We will show a basic argument why Harnack inequality implies Holder continuity. Assume is a weak solution to a uniform elliptic equation
in . Assume we have Harnack inequality (Indeed we do have due to Moser), i.e. for any positive weak solution we will have
where is a constant independent of .
First one will know , see this note.
If is a weak solution, so are and . And they are strictly positive in due to the strong maximum principle, otherwise will be a constant. By Harnack inequality, we will have
Adding these two inequalities up we will have
Denote , and choose such that , by interation we know
Then for any , choose such that , we will have
So Holder continuity follows.