This is a good question with many possible answers.

A good place to start is

J. Johnsen, S.M. Hansen, W. Sickel, Characterisation by local means of anistropic Lizorkin-Triebel spaces with mixed norms:

http://people.math.aau.dk/~jjohnsen/Articles/WEBfiler/localmeans.pdf

See, for example, Section  2.2, starting on page 4: Lizorkin-Triebel spaces with mixed norms.   

Another approach to mixed norms is given in

E.P. Ushakova, Norm inequalities of Hardy and Polya-Knopp types, Ph.D. thesis, Lulea University of Technology, 2006:

http://epubl.ltu.se/1402-1544/2006/53/LTU-DT-0653-SE.pdf

See, for example, formula (46), p. 12.    Yet another approach to mixed norms is given in

M. Milman, On interpolation of 2^n Banach spaces and Lorentz spaces with mixed norms, J. of Functional Analysis 41 (1), 1981, 1-7:

http://www.sciencedirect.com/science/article/pii/0022123681900586

with a reference to

D.L. Fernanez, Lorentz spaces with mixed norms, J. of Functional Analysis 25, 1977, 138-146.

Thank you so so much Mr.James F Peters