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Imbeddings of Brezis-Wainger type. The case of missing derivatives. (English) Zbl 0985.46018

Let \(\alpha > 0\) and let \(\text{Lip} (1, \alpha) (R^n)\) be the collection of all \(f \in C(\mathbb{R}^n)\) such that \[ \|f |Lip (1, \alpha) \|= \|f |C \|+ \sup_{|h|< \frac{1}{2}} \frac{\|\Delta_h f |L_\infty \|}{|h||\log |h||^\alpha } < \infty. \] The authors deal with limiting embeddings in these target spaces in 3 cases.
1. Isotropic case, typically (embeddings of Brezis-Wainger type), \[ B^{\frac{n}{p} + 1}_{pq} (\mathbb{R}^n) \subset \text{Lip} (1, \frac{1}{q'}), \quad 0 < p \leq \infty, \quad 1 < q \leq \infty, \quad \frac{1}{q} + \frac{1}{q'} = 1. \] There are also embeddings of Trudinger type with \(\frac{n}{p}\) in place of \(\frac{n}{p} + 1\) in (1). There are two generalizations:
2. Related limiting embeddings where the source spaces are spaces with dominating mixed derivatives and also mixed metrics.
3. Related limiting embeddings where the source spaces are spaces with missing derivatives, typically of type \[ W^M_{\overline{p}} = \left\{ f \in L_{\overline{p}}: \|f |W^M_{\overline{p}} \|= \sum_{\alpha \in M} \sum_{\beta \leq \alpha} \|D^\beta f |L_{\overline{p}}\|< \infty \right\} , \] where \(\overline{p} = (p_1,\dots, p_n)\) and where \(M\) is a given subset of \({\mathbb N}^n_0\) (muli-indices in \(\mathbb{R}^n\).

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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