Suppose $p=1$ and $q=\infty$, and the right hand side of Holder inequality is finite. Then, Holder inequality is equality iff $|g| = ||g||_\infty$ a.e. on $\{x: f(x) \not=0\}$.
And here is the solution I found on the internet:
I understand this part $\int_{A_\varepsilon} |fg| \le -\delta \varepsilon^2 + \int_{A_\varepsilon} |f |||g||_\infty$. But, to hold the inequality, we have to show $\int_{X\sim A_\varepsilon} |fg| = \int_{X\sim A_\varepsilon} |f|||g||_\infty$. I have $$X\setminus A_\varepsilon = \{x: |f(x)| < \varepsilon\} \cup \{x: |g(x)|> ||g||_\infty -\varepsilon\}.$$ Since we have the reverse inequality, $g(x) = ||g||_\infty$ on $x \in X\setminus A_\varepsilon$. Is this correct?
Also, the last equality does not make sense to me because $-\delta\varepsilon^2 + ||g||_\infty\int_X |f| \ge ||g||_\infty \int_X |f|$.
Lastly, suppose that this equation holds. Why is the condition $\{x: f(x) \not=0\}$ necessary?
I appreciate if you give me some help.