Sobolev embeddings into spaces of Campanato, Morrey, and Hölder type ✩

Necessary and sufficient conditions on a rearrangement-invariant Banach function space X(Q) on a cube Q in Rn, n 2, are given for the corresponding Sobolev space W1X(Q) to be continuously embedded into (generalized) Campanato, Morrey, or Hölder spaces. The optimal such r.i. spaces X(Q) are found. As a by-product, sharp inclusion relations are proved among Campanato, Morrey, and Hölder type spaces.  2003 Elsevier Science (USA). All rights reserved.