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

Černý, Robert

Commentationes Mathematicae Universitatis Carolinae, Tome 53 (2012) no. 4, pp. 557-571.

Résumé

Let $n\geq 2$ 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 $t^n\log^{\alpha}(t)$, $\alpha$, 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 $W_0^mL^{\frac{n}{m},q}\log^{\alpha}L(\Omega)$, $m n$, $q\in (1,\infty]$, $\alpha\frac{1}{q'}$, embedded into the Zygmund space $Z_0^{\frac{1}{q'}-\alpha}(\Omega)$.

Détail

Classification : 26D10, 46E30, 46E35
Keywords: Orlicz-Sobolev spaces; Lorentz-Sobolev spaces; Trudinger embedding; Moser-Trudinger inequality; best constants