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Keywords:
Orlicz-Sobolev spaces; Lorentz-Sobolev spaces; Trudinger embedding; Moser-Trudinger inequality; best constants
Orlicz-Sobolev spaces; Lorentz-Sobolev spaces; Trudinger embedding; Moser-Trudinger inequality; best constants
Summary:
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 $t^n\log^{\alpha}(t)$, $\alpha<n-1$, 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 $W_0^mL^{\frac{n}{m},q}\log^{\alpha}L(\Omega)$, $m< n$, $q\in (1,\infty]$, $\alpha<\frac{1}{q'}$, embedded into the Zygmund space $Z_0^{\frac{1}{q'}-\alpha}(\Omega)$.
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