theoretical measure – Hausdorff dimension and functions $ W ^ {1,1} $

What can be said of the Hausdorff dimension of the image of a set by a $ W ^ {1,1} $ map?

In other words,
what is the relationship between
$ mathrm {dim} _H f (A) $ and $ mathrm {dim} _H A $, or $ f in W ^ {1,1} $?

Is the result also valid if $ f $ is a BV function?