Czechoslovak Mathematical Journal

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

A sharp form of an embedding into multiple exponential spaces



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 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    D.R. Adams and L. I. Hedberg: Function Spaces and Potential Theory. Springer, 1996.Google Scholar
  2. [2]
    A. Cianchi: A sharp embedding theorem for Orlicz-Sobolev spaces. Indiana Univ. Math. J. 45 (1996), 39–65.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    D.E. Edmunds, P. Gurka and B. Opic: Double exponential integrability of convolution operators in generalized Lorentz-Zygmund spaces. Indiana Univ. Math. J. 44 (1995), 19–43.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    D.E. Edmunds, P. Gurka and B. Opic: Double exponential integrability, Bessel potentials and embedding theorems. Studia Math. 115 (1995), 151–181.MATHMathSciNetGoogle Scholar
  5. [5]
    D.E. Edmunds, P. Gurka and B. Opic: Sharpness of embeddings in logarithmic Bessel-potential spaces. Proc. Roy. Soc. Edinburgh 126A (1996), 995–1009.MathSciNetGoogle Scholar
  6. [6]
    D.E. Edmunds, P. Gurka and B. Opic: On embeddings of logarithmic Bessel potential spaces. J. Functional Analysis 146 (1997), 116–150.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    D.E. Edmunds, P. Gurka and B. Opic: Norms of embeddings in logarithmic Bessel- potential spaces. Proc. Amer. Math. Soc. 126 (1998), 2417–2425.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    D.E. Edmunds and M. Krbec: Two limiting cases of Sobolev imbeddings. Houston J. Math. 21 (1995), 119–128.MATHMathSciNetGoogle Scholar
  9. [9]
    N. Fusco, P. L. Lions and C. Sbordone: Sobolev imbedding theorems in borderline cases. Proc. Amer. Math. Soc. 124 (1996), 561–565.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    L. I. Hedberg: On certain convolution inequalities. Proc. Amer. Math. Soc. 36 (1972), 505–512.MathSciNetGoogle Scholar
  11. [11]
    S. Hencl: A sharp form of an embedding into exponential and double exponential spaces. J. Funct. Anal. 204 (2003), 196–227.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    J. Moser: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20 (1971), 1077–1092.CrossRefGoogle Scholar
  13. [13]
    B. Opic and L. Pick: On generalized Lorentz-Zygmund spaces. Math. Ineq. Appl. 2 (July 1999), 391–467.MATHMathSciNetGoogle Scholar
  14. [14]
    M.M. Rao and Z.D. Ren: Theory of Orlicz Spaces. Pure Appl. Math., 1991.Google Scholar
  15. [15]
    R. S. Strichartz: A note on Trudinger’s extension of Sobolev’s inequality. Indiana Univ. Math. J. 21 (1972), 841–842.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    G. Talenti: Inequalities in rearrangement invariant function spaces. Nonlinear Analysis, Function Spaces and Applications 5 (1994), 177–230. Prometheus Publ. House Prague.MathSciNetGoogle Scholar
  17. [17]
    N. S. Trudinger: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17 (1967), 473–484.MATHMathSciNetGoogle Scholar
  18. [18]
    V. I. Yudovich: Some estimates connected with integral operators and with solutions of elliptic equations. Soviet Math. Doklady 2 (1961), 746–749.MATHGoogle Scholar

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

Personalised recommendations