On Lipschitz-type maximal functions and their smoothness spaces

In a recent monograph (cf. No. 293 of the Memoirs of the Amer. Math. Soc. 47 (1984)) DeVore and Sharpley study maximal functions of integral type and their related smoothness spaces. One of their central results gives an embedding theorem for the smoothness spaces in terms of Besov spaces. In this paper we consider the related problem when the Besov spaces are substituted by the so-called A-spaces introduced by Popov (take the τ-modulus instead of the ω-modulus). We will define Lipschitz-type maximal functions whose smoothness spaces satisfy a corresponding embedding theorem in terms of A-spaces. By well-known results new insights can only be expected for functions satisfying low order smoothness conditions and, therefore, only function spaces generated by first order differences are considered.