Function Spaces with Dominating Mixed Smoothness Characterization by Differences

This paper deals with function spaces with dominating mixed smoothness properties of Besov and Lizorkin-Triebel type on R as well as on the d-torus T. The main result is the characterization of these classes in terms of integral means of differences for the largest possible range of parameters. Moreover we obtain further characterizations based on this result using well-known classical moduli of smoothness.