I don't know the meaning of these:
Where is the ball of radius .
I don't know what is
Please read my Latex because the symbol really don't show correctly.
Anyway, given that you know the center of the balls you're considering, the first would mean that we take the supremum of |f(x,y)| for x in (if y is fixed), or conversely. But this one is not very clear...
The second one means that you integrate in the area , that is to say for and