Musielak-Orlicz Besov-type and Triebel-Lizorkin-type spaces

Let be two Musielak-Orlicz functions that, on the space variable, belong to the Muckenhoupt class uniformly on the time variable. In this paper, the authors introduce Musielak-Orlicz Besov-type spaces and Musielak-Orlicz Triebel-Lizorkin-type spaces , and establish their -transform characterizations in the sense of Frazier and Jawerth. The embedding and lifting properties, characterizations via Peetre maximal functions, local means, Lusin area functions, smooth atomic and molecular decompositions of these spaces are also presented. As applications, the boundedness on these spaces of Fourier multipliers with symbols satisfying some generalized Hormander condition are obtained. These spaces have wide generality, which unify Musielak-Orlicz Hardy spaces, unweighted and weighted Besov(-type) and Triebel-Lizorkin(-type) spaces as special cases.