So in an excersise we derived a so called "explicit form" of the inequality.
Now later in the book the Harnack inequality is defined as
How are these two statements related to each other? I just dont see how you get from the upper statement to the bottom one and the book does not cover it either
The first equation is a special case of the second one. Pick any x in $B⁰(0, r)$. Clearly $u(x) \leq \sup_V u(\cdot)$ holds and, due to Harnack’s inequality, $u(x) \leq \sup_V u(\cdot) \leq C \inf_V u(\cdot)$ is true. Since $\inf_V u(\cdot) \leq u(y)$ holds for any $y$, we can conclude $u(x) \leq C u(0)$. If we interchange the roles of $x$ and $0$, we can $u(0) \leq C u(x)$ or $\frac{1}{C} u(0) \leq u(x)$. Putting this together, we get $\frac{1}{C} u(0) \leq u(x) \leq C u(0)$. This is the first statement.
Addition: This is not exactly the first statement since the constants aren’t reciprocal to each other. The first equation can therefore be regarded as a more sophisticated variant. But I think the idea should be clear from the above.