This is a follow-up question on On the equality case of the Hölder and Minkowski inequalites.
What is the conditions for $$\int \vert fg\vert =\Vert f \Vert_p\Vert g \Vert_q$$ when $p=1, q= \infty$?
It's seems like that the same proof won't work, as $q = \infty $.
If $\displaystyle \int|fg|=\|f\|_1\,\|g\|_\infty$, this can be written as $$ \int|fg|=\int|f|\,\|g\|_\infty. $$ Rewrite as $$ \int |f|\,(\|g\|_\infty-|g|)=0. $$ So a necessary and sufficient condition for the equality is that $|g|=\|g\|_\infty$ a.e. whenever $f\ne0$.
