caa a22 4500 605496706 CHVBK 20210128100537.0 cr unu---uuuuu 210128e20150401xx s 000 0 eng 10.1007/s10231-013-0383-1 doi (NATIONALLICENCE)springer-10.1007/s10231-013-0383-1 Embeddings of Sobolev-type spaces into generalized Hölder spaces involving $$k$$ k -modulus of smoothness [Elektronische Daten] [Amiran Gogatishvili, Susana Moura, Júlio Neves, Bohumír Opic] We use an estimate of the $$k$$ k -modulus of smoothness of a function $$f$$ f such that the norm of its distributional gradient $$|\nabla ^kf|$$ | ∇ k f | belongs locally to the Lorentz space $$L^{n/k, 1}({\mathbb {R}}^n),\,k \in {\mathbb {N}},\,k\le n$$ L n / k , 1 ( R n ) , k ∈ N , k ≤ n , and we prove its reverse form to establish necessary and sufficient conditions for continuous embeddings of Sobolev-type spaces. These spaces are modelled upon rearrangement-invariant Banach function spaces $$X({\mathbb {R}}^n)$$ X ( R n ) . Target spaces of our embeddings are generalized Hölder spaces defined by means of the $$k$$ k -modulus of smoothness $$(k\in {\mathbb {N}})$$ ( k ∈ N ) . General results are illustrated with examples. Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg, 2013 Rearrangement-invariant Banach function space nationallicence Modulus of smoothness nationallicence Distributional gradient nationallicence Lorentz space nationallicence Sobolev-type space nationallicence Banach lattice nationallicence Hölder-type space nationallicence Embeddings nationallicence Gogatishvili Amiran Institute of Mathematics, Academy of Sciences of the Czech Republic, Z̆ itná 25, 11567, Prague 1, Czech Republic aut Moura Susana Department of Mathematics, CMUC, University of Coimbra, Apartado 3008, 3001-454, Coimbra, Portugal aut Neves Júlio Department of Mathematics, CMUC, University of Coimbra, Apartado 3008, 3001-454, Coimbra, Portugal aut Opic Bohumír Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 18675, Prague 8, Czech Republic aut Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/2(2015-04-01), 425-450 0373-3114 194:2<425 2015 194 10231 https://doi.org/10.1007/s10231-013-0383-1 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10231-013-0383-1 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Gogatishvili Amiran Institute of Mathematics, Academy of Sciences of the Czech Republic, Z̆ itná 25, 11567, Prague 1, Czech Republic aut NATIONALLICENCE 700 1- Moura Susana Department of Mathematics, CMUC, University of Coimbra, Apartado 3008, 3001-454, Coimbra, Portugal aut NATIONALLICENCE 700 1- Neves Júlio Department of Mathematics, CMUC, University of Coimbra, Apartado 3008, 3001-454, Coimbra, Portugal aut NATIONALLICENCE 700 1- Opic Bohumír Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 18675, Prague 8, Czech Republic aut NATIONALLICENCE 773 0- Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/2(2015-04-01), 425-450 0373-3114 194:2<425 2015 194 10231