x)\right|\}\{K\}\Big )^\{n/(n-1)\}\Big )\colon f\in W^\{1,n\}\_0(\Omega ),\Vert \nabla f\Vert \_\{L^n\}\le 1\bigg \rbrace <\infty . \] We extend this result to the situation in which the underlying space $L^n$ is replaced by the generalized Zygmund space $L^n\log ^\{n-1\}L \log ^\{\alpha \}\log L$$(\alpha <n-1)$, the corresponding space of exponential growth then being given by a Young function which behaves like $\exp (\exp (t^\{n/(n-1-\alpha )\}))$ for large $t$. We also discuss the case of an embedding into triple and other multiple exponential cases.},
author = {Černý, Robert, Mašková, Silvie},
journal = {Czechoslovak Mathematical Journal},
keywords = {Orlicz spaces; Orlicz-Sobolev spaces; embedding theorems; sharp constants; Orlicz space; Orlicz-Sobolev space; embedding theorem; sharp constant},
language = {eng},
number = {3},
pages = {751-782},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A sharp form of an embedding into multiple exponential spaces},
url = {http://eudml.org/doc/38040},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Černý, Robert
AU - Mašková, Silvie
TI - A sharp form of an embedding into multiple exponential spaces
JO - Czechoslovak Mathematical Journal
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 3
SP - 751
EP - 782
AB - Let $\Omega $ be a bounded open set in $\mathbb {R}^n$, $n \ge 2$. In a well-known paper Indiana Univ. Math. J., 20, 1077–1092 (1971) Moser found the smallest value of $K$ such that \[ \sup \bigg \lbrace \int _{\Omega } \exp \Big (\Big (\frac{\left|f(x)\right|}{K}\Big )^{n/(n-1)}\Big )\colon f\in W^{1,n}_0(\Omega ),\Vert \nabla f\Vert _{L^n}\le 1\bigg \rbrace <\infty . \] We extend this result to the situation in which the underlying space $L^n$ is replaced by the generalized Zygmund space $L^n\log ^{n-1}L \log ^{\alpha }\log L$$(\alpha <n-1)$, the corresponding space of exponential growth then being given by a Young function which behaves like $\exp (\exp (t^{n/(n-1-\alpha )}))$ for large $t$. We also discuss the case of an embedding into triple and other multiple exponential cases.
LA - eng
KW - Orlicz spaces; Orlicz-Sobolev spaces; embedding theorems; sharp constants; Orlicz space; Orlicz-Sobolev space; embedding theorem; sharp constant
UR - http://eudml.org/doc/38040
ER -