Embeddings of function spaces: a criterion in terms of differences

Abstract

We study criteria for the embedding , 0 < p, q ≤ ∞, 1 < r < ∞, s > 0 with , in terms of inequalities for iterated differences and moduli of smoothness. The article was inspired by the recent paper by W. Trebels [W. Trebels, Inequalities for moduli of smoothness versus embeddings of function spaces, Arch. Math. 94 (2010), pp. 155–164], although we deal with the inhomogeneous setting here. Another motivation came from the new characterization of Sobolev spaces by H. Triebel [H. Triebel, Sobolev-Besov spaces of measurable functions, Studia Math. 201(1) (2010), pp. 69–86]. We also collect some consequences formulated in the spirit of inequalities of Ul'yanov type. In the end an interesting decomposition technique is presented.

Keywords:

• Besov spaces
• Sobolev spaces
• embeddings
• differences
• modulus of smoothness

AMS Subject Classification:

• 46E35

Acknowledgement

Research of D.D. Haroske was supported by Heisenberg grant Ha 2794/1-2.