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Keywords:
Orlicz spaces; Orlicz-Sobolev spaces; embedding theorems; sharp constants
Orlicz spaces; Orlicz-Sobolev spaces; embedding theorems; sharp constants
Summary:
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 $L^n\log ^{n-1}L \log ^{\alpha }\log L$ $(\alpha <n-1)$, 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.
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