Origin of the mixed norm One can define the ($\alpha, \beta$)-mixed norm for a matrix $A$ as 
\[ \|A \|_{\alpha, \beta} = (\sum \|a_i\|_\alpha^\beta)^{1/\beta} \]
where $a_i$ is the $i^{\text{th}}$ column of $A$ and $\|a_i\|_\alpha$ is the usual $\alpha$-norm
\[ \|v\|_\alpha = (\sum_i v_i^\alpha)^{1/\alpha} \]
Is this a well-known norm in that there's a reference for where it's first defined ?  
 A: These mixed norms are well known, but probably more in the context of vectors norms (where you partition a vector, put different norms on the subvectors and then apply another norm to the resulting vector of numbers). 
Indeed, mixed norms play an important role in the theory of function spaces such as Besov spaces or Triebel-Lizorkin spaces. One definition of Besov spaces uses a mixed norm of the sequence of wavelet coefficients another one uses a kind of partition of unity to split the Fourier-transform of the function into different dyadic frequency bands, measures the "energy" in each frequency band by some $L^p$-norm and then takes an $\ell^q$ norm of the resulting sequence of numbers.
Probably the books by Triebel can be considered (but are hard to read - but basically any book on Besov spaces is hard to read...).
However, the search word "mixed norm space" will lead you to many references.
Edit: I should add that there are two specific well known matrix norms of this type, namely the norms induced by the 1-norm and the $\infty$-norm, resulting in the maximum-column-sum-norm and the maximum-row-sum-norm, see, e.g. here.
