Czechoslovak Mathematical Journal

, Volume 60, Issue 3, pp 751–782 | Cite as

A sharp form of an embedding into multiple exponential spaces

  • Robert Černý
  • Silvie Mašková


Let Θ be a bounded open set in ℝ n , n ⩾ 2. In a well-known paper Indiana Univ. Math. J., 20, 1077–1092 (1971) Moser found the smallest value of K such that
$$ \sup \left\{ {\int_\Omega {\exp \left( {\left( {\frac{{\left| {f(x)} \right|}} {K}} \right)^{{n \mathord{\left/ {\vphantom {n {(n - 1)}}} \right. \kern-\nulldelimiterspace} {(n - 1)}}} } \right):f \in W_0^{1,n} (\Omega ),\left\| {\nabla f} \right\|_{L^n } \leqslant 1} } \right\} < \infty $$
. We extend this result to the situation in which the underlying space L n is replaced by the generalized Zygmund space L n logn−1 L log α log L (α < n − 1), the corresponding space of exponential growth then being given by a Young function which behaves like exp(exp(t n/(n−1−α))) for large t. We also discuss the case of an embedding into triple and other multiple exponential cases.


Orlicz spaces Orlicz-Sobolev spaces embedding theorems sharp constants 

MSC 2010

46E35 46E30 


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Copyright information

© Mathematical Institute, Academy of Sciences of Czech Republic 2010

Authors and Affiliations

  1. 1.Faculty of Mathematics and Physics, Department of Mathematical AnalysisCharles UniversityPraha 8Czech Republic
  2. 2.Faculty of Mathematics and Physics, Department of Condensed Matter PhysicsCharles UniversityPraha 2Czech Republic

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