1. The problem statement, all variables and given/known data Ok, so we're in R^N, and we're looking at the unit ball. I wanna prove that the function g(x)=log(log(1+1/r)) is in L^N of the unit ball, where r=|x|. I also want to prove that its partial derivative are there, but like we say here, one cow at a time. 2. Relevant equations None, really. 3. The attempt at a solution Well, I don't remember if it's true, but maybe if xf(x) tends to zero when x tends to zero and f>0, then its integral is finite? I could then try and use it on |g|^N... Any help would be immensly appreciated.