Spaces of Variable Smoothness and Integrability: Characterizations by Local Means and Ball Means of Differences

We study the spaces and of Besov and Triebel-Lizorkin type as introduced recently in Almeida and Hasto (J. Funct. Anal. 258(5):1628-2655, 2010) and Diening et al. (J. Funct. Anal. 256(6):1731-1768, 2009). Both scales cover many classical spaces with fixed exponents as well as function spaces of variable smoothness and function spaces of variable integrability. The spaces and have been introduced in Almeida and Hasto (J. Funct. Anal. 258(5):1628-2655, 2010) and Diening et al. (J. Funct. Anal. 256(6):1731-1768, 2009) by Fourier analytical tools, as the decomposition of unity. Surprisingly, our main result states that these spaces also allow a characterization in the time-domain with the help of classical ball means of differences. To that end, we first prove a local means characterization for with the help of the so-called Peetre maximal functions. Our results do also hold for 2-microlocal function spaces and which are a slight generalization of generalized smoothness spaces and spaces of variable smoothness.