Inequalities for moduli of smoothness versus embeddings of function spaces

The so-called sharp Marchaud inequality and some converse of it, as well as the Ulyanov and Kolyada inequalities are equivalent to some embeddings between Besov and potential spaces. Peetre’s (modified) K-functional, its characterization via moduli of smoothness (also of fractional order), and limit cases of the Holmstedt formula are essentially used.