  In this paper, we introduce a family of function spaces of Besov and
Triebel-Lizorkin type. We present the $\varphi $-transform characterization of
these spaces in the sense of Frazier and Jawerth and we prove their Sobolev and
Franke-Jewarth embeddings. Also, we establish the smooth atomic, molecular and
wavelet decomposition of these function spaces. Characterizations by ball means
of differences are given. Finally, we investigate a series of examples which
play an important role in the study of function spaces of
Besov-Triebel-Lizorkin type.
