Can some one tell me the reference or any idea how to take the Discrete Sobolev space work defined for a scalar valued map to the space of maps which are vector valued.Let's say   

 $f:\Omega \rightarrow R^n $ where $\Omega \subset R^d$. A Sobolev space of vector valued maps are defined. Though it looks pretty straightforward to lift the discrete Sobolev space of scalar valued maps to vector valued analogue but does there any work exists ?



Discrete Sobolev space of $R^n$ valued maps
well, there are objects called bochner-lebesgue spaces which are nothing but lebesgue spaces of vector-valued functions. if the target space is finite dimensional, the theory is absolutely trivial, in the general case it is much more involved but still very well understood (see for instance [this book](http://books.google.es/books/about/Vector_Measures.html?id=NCm4E2By8DQC&redir_esc=y); but the rule of thumb is that everything still is very easy if the target space is a separable, reflexive banach space). because this theory works well for a very large class of measures, including atomic ones, there is no problem carrying everything over to the case of sobolev spaces on graphs/point sets.


(i was reflecting on similar problems recently while writing [this](http://arxiv.org/pdf/1207.1996.pdf), which you might want to take a look at).


