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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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