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Continuity envelopes of spaces of generalised smoothness: a limiting case; embeddings and approximation numbers. (English) Zbl 1079.46019
Summary: Continuity envelopes for the spaces of generalised smoothness $$B_{pq}^{(s,\Psi)}(\mathbb{R}^n)$$ and $$F_{pq}^{(s,\Psi)} (\mathbb{R}^n)$$ are studied in the so-called supercritical case $$s=1+n/p$$, paralleling recent developments for a corresponding limiting case for local growth envelopes of spaces of such a type. In addition, the power of the concept is used in proving conditions for some embeddings between function spaces to hold, as well as in the study of the asymptotic behaviour of approximation numbers of related embeddings.

##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 41A46 Approximation by arbitrary nonlinear expressions; widths and entropy 47B06 Riesz operators; eigenvalue distributions; approximation numbers, $$s$$-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
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