Math. Nachr. 280, No. 9–10, 1083 – 1093 (2007) / DOI 10.1002/mana.200510537 Sharpness and non-compactness of embeddings of Bessel-potential-type spaces Amiran Gogatishvili ∗1 ,J´ ulio Severino Neves ∗∗2 , and Bohum´ ır Opic ∗∗∗ 1 1 Mathematical Institute, Academy of Sciences of the Czech Republic, ˘ Zitn´ a 25, 11567 Prague 1, Czech Republic 2 CMUC, Department of Mathematics, University of Coimbra, Apartado 3008, 3001-454 Coimbra, Portugal Received 10 December 2005, revised 21 July 2006, accepted 29 July 2006 Published online 6 June 2007 Key words Slowly varying functions, Lorentz–Karamata spaces, rearrangement-invariant Banach function spaces, (fractional) Sobolev-type spaces, H¨ older-type spaces, Bessel potentials, embedding theorems MSC (2000) 46E35, 46E30, 26D15 Dedicated to Professor Frank-Olme Speck on the occasion of his 60th birthday We establish embeddings for Bessel potential spaces modeled upon Lorentz–Karamata spaces with order of smoothness less than one. The target spaces are of H¨ older-continuous type. In the super-limiting case we also prove that the embedding is sharp and fails to be compact. c  2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction In a series of recent papers [7]–[10] a systematic research of embeddings of Bessel potential spaces with order of smoothness σ ≥ 1 and modeled upon generalized Lorentz–Zygmund (GLZ) spaces was carried out. The authors of those papers established embeddings of such spaces either into GLZ-spaces or into H¨ older-type spaces C 0,λ(·) ( Ω) and showed that their results are sharp (within the given scale of target spaces) and fail to be compact. They also clarified the role of the logarithmic terms involved in the quasi-norms of the spaces mentioned. This role proved to be important especially in limiting cases. In particular, they obtained refinements of the Sobolev embedding theorems, Trudinger’s limiting embedding as well as embeddings of Sobolev spaces into λ(·)-H¨ older continuous functions including the result of Br´ ezis and Wainger about almost Lipschitz continuity of elements of the (fractional) Sobolev space H 1+n/p p (R n ) (cf. [5]). Although GLZ-spaces form an important scale of spaces containing, for example, Zygmund classes L p (log L) α , Orlicz spaces of multiple exponential type, Lorentz spaces L p,q , Lebesgue spaces L p , etc., they are a particular case of more general spaces, namely the Lorentz–Karamata (LK) spaces. The embeddings mentioned above were extended in [20] and [21] to the case when Bessel-potential spaces are modeled upon LK-spaces. Since Neves considered more general targets (besides LK-spaces and H¨ older-type spaces also generalized H¨ older spaces), in several cases he obtained improvements of embeddings from [7]–[10]. The sharpness and non-compactness of these embeddings were proved in [15] and [16]. In [11] and [12], the authors analyzed the situation when the order of smoothness is less than one. In such a case one cannot use the method in which a lifting argument (based on [9, Lemma 4.1] and [16, Lemma 4.5], which extend the Calder´ on result [6, Theorem 7]) is applied to reduce the superlimiting case to the sublimiting one, and a new approach was used. Although many results were obtained, the research is not yet complete. Here we extend some results of [11] and [12]. Nevertheless, there are still open questions which are under investigation. ∗ e-mail: gogatish@math.cas.cz, Phone: +420222090786, Fax: +420222211638 ∗∗ Corresponding author: e-mail: jsn@mat.uc.pt, Phone: +35 1239791150, Fax: +35 1239793069 ∗∗∗ e-mail: opic@math.cas.cz, Phone: +420222090745, Fax: +420222211638 c  2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim