Geodesic
A sharp form of an embedding into multiple exponential spaces
Czechoslovak Mathematical Journal, Tome 60 (2010) no. 3, pp. 751-782 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Let $\Omega $ be a bounded open set in $\mathbb R^n$, $n \geq 2$. In a well-known paper {\it Indiana Univ. Math. J.}, 20, 1077--1092 (1971) Moser found the smallest value of $K$ such that $$ \sup \bigg \{\int _{\Omega } \exp \Big (\Big (\frac {\left |f(x)\right |}K\Big )^{n/(n-1)}\Big )\colon f\in W^{1,n}_0(\Omega ),\|\nabla f\|_{L^n}\leq 1\bigg \}\infty . $$ 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 $, 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.
Let $\Omega $ be a bounded open set in $\mathbb R^n$, $n \geq 2$. In a well-known paper {\it Indiana Univ. Math. J.}, 20, 1077--1092 (1971) Moser found the smallest value of $K$ such that $$ \sup \bigg \{\int _{\Omega } \exp \Big (\Big (\frac {\left |f(x)\right |}K\Big )^{n/(n-1)}\Big )\colon f\in W^{1,n}_0(\Omega ),\|\nabla f\|_{L^n}\leq 1\bigg \}\infty . $$ 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 $, 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.
Classification : 46E30, 46E35
Keywords: Orlicz spaces; Orlicz-Sobolev spaces; embedding theorems; sharp constants 
@article{CMJ_2010_60_3_a11,
     author = {\v{C}ern\'y, Robert and Ma\v{s}kov\'a, Silvie},
     title = {A sharp form of an embedding into multiple exponential spaces},
     journal = {Czechoslovak Mathematical Journal},
     pages = {751--782},
     year = {2010},
     volume = {60},
     number = {3},
     mrnumber = {2672414},
     zbl = {1224.46064},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2010_60_3_a11/}
}
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Černý, Robert; Mašková, Silvie. A sharp form of an embedding into multiple exponential spaces. Czechoslovak Mathematical Journal, Tome 60 (2010) no. 3, pp. 751-782. http://geodesic.mathdoc.fr/item/CMJ_2010_60_3_a11/