Sharp constants for Moser-type inequalities concerning embeddings into Zygmund spaces

Robert Černý

Commentationes Mathematicae Universitatis Carolinae (2012)

  • Volume: 53, Issue: 4, page 557-571
  • ISSN: 0010-2628

Abstract

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Let n ≥ 2 and Ω ⊂ ℝ n be a bounded set. We give a Moser-type inequality for an embedding of the Orlicz-Sobolev space W 0 L Φ ( Ω ) , where the Young function Φ behaves like t n log α ( t ) , α < n - 1 , for t large, into the Zygmund space Z 0 n - 1 - α n ( Ω ) . We also study the same problem for the embedding of the generalized Lorentz-Sobolev space W 0 m L n m , q log α L ( Ω ) , m < n , q ∈ ( 1 , ∞ ] , α < 1 q ' , embedded into the Zygmund space Z 0 1 q ' - α ( Ω ) .

How to cite

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Černý, Robert. "Sharp constants for Moser-type inequalities concerning embeddings into Zygmund spaces." Commentationes Mathematicae Universitatis Carolinae 53.4 (2012): 557-571. <http://eudml.org/doc/252520>.

@article{Černý2012,
abstract = {Let $n\ge 2$ and $\Omega \subset \mathbb \{R\}^n$ be a bounded set. We give a Moser-type inequality for an embedding of the Orlicz-Sobolev space $W_0L^\{\Phi \}(\Omega )$, where the Young function $\Phi $ behaves like $t^n\log ^\{\alpha \}(t)$, $\alpha <n-1$, for $t$ large, into the Zygmund space $Z_0^\{\frac\{n-1-\alpha \}\{n\}\}(\Omega )$. We also study the same problem for the embedding of the generalized Lorentz-Sobolev space $W_0^mL^\{\frac\{n\}\{m\},q\}\log ^\{\alpha \}L(\Omega )$, $m< n$, $q\in (1,\infty ]$, $\alpha <\frac\{1\}\{q^\{\prime \}\}$, embedded into the Zygmund space $Z_0^\{\frac\{1\}\{q^\{\prime \}\}-\alpha \}(\Omega )$.},
author = {Černý, Robert},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Orlicz-Sobolev spaces; Lorentz-Sobolev spaces; Trudinger embedding; Moser-Trudinger inequality; best constants; Orlicz-Sobolev space; Lorentz-Sobolev space; Moser-Trudinger inequality},
language = {eng},
number = {4},
pages = {557-571},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Sharp constants for Moser-type inequalities concerning embeddings into Zygmund spaces},
url = {http://eudml.org/doc/252520},
volume = {53},
year = {2012},
}

TY - JOUR
AU - Černý, Robert
TI - Sharp constants for Moser-type inequalities concerning embeddings into Zygmund spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2012
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 53
IS - 4
SP - 557
EP - 571
AB - Let $n\ge 2$ and $\Omega \subset \mathbb {R}^n$ be a bounded set. We give a Moser-type inequality for an embedding of the Orlicz-Sobolev space $W_0L^{\Phi }(\Omega )$, where the Young function $\Phi $ behaves like $t^n\log ^{\alpha }(t)$, $\alpha <n-1$, for $t$ large, into the Zygmund space $Z_0^{\frac{n-1-\alpha }{n}}(\Omega )$. We also study the same problem for the embedding of the generalized Lorentz-Sobolev space $W_0^mL^{\frac{n}{m},q}\log ^{\alpha }L(\Omega )$, $m< n$, $q\in (1,\infty ]$, $\alpha <\frac{1}{q^{\prime }}$, embedded into the Zygmund space $Z_0^{\frac{1}{q^{\prime }}-\alpha }(\Omega )$.
LA - eng
KW - Orlicz-Sobolev spaces; Lorentz-Sobolev spaces; Trudinger embedding; Moser-Trudinger inequality; best constants; Orlicz-Sobolev space; Lorentz-Sobolev space; Moser-Trudinger inequality
UR - http://eudml.org/doc/252520
ER -

References

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