Entropy Numbers of Embeddings Between Logarithmic Sobolev Spaces

António M. Caetano

Abstract: Let $\Omega$ be a bounded domain in $\R^n$ and $id$ be the natural embedding
$$ H^{s_1}_{p_1}(\log H)_{a_1}(\Omega) \rightarrow H^{s_2}_{p_2}(\log H)_{a_2}(\Omega) $$
between these logarithmic Sobolev spaces, where $-\infty<s_2<s_1<\infty$, $0<p_1<p_2<\infty$, with $s_1-n/p_1=s_2-n/p_2$, and $-\infty<a_2\leq a_1<\infty$. We show that if the real numbers $a_1$ and $a_2$ satisfy the conditions $a_1>0$, $a_1\not\in ]1/\min\{1,p_2\},2(s_1-s_2)/n+1/\min\{1,p_2\}]$ and $a_2<a_1-2(s_1-s_2)/n-1/\min\{1,p_2\}$ then there exist $c_1,c_2>0$ such that, for all $k\in\N$,
$$ c_1 k^{-(s_1-s_2)/n} \leq e_k(id) \leq c_2 k^{-(s_1-s_2)/n}, $$
where the $e_k$ stand for entropy numbers. This improves earlier results of Edmunds and Triebel [4].

Keywords: Entropy; embeddings; limiting embeddings; logarithmic Sobolev spaces; interpolation; multipliers; Triebel-Lizorkin spaces.

Classification (MSC2000): 46E35.

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