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Sharp embeddings of Besov-type spaces. (English) Zbl 1132.46022
This paper gives a rather elementary approach to sharp embeddings of Besov spaces $$B^{\sigma,b}_{p,r}(\mathbb{R}^n)$$, with a slowly varying function $$b$$, into Lorentz–Karamata spaces $$L_{p,q; b}(\Omega)$$, where $$\Omega\subset \mathbb{R}^n$$ is a measurable subset of $$\mathbb{R}^n$$. Both the so-called sub-critical case $$\sigma<\frac{n}{p}$$ and the critical case $$\sigma=\frac{n}{p}$$ are studied. The main results, contained in Theorems 3.1, 3.3 (for the sub-limiting case) and Theorems 3.5, 3.7, 3.9, 3.11 (for the limiting case), are of the form that the inequality
$\bigg(\int\limits_0^1 \left[ t^\frac1q b\left(t^\frac1n\right) \kappa(t) f^\ast(t)\right]^s\frac{d t}{t}\bigg)^{1/s} \leq \;C \left\| f \right\| _{B^{\sigma, b}_{p,r}(\mathbb{R}^n)}$ holds for some $$C>0$$ and all $$f\in B^{\sigma,b}_{p,r}(\mathbb{R}^n)$$ if, and only if, $$\kappa$$ is bounded on $$(0,1)$$. Here $$1\leq r\leq s\leq \infty$$, $$1\leq p<\infty$$, $$0<\sigma<\frac{n}{p}$$, $$\frac1q = \frac1p-\frac{\sigma}{n}$$, and $$b$$ slowly varying. It can be shown that for $$\kappa\equiv 1$$ this is true if, and only if, $$s\geq r$$. In the critical case, the outcome is of a similar structure. The authors link these results with assertions on growth envelopes.
Essentially, the authors extend their results in [P. Gurka and P. B. Opic, Rev. Mat. Comput. 18, No. 1, 81–110 (2005; Zbl 1083.46018)], where they dealt with similar questions but for special functions $$b$$ (of logarithmic type) only. The present work demonstrates the strength of the methods developed before. There are close links to [A. Gogatishvili, B. Opic and P. J. S. Neves, Proc. R. Soc. Edinb., Sect. A, Math. 134, No. 6, 1127–1147 (2004; Zbl 1076.46021)] and the earlier papers by A. M. Caetano and S. D. Moura [Math. Nachr. 273, 43–57 (2004; Zbl 1070.46020) and Math. Inequal. Appl. 7, No. 4, 573–606 (2004; Zbl 1076.46025)], where the last-mentioned results cover less general situations (dealing with admissible instead of slowly varying functions). On the other hand, the stressed arguments are essentially different: whereas the last-mentioned papers rely on atomic decomposition techniques and interpolation methods, the present one contains a direct and more elementary approach.

##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 26D10 Inequalities involving derivatives and differential and integral operators
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##### References:
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